MO~Blin~ an~ U~namiGs 01 InfBctious ~iSBaSBS
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Series in Contemporary Applied Mathematics CAM 11
Mo~e in~ an~ ~~namic~ of nfectiou~ ~i~ea~e~ editors
Zhien Ma Yicang Zhou Xi'an }iaotong University, China
jianhong Wu York University, Canada
•
Higher Education Press
,~World Scientific NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI
Zhien Ma, Yicang Zhou
JianhongWu
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f~~m8{Jgtm!:jZ9JjJ,+ = Modeling and Dynamics of
Infectious Diseases: 1J;!;x / .!:b~Jff}" )aJ5(11;-,
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v
Preface
This book contains a carefully chosen and coordinated series of lecture notes at the China-Canada Joint Program on Infectious Disease Modeling, held in Xi'an Jiaotong University, May 10-29, 2006. The joint program consists of a summer school attended by over 100 students from a variety of backgrounds, and a workshop participated by invited speakers from both academic institutes and public health agencies such as US Centers for Disease Control and Prevention (CDC) and Public Health Agency of Canada (PHAC). These contributions are grouped into three categories: lectures notes that briefly introduce the basic concepts and techniques; survey articles that provide reviews on some specific diseases or issues; and research papers dedicating to some important problems of current interest in the epidemiological modeling. There are also two articles describing some recent progresses by a Chinese and a Canadian team. The aim of this book is to provide fundamental methods and techniques for students who are interested in epidemiological modeling, and to guide junior research scientists to some frontiers in the interface of mathematical modeling and public health. Contributions are provided from different and complementary angles, with the balance between the theory and applications, between mathematical modeling and its applications to public health policy. It is hoped that this book can help in increasing the awareness of the importance of mathematical modeling in the study of infectious disease transmission, and in bridging the gap between mathematical modelers in basic theoretical research and medical scientists and public health policy makers working in health research institutes. There has been a long history of mathematical epidemiology and there are many successful stories in applying mathematical modeling to optimal design of feasible public health policy for disease prevention, control and management. Some emerging and re-emerging infectious diseases such as HIV, FMD, SARS and pandemic influenza have generated substantial renewed interest, and have been continuing to challenge modelers for effective mathematical and computational models. Covering a comprehensive range of topics, this book hopefully provides an alternative and additional textbook for graduate students in applied mathematics, health informatics, applied statistics and qualitative public health, and a useful resource for researchers in these areas.
vi
Preface
The book provides complementary approaches from deterministic, to statistical, to network modeling, and it seeks view points of the same issues from different angles from mathematical modeling, to statistical analysis, to computer simulations, and to concrete applications. For example, we have included a chapter that introduces the network models describing the beginning of a disease outbreak in terms of the degree distribution of a branching process, in comparison with the chapter that introduces the basic deterministic models along with the instructions how to calculate the basic reproduction number and the final size of an epidemic. Other chapters deal with mathematical analysis for disease transmission involving structured population; a chapter develops mathematical approaches for analysis of epidemic models with time delays; a chapter for age structured population models with applications to epidemiology, and age structured epidemic models; and a chapter deals with the uniqueness and global stability of endemic equilibria of multi-group epidemic models of SEIR type. Disease spread in a heterogeneous environment is an important issue addressed in a chapter which uses metapopulation models consisting of graphs, with systems of differential equations in each vertex, to address the issue of spatial dispersal of diseases. This is further complemented by a chapter that deals with various issues involving stochastic processes for disease spread. A chapter is also included to detail two complimentary mathematical approaches for incorporating evolution into epidemiological models, and a chapter is dedicated to the investigation of the effects of the reservoir on the time course of the disease and on endemic states. The coexistence of a vertically and a horizontally transmitted parasite strain under complete cross protection is addressed as well. Various chapters deal with the evaluation of different control measures. For instance, there is a chapter that studies the effectiveness of quarantine and isolation as control measures for the spread of infectious diseases, and general integral equation models which assumes an arbitrarily distributed disease stage for both the latent and the infectious stages. Another chapter discusses the pulse vaccination SIR model with periodic infection rate. Other chapters deal with specific diseases of current interest. One such chapter describes the estimation of congenital rubella syndrome from disease or serological surveillance and demographic, and possible strategies for mitigating the burden of congenital rubella syndrome. Another chapter examines the estimate of turning points and case numbers of the 2003 severe acute respiratory syndrome outbreaks in Taiwan, Beijing, Hong Kong, Toronto, and Singapore. Added to these materials are the chapter that studies HIV transmission and disease progression, and a detailed case study of the West Nile Virus in Southern Ontario Canada. The book also contains a contribution that depicts the man-
Preface
vii
ner in which the pandemic develops in a specific community, and the affection of antiviral treatment. This is supplemented by two chapters that briefly summarizes progresses and adventure of the Xi'an Jiatong University group, and a MITACS team for HIV, SARS, West Nile Virus, pandemic influenza and other emerging infectious diseases. We wish to thank all contributors for their excellent contributions without which this book is impossible, we wish to express our sincere appreciation to the staff members and students of the Xi'an Jiatong University for their hospitality and hard working that made the CanadaChina program a successful event and an enjoyable experience. We wish to thank Professor Ta-Tsien Li for encouraging us to include this book in the Series in Contemporary Applied Mathematics by Higher Education Press, and would like to acknowledge the support of Mathematics for Information Technology and Complex Systems (MITACS) for the Canada-China Joint Program. Zhien Ma and Yicang Zhou, Xi'an Jiaotong University Jianhong Wu, York University
ix
Contents Preface
Zhien Ma: Some Recent Results on Epidemic Dynamics Obtained by Our Group........................................ 1
Fred Brauer, Jianhong Wu: Modeling SARS, West Nile Virus, Pandemic Influenza and Other Emerging Infectious Diseases: A Canadian Team's Adventure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36
Julien A rino: Diseases in Metapopulations . . . . . . . . . . . . . . . . . . . . . . ..
64
Fred Brauer: Modeling the Start of a Disease Outbreak........... 123 Troy Day: Mathematical Techniques in the Evolutionary Epidemiology of Infectious Diseases. . . . . . . . . . . . . . . . . . . . . . . . . .. 136
Zhilan Feng, Dashun Xu, Haiyun Zhao: The Uses of Epidemiological Models in the Study of Disease Control. . . . . .. 150
John W. Glasser, Maureen Birmingham: Assessing the Burden of Congenital Rubella Syndrome and Ensuring Optimal Mitigation via Mathematical Modeling. . . . . . . . . . . . . .. 167
Thanate Dhirasakdanon, Horst R. Thieme: Persistence of Vertically Transmitted Parasite Strains which Protect against More Virulent Horizontally Transmitted Strains ...... 187
Ying-Hen Hsieh: Richards Model: A Simple Procedure for Real-time Prediction of Outbreak Severity. . . . . . . . . . . . . . . . . . .. 216
x
Contents
James Watmough: The Basic Reproduction Number and the Final Size of an Epidemic ................................... "
237
K.P. Hadeler: Epidemic Models with Reservoirs.................. 253 Hongbin Guo, Michael Y. Li, Zhisheng Shuai: Global Stability in Multigroup Epidemic Models.............. 268
Wendi Wang: Epidemic Models with Time Delays ................ 289 Shenghai Zhang: A Simulation Approach to Analysis of Antiviral Stockpile Sizes for Influenza Pandemic. . . . . . . . . . . . ..
315
Peter Buck, Rongsong Liu, Jiangping Shuai, Jianhong Wu, Huaiping Zhu: Modeling and Simulation Studies of West Nile Virus in Southern Ontario Canada........... ......
331
1
Some Recent Results on Epidemic Dynamics Obtained by Our Group Zhien Ma Department of Applied Mathematics, Xi'an Jiaotong University Shaanxi 710049, China E-mail:
[email protected]
Abstract The goal of this synthetic paper is to introduce a part of research directions on epidemic dynamics investigated by our group and our main results during the past several years. Before this, some basic knowledge on epidemic dynamics will be introduced which may be helpful to those readers who are not familiar with the mathematical modeling on Epidemiology.
1
Basic knowledge on epidemic dynamics
Epidemic dynamics is an important method of studying the spread of infectious disease qualitatively and quantitatively. It is based on the specific property of population growth, the spread rules of infectious diseases, and the related social factors, etc., to construct mathematical models reflecting the dynamic properties of infectious diseases, to analyze the dynamical behavior and to do some simulations. The research results are helpful to predict the developing tendency of the infectious disease, to determine the key factors of the spread of infectious disease and to seek the optimum strategies of preventing and controlling the spread of infectious diseases. In contrast with classic biometrics, dynamical methods can show the transmission rules of infectious diseases from the mechanism of transmission of the disease, so that people may know some global dynamic behavior of the transmission process. Combining statistics methods and computer simulations with dynamic methods could make modeling and the original analysis more realistic and more reliable, make the comprehension for spread rule of infectious diseases more thorough. Now, the popular epidemic dynamic models are still so called compartmental models which were constructed by Kermack and Mckendrick in 1927[lJ and is developed by many other biomathematicians. In the
Zhien Ma
2
K-M model, the population is divided into three compartments: susceptible compartment S, in which all individuals are susceptible to the disease; infected compartment I, in which all individuals are infected by the disease and have infectivity; removed compartment R, in which all the individuals recovered from the class I and have permanent immunity. Three assumptions they did as follows: (1) The disease spread in a closed environment, no emigration and immigration, and is no birth and death in population, so the total population remains a constant k, i.e. Set) + I(t) + R(t) == k. (2) The infective rate of an infected individual is proportional to the number of susceptible, the coefficient of the proportion is a constant /3, so that the total number of new infected at time t is /3S(t)I(t). (3) The recovered rate is proportional to the number of infected, and the coefficient of proportion is a constant '"'(. So that the recovered rate at time t is '"'(I(t). According to the three assumptions above, it is easy to establish the epidemic model as follows
(K-M)
dS = -/3S1 dt ' dI dt = /3S1 - '"'(I,
Set)
+ I(t) + R(t) == k.
dR _ I dt - '"'( , Now, let us explain some basic concepts on epidemilological dynamics.
1.1
Adequate contact rate and incidence
It is well-known that the infections are transmitted through the contact. The number of times an infective individual contacts the other members in unit time is defined as contact rate, which often depends on the number N of individuals in the total population, and is denoted by function U(N). If the individuals contacted by an infected individual are susceptible, then they may be infected. Assume that the probability of infection by every time contact is /30, then function /30 U (N) is called the adequate contact rate, which shows the ability of an infected individual infecting others (depending on the environment, the toxicity of the virus or bacterium, etc.). Since, except the susceptible, the individuals in other compartments of the population can't be infected when they contact with the infectives, and the fraction of the susceptibles in total population is SIN, so the mean adequate contact rate of an infective to the susceptible individuals is /3oU(N)SIN, which is called the infection
Some Recent Results on Epidemic Dynamics ...
3
rate. Further, the number of new infected individuals yielding in unit time at time t is (3oU(N)S(t)J(t)/N(t), which is called the incidence of the disease. When U(N) = kN, that is, the contact rate is proportional to the size of total population, the incidence is (3okS(t)l(t) = (3S(t)l(t) (where (3 = (3ok is defined as the transmission coefficient) which is called bilinear incidence or simple mass-action incidence. When U(N) = k', that is, the contact rate is a constant, the incidence is (3ok' S(t)J(t)/ N(t) = (3S(t)l(t)/ N(t) (where (3 = (3ok') which is called standard incidence, for instance, the incidence formulating the sexually transmitted disease is often of standard type. Two types of incidence mentioned above are often used, but they are special for the real cases. In recent years, some cOIl-tact rates with saturate feature between them are proposed, such as U(N) = aN/(I+wN)[2], U(N) = aN/(I+bN+V1+2bN)[3]. In general, the saturate contact rate U(N) satisfies the following conditions:
U(O) = 0, U'(N) ~ 0, (U(N)/N)' ~ 0, lim U(N) = Uo. N-+oo Besides, some incidences, which are much more plausible for some special cases, are also introduced, such as (3SP lq, (3SP lq / N[4,5].
1.2
Basic reproduction number
Basic reproduction number, denoted by Ro, represents the average number of secondary infectious infected by an individual of infectives during whose whole course of disease in the case that all the members of the population are susceptible. According to this meaning, it is easy to understand that if Ro < 1 then the infectives will decrease so that the disease will go to extinction; if Ro > 1 then the infectives will increase so that the disease can not be eliminated and usually develop into an endemic. From the mathematical point of view, usually when Ro < 1, the model has only disease free equilibrium Eo(So,O) in the SOl plane, and Eo is globally asymptotically stable; when Ro > 1, the equilibrium becomes unstable and usually a positive equilibrium E* (S*, 1*) appears. E* is called an endemic equilibrium and in this case it is stable. Hence, if all the members of a population are susceptible in the beginning, then Ro = 1 is usually a threshold whether the disease go to extinction or go to an endemic.
Zhien Ma
4
Example
Consider the following model:
dS = A - j3S I - bS dt ' dI dt =j3SI-bI-1 I , dR
-
dt
=
'VI -bR ' ,
where b is the natural death rate, 1 is the recovered rate, A is recruitment. Let
~=k b
'
consider the first two equation we have
(M{) :
~
dS = bk - j3S1 - bS dt ' { dI dt = j3S1 - (b + 1)1.
< 1, the system has only one disease free equilibrium Eo(k, 0) and it is stable; when Ro > 1, · . . ·l·b . E* (b + 1 b[j3k - (b + 1)]) b eSI·des E 0 t h ere IS a pOSItive eqUl 1 num -13-' j3(b + 1) , Let Ro =
b+1'
it is easy to see that when Ro
and, in this case, Eo is unstable, E* is stable, the endemic appears. So Ro = 1 is a threshold to distinguish the disease extinction or persistence. From model (Md we can see that
d:
=
b(k _ N), N(t)
=
Set)
+ I(t) + R(t).
Hence, the total number of the population is k, and 13k should be the number of secondary infectious infected by an individual of infectives per unit time when the number of susceptible is k. From the second equation of the system M{ we can see that l/(b + 1) is the average course of the disease. Therefore, Ro = j3k/(b + 1) is the average secondary infectious infected by an individual of the infectives during whose whole course of disease, that is just the reproduction number. It should be indicated that the reproduction number is not always equivalent to the threshold mentioned above.
2
Epidemic models with vaccination
So far, there are two effective methods to prevent and control the spread of infection, which are vaccination and quarantine. To model the transmission of the infection under vaccination, ordinary differential equations, delay differential equations, and pulse differential equations are often used.
Some Recent Results on Epidemic Dynamics ...
5
For investigating dynamic behavior of an epidemic model with vaccination, one usually use a SIR compartment model and remove a part of newborns or susceptibles from susceptible class S directly into the removed class R due to vaccination. But if the immunity caused by the vaccination is temporary and the periods of immunity loss from vaccinated and recovered are not the same, then another compartment V should be introduced.
(1)
SIS-VS model
The following Figure 2.1 describes an SIS-VS model, where A is newborns per unit time, q is a fraction of vaccinated for the newborns, p is the proportional coefficient of vaccinated for the susceptibles, Q( t) is the probability that an individual remains in the class Vat least t time units before returning to the class S, d and a are the natural death rate and death rate due to disease, respectively. rl
",I (1-q)A
L
S
~
~
q(A)
1- Q(t)
..
1
1
/3S1
",I
J
I
I
I
t t
ds
dI
V
I
1v
al
pS
Figure 2.1: The flowchart of an SIS-VS model. From the flowchart, we may write down the model as follows
dI dt =(3SI-(d+a+,)I,
V = Vo(t)
+ fat [qA + pS(u)]Q(t - u)e-d(t-u)du,
dN = A-dN -aI dt ' where Vo(t) is the number of individuals who have already remained in the class V at time t = O. If the probability Q is exponential distribution, i.e., Q = e- ct , constant 10 > 0 is the immunity loss rate, then the model (M2) becomes
dS dt = (1 (Mi) :
q)A - {3SI - (p + d)S +,I + cV,
dI = {3SI - (d dt dV
dt
+ a + ,)I,
= qA + pS -
(10
+ d)V.
Zhien Ma
6
If Q = {I, t E [0, T), then model (M2) becomes 0, t ~ T d8 = (1- q)A - (p + d)8 - (381 dt +[qA + p8(t - T)]e- dT ,
-
dI dt
(d + a
- = (381 dV dt
= qA + p8 -
where T is the period of immunity. For model (MJ), let N = 8 + I
~~
=
+ "()I,
[qA
+ V,
+ "(I
+ p8(t - T)]e- dT
-
dV,
we consider its replacement:
I[(3(N - I - V) - (d + "( + a)],
dV dt = qA + p(N -
1) - (p
+ d + c)V,
dN =A-dN-aI. dt
The following are the main results of the system (MJ). A(3[c + d(l - q)] d(d + "( + a)(d + c + p) ~ 1 then the system ('M"J) has only a disease free equi-
Theorem 1[6] If
ROl
librium Eo (0,
Let
ROl
=
d~(; c+:~)' ~),
and it is globally asymptotically sta-
ble; if ROl > 1, Eo is unstable, and there is an endemic equilibrium E* (1*, V*, N*) which is locally stable. Moreover, E* is globally asymptotically stable if ROl > 1 and there exist two positive constants m and n such that the matrix M is positive definite, where (3m M=
(3m + n(3 2 a- (3m 2
(3m + n(3 2 n(p + d + c) np
2
a- (3m 2 np 2 d
For the model (Mi), since V does not appear explicitly in the first two equations, we need only to discuss the system consisted of the first two
Some Recent Results on Epidemic Dynamics ...
7
equations.
l
(M?) :
Theorem
=
+[qA + p8(t - r)]e- dr , dI dt ={JSI-(d+"f+a)I.
Let
2[7]
R02
~~ =A(l-q)-{JSI-(d+p)S+"fI
(JA[l - q(l - e- dr )] (d + a + "f)[d + p(l - e- dr )]
If ROl ~ 1, the system (M:]) has only a disease free equilibrium
Eo(S02, 0), it is globally asymptotically stable; if R02 > 1, Eo is unstable, and the endemic equilibrium E* (8*, I*) appears, which is globally asymptotically stable, where
(2) SIS-VS model with efficiency of vaccine In the reality, the efficiency of every type of vaccines may not be 100%, which means that even some of susceptibles have been vaccinated, they still have a certain probability to be infected by the disease. In this case the flowchart of the epidemic may be described by Figure 2.2. ev
a/3 ~I
--
qrN
r(l-q)N
f(N)s
aI
feN) V
ps
Figure 2.2: The flowchart of an SIS-VS model.
In this model, we assume that the natural death rate is density dependent to the population, i.e., it is a function of N; the disease spreads in the form of standard incidence; the average number of adequate contact of an infective and a vaccinated individual per unit time are {J and a{J respectively, 0 ~ a ~ 1, the fraction a reflects the effect of reducing
Zhien Ma
8
the infection due to vaccination, fY = 0 means that the vaccine is completely effective in preventing infection. The other parameters are easy to be understood as before. According to the flowchart, the model can be written as
dS SI = r(1 - q)N - (3- - [p + f(N)]S + 'YI + 1OV, dt N dI I dt = (3(S + fYV) N - b + a + f(N)]I,
-
dV
dt = rqN + pS -
IV fY(3j1j - [10 + f(N)]V.
Adding the three equations together gives
dN = N[r - feN)] - aI. dt
-
Taking the transformation x
=
!,
y
= ~, z = ~,
we obtain
dx dt = r(1 - q) - «(3 - a)xy - (p + r)x + 'YY + 1oZ, dy dt = y[(3x + ay + (3fYZ - (r dz dt = rq + px - (10 + r)z
+ a + 'Y)],
+ (a - fY(3)yz,
x+y+z=l. We need only to consider the system consisted of the second and third equations, denote it by (M4). Theorem 3
[8]
Let Ro = (3[10 + fYP
+ r(1 - (1 - fY)q)] (a+r+'Y)(p+1O+r)
i) If Ro > 1, the endemic equilibrium E*(y*, 1*) exists and globally asymptotically stable; ii) If Ro < 1, a < fY(3, (3 > r + 'Y + a, B > 2VAC, there exist two endemic equilibria Ei (yi, zi) and E2 (Y2, Z2J (yi < Y2, zi > Z2) and two stable manifolds of P{ which divide the region D = {(y, z)ly ): 0, z > 0, y + z < I} into two parts Dl and D 2 , where P{ ~ D 1 , P{ E D2 such that lim (y(t), z(t)) = (0, zo) when (y(O), z(O)) E D 1 , and t-->oo
Some Recent Results on Epidemic Dynamics ... lim (y(t),z(t))
t ...... oo
= (Y2,z2)
9
when (y(O),z(O)) E D 2 , where
A = (a - a(3)({3 - a),
B
=
a(p + c + I' + a
+ 2r)
-(3[(a + r + c) - a({3 - r - a
+ I' -
p)],
G = (3(p + r + c) - (3(1 - a)(rq + p) - (p + r + c)(r + a
+ 1')
= (p+r+c)(r+a+I')(Ro -1). iii) If Ro < 1, a < a{3, (3 > r + I' + a, B = 2v'AG, there exist an endemic equilibrium Ej and two stable mainifolds of equilibrium Ej, which divide the region D into two parts Dl and D2 where Dl is above D2, such that lim (y(t), z(t)) = (0, zo) when (y(O), z(O)) E D 1 , and t ...... oo
= (y*, z*) when (y(O), z(O)) E D 2 . If Ro = 1, a < a{3, B > 0, then there exists
lim (y(t), z(t))
t ...... oo
iv) an endemic equilibrium E 4, which is globally asymptotically stable. v) If the parameters of model (M~) don't satisfy the cases of i)iv), then the disease free equilibrium Eo(O, zo) is globally asymptotically stable.
Figure 2.3: The diagram of backward bifurcation. It is worth to indicate that when
the two endemic equilibria still exist even if Ro < 1. In this case the dis. . '1 ,8[e: + (YeP + r(1 - (1 - (Ye)q)] ease can stIll eXIst untl Ro < R e , where Re = ( )( )' a+r+'Y p+e:+r
is a root of the equation B = 0, (a2 < a e < al < 1). This phenomenon is called backward bifurcation. Hence, in order to prevent and control the spread of disease, estimating accurately the efficiency of the vaccine is necessary and important. (Ye
Zhien Ma
10
(3) The impulsive vaccination Suppose that the vaccination is given to the susceptible group in a time sequence, then the model should be considered as an impulsive differential system. The following is an SIR model with impulsive vaccinations.
dS
dt (M5) :
=
p,k - (3SI -
S(k+)
dI dt = (3SI - (p, + ex)I dR
dt = ).J t---+k+
(1 - p)S(k)
I(k+) = I(k) ).J
p,R (t -=I k, k
here J(k+) = lim J(t),
=
/1,8
R(k+) = R(k)
= 1,2, ... )
J(k) = lim J(t). t---+k-
+ pS(k),
k = 0,1,2""
p is a proportion of inocu-
lation, p, and ex are the death rates due to nature and disease respectively, ,\ is the recovered rate. For the model (M5) we obtained the following results
Theorem 4[9]
Let Ro =
(3 ,\ p,+ex+
r So(t)dt = k _ 1
Jo
11
So(t)dt, where
0
kp(eIJ. - 1) , p,(eIJ.-1+p)
(So(t), 0, Ro(t)) is a periodic solution of the model (M5) with the period l. If Ro < 1, then the disease free periodic solution (So(t), 0, Ro(t)) is globally asymptotically stable. We also investigated the SIR and SIRS models with impulsive vacci.nation and obtained some sufficient conditions for the stability of disease free periodic solutions.
3
Epidemic models with quarantine strategy
Quarantine to the infective individuals is another effective measure to prevent and control the spread of infection. The earliest study on the effects of quarantine on the transmission of the infection is achieved by Feng and Thieme[lO,ll] and Wu and Feng[12]. In those papers, they introduced a quarantine compartment Q, and assume that all the infective individuals must pass through the quarantined compartment before going to the removed compartment or back to the susceptible compartment. We considered some more realistic cases[12]: a part of infective
Some Recent Results on Epidemic Dynamics ...
11
individuals are quarantined and the others are not quarantined and enter into the susceptible class or directly enter into removed class when they recovery. We analyzed SIQS and SIQR two types models with three kinds of incidence: simple mass action incidence, standard incidence, and · d·JUS t ed··d quarant me-a mCl ence N(381 _ Q. Figure 3.1 is the flowchart of an SIQS model with simple mass action incidence. The corresponding model is
d8 dt dI -dt
-
dQ dt
= A - (381 - d8 + "(I + cQ, =
b + 8 + d + a)]I,
[(38 -
= 8I - (c + d + a) Q. £Q
A
-I
1S t ~
YI. PSI 1
!
-I
8I
(d+ a,)I
ds
-I
J
(d+a2)Q
Figure 3.1: The flowchart of an SIQS model.
(3~ Theorem
5[13]
Let Ro
d
=
,,(+8+d+a If Ro ::( 1, then the disease free equilibrium Eo(80 , 0, 0) of the model (M6) is globally asymptotically stable; if Ro > 1, Eo is unstable and there exists an endemic equilibrium E* (8*,1*, Q*) which is globally asymptotically stable. For the SIQR model with quarantine-adjusted incidence we obtained the following results. Theorem 6[13] Consider the model A_
d8 dt
=
dI dt
= [
(381 _ d8 8+1 +R ' (38
8+R+I
_
b + t5 + d + ad]
~~ =8I-(c+d+a2)Q, dR
dt
=
"(I + cQ - dR.
I,
12
Zhien Ma Let Ro
Eo (
=
f3
'Y + 8 + d + a1
~,O, 0, 0)
. If Ro :::; 1 then the disease free equilibrium
of the model (M7) is locally stable; if Ro > 1 then Eo
is unstable, the disease is uniformly persistent, and there is an unique endemic equilibrium E* which is usually locally stable, but Hopf bifurcation can occur for some parameter values, so that E* is sometimes an unstable spiral and a periodic solution around E* can occur. [13J analyzed all six cases (SIQS, SIQR two types models with three kinds of incidence) and found that only the SIQR model with the quarant ine-adjusted incidence may exist a periodic solution around the endemic equilibrium, which is produced by Hopf bifurcation, for all the other five models, the endemic equilibrium is always globally asymptotically stable.
4
Epidemic models with complicated structures
(1) SEIR models with saturating contact rate and more general contact rate. In general SEIR and SEIRS models cannot be reduced to two dimensional system. For the competitive system the global stability may be obtained by means of the orbital stability, the second additive compound matrix and some methods of ruling out the existence of periodic solution proposed by Muldowney and Li[14-16]. Using those methods we investigated the following model and the complete results were gained. Theorem 7[17] Consider an SEIR model with saturating contact bN rate C(N) = as follows 1 + bN + VI + 2bN
dS = A _ aoSI _ S dt h(N) IL, dE aoSI dt = h(N) - coE - ILE,
(Ms) :
dI dt = coE - 'YoI - ILl - aoI, dR
dt = 'YoI -
ILR,
where A is recruitment, E is the exposed class or the latent class. is the period of latent,
~o
1 co is the course of disease, IL and ao are the
Some Recent Results on Epidemic Dynamics· ..
f' ·+
13
+
0PC(N)i-I~~
A
----'-=--~~.
J.lS
J.lE
~ I~
J.lI
aoI
J.lR
Figure 4.1: The flowchart of an SEIR model. death rates due to nature and disease, respectively, h(N) = 1 + bN VI + 2bN, ao = {3b. Let
Ro
=
(3C(~)
/1 (/1 + 10
co
+ 0'.0)(/1 + co)
+
.
If Ro ::;; 1 then the disease free equilibrium Po is globally asymptotically stable; If Ro > 1, then Po is unstable and there exists a unique endemic equilibrium P* which is globally asymptotically stable. We also investigated an SEIR model with a general contact rate and obtained similar results. (2) Epidemic models with different infective groups For a certain disease, different infectives may have different infectivities and different recovered rate, in this case, we may partition the infected compartment I into n sub-compartments, denoted by Ii (i = 1,2,··· , n). We suppose that the individuals in each of the infected sub-compartments may contact and infect the susceptibles and the secondary infectives will enter into different group Ii according to a certain n
proportion Pi,
L
Pi = 1. Hence the flowchart and corresponding model
0<=1
are shown in Figure 4.2 and (Mg).
(Mg) :
Theorem
8[19J
Let Ro
= SO
t
i=1
(3iPi . If Ro < 1, then the disease
/1 + Ii
free equilibrium of the model (MlO), Eo(SO, 0, ... ,0) is globally asymptotically stable; if Ro > 1, then Eo is unstable and there exists a unique endemic equilibrium E* (S*,1; , ... , I~) which is globally asymptotically stable. It should be indicated that for the model (Mg ), instead of mass action law incidence if we use standard incidence, then the similar results have
14
Zhien Ma
Figure 4.2: The fiowchar of a model with different infective groups. also been obtained provided the reproduction number is taken into Ro = (JiPi . i=l f-t + "Ii For SIS models, if we divide I into n sub-compartments, and suppose that there is no immunity for the individuals in each group Ii; the recoveries will return to the susceptible group in the different recovered rates. In this case, the model is similar, but analysis for this n + I-dimensional space is very complicated. We only considered the special case that I is divided only into two sub-compartments and obtained the reproductive number and complete results for the stability of equilibria. Moreover, we also added more demographic effects to assume density-dependent birth and death rates for the population in this simple case, and obtained the similarly complete results as well. For the SIR or SIS models with n-stages of infections as shown in Figure 4.3.
f:
Figure 4.3: The flowchart of the SIR or SIS models with n-stages of infections. If the individuals of infectives are divided into n-groups Ii according to their courses of disease, and suppose that each group Ii may contact
Some Recent Results on Epidemic Dynamics ...
15
with susceptibles and infects them with different infectivities. Of course, all the new infective will enter into the first group It and then develop to the groups 12 ,13 , .. , ,In successively depending on the courses of disease. For this model, we obtained some results only for three stages.
(3) Epidemic models with different susceptible groups and different infective groups.' Since different susceptibles may have different immunities against the disease, besides different infective groups we sometimes also need to partition the susceptibles into several sub-groups. [20] investigated an epidemic model in a homosexual population which consists of susceptible and infective individuals. The assumptions are the following: there are two groups of individuals who have different response to disease due to the difference of sexual activities, genetics, immune systems or other factors; the infectives in each group are divided into 2 classes based on the infecting pathogen strains and that susceptibles infected by individuals with a certain pathogen strain have the same pathogen strain; there is no superinfection such that an individual can be infected only by one strain if this individual is infected. We use Sk, k = 1,2 to denote the susceplibles will have sexual activity Tk, which is the number of contacts per individual in group K per unit of time, and use hand Jk to denote the infectives with sexual activity k and infected by strain 1 and strain 2, respectively. The dynamics of the disease transmission then are governed by the following model.
k
(MlO)
= 1,2,
where 2
2
L
L"tiIj I
Bk
=
I j=l Sk"fkf3 .0...2,-------
J
Bk
= Sk"fkf3
"fjJj
J j=l -'-:2:---
L"fjTj
L"fjTj
j=l
j=1
are the incidences with Tk = Sk + h + Jk being the population size of group k, and other parameters are easy to be understood. Since
Zhien Ma
16 the equilibrium for Tk is
S2, thus the limiting system of model (MlO) is
(Mn)
where
Vk Theorem 9[20]
Let
RU
=
:= ILk
u uSa
1')'1
V 2 0"1
+ ')'k'
u = I, J.
+ V 1u0"2uSa2')'2
vfv~
a
O"f
0
O"~
= Sl')'l--U+S2'Y2--U'
u=I,J.
PI ')'2 If RI ~ 1 and R J ~ 1, then the disease free equilibrium Eo(O, 0, 0, 0) is globally asymptotically stable; if RI > 1 or RJ > 1, then Eo is unstable. There exist two types of endemic equiliria for model (M11 ) , one of which consists of either EI (IP, Ig, 0, 0) or E J (0,0, Jp, Jg) and another has all components positive, E* (Ii ,Ii , Ji, J2). We call the first type of endemic equilibria boundary equilibrium, and the second type coexistence endemic equilibrium. Theorem 10[20] The boundary equilibrium EI (EJ) exists if and only if Rl > 1 (RJ > 1). If Rl > l(RJ > 1) and R J ~ l(Rl ~ 1) then the boundary equilibrium E J (El) does not exist and EI (EJ) is globally asymptotically stable. If both Rl > 1 and R J > 1, then when Dk < l(Dk > 1), El(EJ) is globally asymptotically stable and EJ(EI) is unstable, where Dk
=
%, k = 1,2. k
5
Epidemic models with natural age and infection age structures
As we know, chronological and infection age are very important factors in disease spread and a number of papers have already investigated some epidemic models involved these two age structures[21,22]. But previous dynamical analysis for many age-structure models were incompleted. The local stability for disease-free steady-state is easy to establish for most age-structure models when the basic reproduction number is less than a unity. The global stability of a stable age distribution, however, is very difficult in general. The following SIS model investigated by us
Some Recent Results on Epidemic Dynamics ...
17
focuses on the study of the global dynamics of the two-age structure[23].
oS oa
oS
+ at = -JL(a)S(a,t) -
r
G(a,t) +')'(a) lo i(a,e,t)de
S(O, t) = fA2 b(a, P(t))p(a, t)da
lAl
S(a, 0) = So(a), oi oa
oi
SeA, t) = 0,
oi
+ oe + ot = -(JL(a) +')'(a))i(a,e,t),
i(a, 0, t) = G(a, t), i(A, e, t) = 0,
i(a, e, 0) = io(a, e),
where a and e are the chronological age and infection age respectively, the total numbers of the susceptibles Set) and infectives let) at time tare given by Set) = faA sea, t)da,I(t) = faA faa i(a, e, t)deda, respectively. A is the maximum age and the total population size is pet) = S(t)+l(t). It is assumed that all newborns are susceptibles and the disease is not fatal, pea, t) = sea, t) + faa i(a, e, t)de is the entire population density at time
t, pet) = faA pea, t)da is the total population size at time t, b(a, pet)) is the density-dependent age-specific birth rate, So(a) and io(a, e) are the initial distributions, [AI, A2J is the fecundity period, < Al < A2 < A, JL(a) the age-specific mortality rate, ')'(a) the age-specific recovery rate, G(a, t) is the rate at which susceptible individuals of age a move over into the infective group per capita and per unit time, and G(a, t)
°
satisfies
G(a, t) = (poo(a) - faa G(a - e, t - e)n(a', e)de) . faA faa' A(a, a', e)G(a' - e, t - e)n(a', e)deda' , where Poo (a) is the total population at its demographic steady-state,
n(a', e) = exp( - fa~'_c(JL(e) + ')'(e))de) , A(a, a', e) is the adequate contact rate. Under the assumption that A(a, a', e) production number is defined to be
Ro =
10 lor' AI(a' A
= AI(a)A2(a',e),
the basic re-
e)A2(a',e)poo(a' - e)n(a',e)deda',
and the following two main results have been proved [23]. Theorem 11[23] Under some assumptions (see reference [23]) if A(a,a',e) = AI(a)A2(a',e), then the disease free steady-state is globally
Zhien Ma
18
asymptotically stable if Ro :::;; 1, whereas it is unstable and there exists a unique endemic steady-state if Ro > 1. Theorem 12[23] Under the same assumptions in Theorem 11, if A(a,a',c) = Al(a)A2(a')e- 6c , then the endemic solution is globally asymptotically stable if Ro > 1. Discrete models in population dynamics have been extensively studied, but the formulation and analysis of discrete models in epidemiology are still in its infancy, especially for discrete epecimic models with agestructure. The following is a general discrete age-structured SIS epidemic model. The basic reproduction number, global stability of the disease free equilibrium and bifurcation of the endemic equilibrium have been investigated [24].
So(t+l) = No, Sj+l(t + 1) j
=
Io(t+l)=O,
t=0,1,2,'"
m Sj(t) pjSj(t) - Aj L{3k 1k(t) N(t)
= 0,1, ... ,m - 1,
k=O
+ "(jlj(t),
J
j = 0,1"" ,m -1,
where {3kAj is the transmission rate between an infective of group k and a susceptible of group j, "(j is the recovery rate of group j. Let
f(x)
=
{31Ao + (32(A 1 + AOql(X)) + (33(A2 + Alq2(X) + AOql(X)q2(X) + ... +(3m[Am- 1 + Am-2qm-l(X) + Am-3qm-2(X)qm-l(X) + ... +AIQ2(X)q3(X)'" Qm-l(X)
+ AOQ1(X)Q2(X)'"
Qm-l(X)],
where Qj = Qj(x) = Pj - "(j - xAj/Nj , j = 1,2,'" ,m - 1. Define the reproduction number Ro = f(O), then we have Theorem 13[24] For the SIS model (M13 ), the disease free equilibrium is globally asymptotically stable if Ro < 1, and it is unstable if Ro > 1. When Ro > 1 and Ro - 1 sufficient small, there exists a small endemic equilibrium. The dynamical behavior of the general SIS model (M13 ) is quite complicated. There is no satisfied result on the uniqueness and global stability of the endemic equilibrium. But for the special case m = 2, the results we got are quite complete, The basic reproduction number R02 has been obtained, and we proved that the disease free equilibrium
Some Recent Results on Epidemic Dynamics ...
19
is globally stable if R02 > 1. For m = 3, we also obtained some results although the behavior is much more complicated [24].
6
Epidemic models with time dependent coefficients
In the reality, the growth of population and the transmission of disease often depend on seasons. This implies that the coefficients of epidemic models are often time dependence. In this case the corresponding models become non-autonomous differential systems. First, let us consider an SIR model with births and deaths as follows
dS dt = p(t) - p(t)S - (3(t)SI, dI dt = (3(t)SI - 'Y(t)I - p(t)I, dR
dt = 'Y(t)I -
p(t)R,
where we suppose that p(t), (3(t), 'Y(t) are all continuous, have upper bounds and positive lower bounds, and S(t) + I(t) + R(t) = N(t) = 1. From the second equation of the model (M14) we can see that
dI dt
(3(t)S
- = [ () "I t
( ) - l]b(t) + p(t)]I(t).
+P t
So if there exists a to such that ,(t~~~(to) < 1 and s(to) :::::: 1, then I(t) decreases at a neighborhood of to. Hence if we want I(t) decreasing with t for any initial value, we need
(3(t) Rmax =
max[ () t
( )] < 1.
"It +pt
But to make Rmax < 1 we will spend much more energy and cost. In the following, we are going to find a preciser condition. Theorem 14[25] Let R = (f)(!)(/1-)' If R > 1 then the disease free solution of the model M (denoted by DFS), S = 1, I = 0, R = 0 is unstable; if R < 1, then DFE is globally asymptotically stable, where (f) = limt->+oo f~ f~U)dU is called long-term average of the function j, and we assume that ((3), b), (p) all exist. It is eary to see that if (3, "I are all constants, then R = ,!/1- is just the basic reproduction number for the corresponding autonomous
20
Zhien Ma
system. For the non-autonomous system (MI4), this R is actually the basic reproduction number of the long-term average system.
dS
dt
=
(f.l) - (f.l)S - ((3)SI,
dI dt = ((3)SI - ('y)I - (f.l)I, dR
dt =
('y)I - (f.l)R.
For the non-automomous SIS model with extra death rate caused by disease and standard incidence, we also proved that the threshold of the corresponding long-term average system (M I5 ), R = 1 is the threshold to distinguish the unstability and global asymptotical stability of the disease free solution of the model. For the following SIRS model,
dN
dt
=
~~
= b(t)N - d(t)S - (3t) SI,
b(t)N - d(t)N - a(t)N + J(t)R,
dI (3(t) dt = N SI - 'Y(t)I - a(t)I - d(t)I, dR
dt
=
'Y(t)I - d(t)R - J(t)R.
We proved that the threshold of the corresponding long-term average system R = (b)+<~?+(-y) is still the threshold to distinguish the unstability and global asymptotical stability of the disease free solution of the mobel (MI6)' For the following simple SEIRS model with latent compartment E:
dS dt = -(3(t)SI + JR, dE
dt = (3(t)SI dI dt = aE - 'Y 1 , dR
dt = 'YI -
JR.
aE,
Some Recent Results on Epidemic Dynamics ... Suppose B(t) + E(t) + 1(t) + R(t) == 1, and (f3) sponding long-term average system is dB
-
dt
= 73 exists.
21 The corre-
-
= -f3B1 +8R
dE - =f3B1 -aB dt
'
'
d1
-=aE--v1 dt
' ,
dR
-dt = "(I - 8R . It is easy to see that for (MI8 ), the reproduction number is Ro = §. and the disease free equilibrium Eo is globally asymptotically stable Ro < 1, is unstable if Ro > 1. For model (M17 ), is Ro = §. still a threshold to distinguish the ptability of disease free solution? 'Unfortunately, it is not true, an example was given in [25J. But we proved the following result: Theorem 15[25J For the model (M17) , if R = (~) < 1, then the disease free solution of the model is globally asymptotically stable. Actually for motel (MI7 ), the threshold should be Ro = a(W) = 1, where w(t) is a solution of the equation ~':' = f3(t) - (a - "()w - aw 2 . It is not solvable, but may be obtained by numerical analysis if it is necessary.
If
,
7
Epidemic models combining with population ecology
The theory and application of epidemiology modeling combining with the population ecology was started more than 20 years ago. The 1981 Dablem conference on the population biology of infectious diseases was a seminal in identifying some key questions about the effects of infectious diseases on naturally fluctuating host populations. In 1982, Anderson and May published a book "population biology of infectious diseases[26J" . They also investigated a predator-prey model with disease transmission only in the prey population and bilinear incidence[27J. The stable periodic oscillation of the two populations has been found in their models. Some other epidemic models of interactive species were discussed in [28 - 31]. Here we just introduce two results obtained by our group. One is a predator-prey model with infectious disease, the model is
Zhien Ma
22
the following:
· N1
N1 = r1(1 - -)N1 - a1N1N2
K1 · a1r1N1 r1 N 1 8 1 = (b 1 )N1 - [d 1 + (1- a1)--l81 - aN2 8 1 K1 K1 -/31 Slv~l + /lh ·
h N2
=
r1 N 1
8 h N1 KaN1N 2 - d2N 2 ,
1 /31--/lh - [d 1 + (1- a1)-K lh - aN21,
=
1
· ~h 82 = K aN1N 2 - a N2 - d 2 82 · 12
8 1 N2
~~
/32 N2 + /2~
8 h -/2 12. N2
2 2 = /32-- d212 + a -2 -
This model has 6 equilibra. By the theoretical analysis, our results imply the following biologic meanings[32j. (1) If there is no prey initially, then there is never any prey, and predator population goes to extinction. (2) Suppose that the feeding efficiency k of the predator population is low enough so that the predator population goes to extinction, and if the basic reproduction number Ro in the isolated prey population is below the threshold, then the disease dies out and the prey population goes to its carrying capacity K 1 ; if Ro is above the threshold, then the disease in the prey approaches the endemic level and the prey population goes to its carrying capacity K 1 . (3) Suppose that the feeding efficiency k of the predator population is high enough so that the predator population persists, and if R1 (R1 is a basic reproduction number in the prey population when the prey and predator populations are at their persistent equilibra) and R2 (R2 is the basic reproduction number for the isolated predator population) are below the thresholds, then the disease dies out and the prey and predator populations go to their usual persistent equilibra; if R1 is below the threshold but R2 is above the threshold, then the disease persists in the predator population, but dies out in the prey population; if R1 is above the threshold, then the disease persists in both the populations. Note that this disease persistence in the predator population occurs even if the basic reproduction number R2 is below the threshold. Thus even if the disease transmission rate a during the predation process is very small, the disease will persist in the predator population whenever it persists in the prey population. We also investigated predator-prey SIS model with mass action incidence, predator-prey SIR model with standard or with mass action
Some Recent Results on Epidemic Dynamics· ..
23
incidence l321 . Another kind of models we investigated are four SIS and SIRS epidemic models of two competitive species with the standard or the mass acting incidence and crossing infection, some complete results were obtained l331 . One of the results shows that under some conditions, the disease can die out finally by cutting off the inter-infections between the two species or decreasing the inter-transmission coefficients between the two species to a fixed value.
8
Epidemic models combining with ecotoxicology
Starting from 1983, T. Hallam, Z. Ma, etc. investigated the dynamic behavior of population in a polluted environment. They got the threshold between weak persistence in the mean and extinction of aquatic populations in a polluted watersI34-371. The mathematical meaning of weak persistence in the mean of a population x(t) is that limsup
f0 x(7)d7 t
t--->+oo
> O.
[38J investigated an SIS model in a polluted environment to see how the toxicant effects the epidemic dynamic behavior. The model they considered is described as dS
SI B(N, Co, Ce)N - D(N, Co, Ce)S - 'x(Co) N
ill = dI dt
+ ')'(Co)I,
SI
= 'x(Co) N - D(N, Co, Ce)I -,),(Co)I,
dN
dt
N
= ')'(Co)N(l - k(C ))' e
dCo -=KCe-(g+m)Co , dt dCe dt where N
=
S
=
-hCe - k1NCe
+ I,
+ glNCo + u(t),
B(Nl' Co, Ce)
=
N b(Co) - ar(Co) K(C ) , r(Co) = e
B(N, Co, C e ) - D(N, Co, C e ). Ce and Co are concentrations of the toxicant in the environment and inside the organism's body respectively, they depend on u(t) which is input rate of the toxicant from outside the environment. Here we assume that besides density dependent, the birth rate and death rate will also be effected by the concentrations of the toxicant Co and C e , the coefficients of the infection rate ,x, recovered rate ,x, recovered rate
Zhien Ma
24
'Y and the intransic growth rate r will be effected by Co, the carrying
capacity k will be effected by Ceo The last two equations describe the interactive influence among the organism, environment and toxicant[35]. When the pollution input rate u(t) is know the model M 20 is a kind of non-autonomous epidemic system. Theorem 16[38] For the model (M20), let
J/ R(T)dT .
(R)* = lim sup 0 t->oo
t
(1) If (R)* < 1 then I(t) goes to extinction; (2) If (R)* > 1 then I(t) is weakly persistent in the mean; (3) If (R)* = 1 then I(t) is at most barely persistent, which means that either lim I(t) = 0 or limsupI(t) > 0 but (I)* = O. t---+oo
t---+oo
We also proved the existence and global stability of the periodic solution for the model M20 under some conditions. The results show that under different conditions, pollution may promote the disease to be spread, but it may also eradicate the disease.
9
The phenomenon of stability switches on some epidemic models with time delay
For the epidemic models with time delay, if we ignore the death rate in the time delay period, then the characteristic equation of the system at the equilibrium usually has the form
P(A)
+ Q(A)e- A = 0, 1"
(9.1)
where P and Q are usually polynomials. Cooke and van den Driessche found that the stability of this characteristic equation may be changed a finite times when T increases[39], this phenomenon is called stability switches, and the stability switches can be determined by a formula deduced by them. But in the reality, when the time delay period is not very short, the death rate should not be ignored. In this case the corresponding characteristic equation is given by
P(A, T)
+ Q(A, T)e- A = O. 1"
(9.2)
Beretta and Kuang found an essential different between the two characteristic equations[40]. For equation (9.1), it must be ultimately unstable; but equation (9.2) may be ultimately stable. Beretta and Kuang also contributed a geometric method to determine the stability switch and
Some Recent Results on Epidemic Dynamics ...
25
the ultimate stability, but their method needs some mathematic software to assist for fixed parameters. We investigated further the ultimate stability of a special type of characteristic equation (9.2) where
= A2 + a(7)A + C(7), Q(\ 7) = b(7)A + d(7).
P(A,7)
(9.3)
This characteristic equation appear often in some biologic systems with time delay. The results obtained by us show that for this type of characteristic equation, ultimate stability, ultimate unstability and permanent alternation of the stability may all happen, and the criterion has been derived to determine these situations directly from the equations instead of by numerical simulation. Theorem 17[41] Under some assumptions (see [41]) which conforms to the common cases, the following is true for the characteristic equation (9.2) with P and Q expressed by (9.3). (1) If the existent interval of y(7) is finite, the equation (9.2) must be ultimately stable; (2) If the existent interval of y( 7) is infinite, (9.2) is ultimately stable provided limsupD(7) < 0, ultimately unstable provided liminf D(7) >
o·,
T-+OO
T-+OO
(3) If the existent interval of y( 7) is infinite, the stability switches of (9.2) will appear forever as 7 increases provided limsupD(7) > 0 and liminf D(7) < O. T->OO
Where ±iY(7) are the pure imaginary eigenvalues, i.e. A(7) = ±iY(7), and y( c) is the positive root of the equation:
We assume that for any 7 E R+ o, the equation F(y, 7) = 0 has at most one positive root y = y(7), and function C2(7) - d2(7) has at most one zero on R+o. Hence if the function c 2 (7) - d2 ( 7) has no zero on R+o then the existent set of y = y(7) is interval (0, +(0) when F(y,7) = 0 has just one positive root y = y( 7); if c2 (7) - d2 (7) has just one zero 7 E R+o, then the existent set of y = y( 7) is interval (0,7') or (7', +(0) and y(7') = 0: D(7) = y(7)7 - B(7) , 27r and B(7)
E
[O,27r] is determined by the equations
. _ -b(7)Y[C(7) - y2] + a(7)d(7)y smB b2(7)y2 + d2(7) , {
cosB
= -
d(7)[C(7) - y2] + a(7)b(7)y2 b2(7)y2 + d2(7) .
26
Zhien Ma
TheoreIll 18[41J Suppose that the existent interval of y( r) is infinite and that lim D(r) = 0, then the following conclusions are true. r->oo
(1) If the number of the roots of D(r) = 0 is even, then (9.2) is ultimately stable; (2) If the number of the roots of D(r) = 0 is odd, then (9.2) is ultimately unstable; (3) If equation D(r) = 0 has infinite number of roots, then the stability switches will appear forever as r increases. To show the applications of our method, [41] gave two examples, one is a Juvenile-adult population model, that the unique positive equilibrium (J*, A *) might be ultimately stable, which has been showed by Beretta and Kuang in terms of some software[40J. Applying our method, it is easy to prove the equilibrium is either always stable, or ultimately stable. Another example is an SEIS epidemic model with time delay. By means of our method, it is easy to prove that in any case the endemic equilibtium of this system is ultimately unstable. We also extended our method to the following more general characteristic equation [42J.
P(>", r)
+ Q(>", r)e- Ar + R(>", r)e- 2Ar = 0,
which may also appear in some epidemic models. The formula, which may differentiate when the stability switches happen, has been deduced, and the method, which determines the ultimate stability for some special characteristic equations, has been obtained.
10
Study on HIV / AIDS
Our group did some work on HIV / AIDS in two ways. One is from the theoretical immunology point of view to investigate the dynamic behavior among the T-cells, antigen presenting cells (APes) and HIV-l. The following Figure 10.1 shows their interaction network[43J. Co+T
~ k, Cr+T
V·
Co+T*
(~.
k.
2k,
nk,
Cn
t
nk •
Cn-r+T*
Figure 10.1: The flowchart of the interaction network of an HIV / AIDS model. Where, the resting T-cells are denoted by T, the healthy and activated T-cells by T* and the conjugate of an APe bound with j T-cells by C j . In detail, an APe, either an antigen-primed dendritic cell or a macrophage, has several T-cell binding sites. After invading into vivo,
Some Recent Results on Epidemic Dynamics ...
27
antigens are detected, taken up, and processed by APes. The interaction of these APes and the specific resting T-cells leads to T-cell activation, and then T -cells translate into class T* from T. The binding rate between T-cells and a free site on an APe is assumed as kb, then the binding rate of T -cells and an APe which have i free sites will be kb multiplied by the number of available free sites, i. After being bound, a T-cell can also dissociate with rate kd, or it can become activated with the rate coefficient k a . To sum up, the dissociation and activation rates per APe are proportional to the total number of T -cells bound to the APe. It was also assumed that the population size of the APe precursors (Cp ) keeps constant according to literatures. Using Ag to represent the concentration of antigen, the description above was translated into the model as follows:
dC
dto = bCpV dC
dt
j
= (n -
j
nkbCoT + (kd
+ 1)kbCj-1T -
+ ka)C1 -
deCo,
[(n - j)kbT + j(kd + ka)]Cj
(M21 )
Using the predigestion technique, we can simplify the system as follows: ds - = a(v - S), dT dx s = a + (-- - l)x, dT l+x dv dT
=
1l'[(8' + 8(p - (1)V)~ _ 82 v], 1 +pv 1+ x
where , s , x , and v are scaled variables which represent the total (scaled) population of antigen presenting sites, the resting T -cells and HIV virus, respectively. Our model emphasizes the impact of APes during HIV infection and the cell-to-cell contact manner in transferring of HIV-l in vivo. The existence and stability of the uninfected steady state and those of the
28
Zhien Ma
infected steady states are discussed. The uniform persistence of the system is also proved. By this model, we found the critical strength of the immunity system, under which the individual will be infected even though the dose of invaded HIV Viruses is very small, but above which, will not be infected for a certain dose. We also obtained the different parameter regions to distinguish the cases where the infected person becomes a rapid progress or a long term survivor[431. We also investigated the effects of cytokinin and therapy strategy by a model of interaction among the healthy CD4+ T-cells, infected CD4+T-cells, CD8+T-cells, IL-2(interleukin-2), CAF(CD8 antiviral factor) and free virus[44, 451. The results show that the effects of IL-2 and CAF in the treatment for the infected are limiting, namely, the curative effect will go to saturation when the does of IL-2 or CAF or both are increased. Our findings also show that for curative effect, CAF is better than IL-2. We also gain some possible reasons for the collapse of the immune system. Another way of our research for HIV / AIDS is to investigate its spreading rules. We constructed a competition model of HIV with recombination effect and found that the principle of competitive exclusion is no longer valid in the competition between the recombination HIV virus and its parental viruses. The recombination effect makes the mosaic virus can either coexist with their parental viruses or survive alone, which depends on the initial state[461. According to this result, we suggest that the most important vaccine is for mosaic virus; and a suggestion of therapy strategy is that first let the parental viruses decrease low enough so that the phase state could be in the attractive region of the equilibrium where the mosaic viruses survive alone, hence the parental viruses may die out by the dynamic behavior itself, then pay attention to eliminate the mosaic virus using the available specific vaccines[ 461. Our results gave a reasonable explanation to the question: why there are two different transmission routes with different B' /0 recombinant strains of HIV in China?[ 461
11
Modeling and study for SARS transmission and control in China
SARS (Severe Acute Respiratory Syndrome) is a newly acute infective disease with high potential of transmission to close contacts. This infection first appeared and was transmitted in China in November 2002, and spread rapidly to 31 countries within half a year. Till June 2003 the cumulative number of diagnosed SARS cases is 8454 with 792 deaths in the whole world[47, 481. It was especially serious in China. The cumulative number of diagnosed cases is 5327 with 343 deaths during about
Some Recent Results on Epidemic Dynamics· ..
29
half a year. In those days of infection peak (middle of May 2003), there were over 100 cases increased per day in Beijing China. In order to provide a reference for the forecast and control to the transmission of SARS in China, our group constructed a model according to the specific situation in China. Our results of research was published to reporters on May 20, 2003, on that day the number of new diagnosed in Beijing still over 100, our report shows that according to the standard of WHO, the travel warning can be removed in the last ten-day period of the June in Beijing, and it was removed on June 23. The number of cumulative diagnosed in the mainland of China estimated by our model is less than 6000, and it is actually 5327. The difficultIes we met in the modeling of SARS were the following: (1) Because SARS is a new disease, the infectious probability is unknown, and whether the individuals in the exposed compartment have infectivity is not sure; (2) how to select the compartments and how to construct the model such that it fits the situation in China? Especially, how to reflect those effective control measures adopted by the government such as various kinds of quarantines? (3) how to get the data of those parameters which are difficult to quantify, for example, the intensity of the quarantine?
As
~".o 6
de~
~9{?
diJ
Ah
g(H, Q, D)
Figure 11.1: The flowchart of our SARS model. According to the general principle of epidemic modeling and the specific situation for SARS outbreak in China, the flowchart of the model we established is shown in Figure 11.1. Where S, E, I, R are the numbers of suscepitables, exposed, infectious and recovered respectively, they are all in the free environment. Q is the number of quarantined in the hospitals who come from two ways: one is from group E, these individuals are infected by SARS but in the latent period; another way is from the group
30
Zhien Ma
8, these individuals without SARS Virus but misdiagnosed as possible SARS patients and they will return to the susceptible group after further medical examinations to rule out SARS Virus, which needs about 10 days. D is the number of diagnosed, who come from both group Q and group I. H is the number of high dangerous susceptibles who are working in the hospitals to take care of SARS patients. In China, the individuals in groups Q, D, and H are all isolated. Because whether the individuals in exposed group have infectivity is not sure during that time, for the sake of safe, we suggested that the infectivity for those individuals is 10 percents comparing with the infectivity of the infectious, On the basis of the flowchart, the model is easily formulated. Let the incidence F(t) = 1(8, E, I, R) = ,B(t)[I(t) + kE(t)], where is the infection rate of an individual in group I, it depends on the probability of transmission of SARS which we do not know during that time, and also depends heavily on the intensity of control measures which are difficult to quantify. But the incidence F(t) = ,B(t)[I(t) + kE(t)] expresses the new infected people at the tth day which can be calculated from the daily report of the Ministry of Health in China (MHC); the individuals in E(t) and I(t) will stay in the free environment for 5 days and 3 days respectively, the cumulative number of E(t) and I(t) at tth day may also be calculated from the daily report. Hence we may get the daily data of the infection rate
,B(t) by ,B(t) = I(t) :(~E(t)' Where K = 0.1. Using this back tracking method we may estimate ,B(t) and all other parameters of our model[49l. For the term -dsqD in the first equation of the model (M2 2), usually it should be -d sq 8, but because 8 is a huge number, the coefficient d sq must be very small, and this small value will make our model to be an ill system. According to some statistic data, one diagnosed will bring 1.3 individuals to the quarantine group, so instead of -d sq we use -dsqD. Figure 11.2 is the graph of ,B(t), where the continuous curve is the smooth approximation of the actual data reported by MHC, which was obtained by regression analysis method. The origin was April 21, 2003. We can see that ,B(t) decreases very fast which shows that the control measures adopted by our government were very efficient and dramatic. Using the curve ,B(t) and other parameters we estimated, we did some simulations from the model established by us according to the flowchart in Figure 11.1. Figure 11.3 shows the number of cumulative diagnosed in the mainland of China; Figure 11.4 shows the number of diagnosed in the hospitals. Both origins are April 21, 2003, we can see from these figures that the number of SARS patients increases rapidly during the first three weeks, reaches the peak between May 11 and May 18, 2003, and with the maximal number in the hospitals between 3164 cases and 3220 cases.
Some Recent Results on Epidemic Dyna~ics ...
31
0.05 i·
o~~~_ --=======,=-~=-=,=d
o
10
20
30
40
50
60
70
80
90
100
Figure 11.2: The graph of the contact rate (3(t).
,....""... ....
5000
~~-~-"-----------------
.l
:t-
o!
4500
,~*
•+
4000
... , ¥
. •
~
3500
oj.
3000
.
~
•¥
2500
.;.
.,'¥ __~~__~__~~__~__~~~ 10 20 30 40 50 60 70 80 90 100
2000~~
o
Figure 11.3: The number of cumulative diagnosed in the mainland of China.
On the basis of the model, we did some simulations on what will happen if the preventive and control measures were relaxed from May 19, June 10, or if the infected individuals were quarantined one or two days later. We also did some theoretical analysis to a continuous model[49] and a discrete model[50] for the SARS spread in China. The reproduction number had been obtained and global stability had been proved.
Zhien Ma
32
3500 .----~--~--~--~--____,
...(\.
3000
-+ :
2500
.
•
•
"\
2000 "
."
1500 ;
•
~
. .
1000
\
500 O~
o
__~____~~~~__~~~~ 50
100
150
200
250
Figure 11.4: The number of diagnosed in the hospitals.
References [1] W.O. Kermack, A.G. Mckendrick: Contributions to the mathematical theory of epidemics. Proc. Roy. Soc. 1927 (A115), 700-72l. [2] K. Dietz: Overall population patterns in the transmission cycle of infectious disease agents. In Population Biology of Infectious Diseases, Springer. 1982. [3] J .A.P. Heesterbeek, J .A.J. Metz: The Saturating contact rate in marrige and epidemic models. J. Math. Biol. 1993(31),529-539. [4] W.M. Liu, S.A. Levin, Y. Iwasa: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 1986(23), 187-204. [5] W.M. Liu, H.W. Hethcote, S.A. Levin: Dynamical behavior of epidemiological model with nonlinear incidence rates. J. Math. Biol. 1987(25), 359-380. [6] J.Q. Li, Z.E. Ma: Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete and Continuous Dynamical Systems Series B. 2004(4),635-642. [7] J.Q. Li, Z.E. Ma: Global analysis of SIS epidemic models with Variable total population size. Math. Comput. Modelling, 2004(39), 1231-1242.
Some Recent Results on Epidemic Dynamics ...
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[8] J.Q. Li, Z.E. Ma, Y.C. Zhou: Global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria. Acta. Math. Sci. 2006(26B) 83-93. [9) Z. Jin, Z.E. Ma: Epidemic models with continuous and impulse vaccination. To appear. [10] Z. Feng, H.R. Thieme: Endemic models with arbitrarily distributed periods of infection, I: general theory. SIAM. J. Appi. Math. 2003(61),803. [11] Z. Feng, H.R. Thieme: Epidemic models with arbitrarily distributed periods of infection, II: fast disease dynamics and permanent recovery. SIAM, J. Appi. Math. 2003(61),983. [12] L-I, Wu, Z. Feng: Homoclinic bifurcation in an SIQR model for Childhood disease. J. Diff. Equ. 2000(168), 150-167. [13] H. Hethcote, Z.E. Ma, S. Liao: Effects of quarantine in six endemic models for infectious disease. Math. Biosci. 2002(180), 141-160. [14] J.S. Muldowney: Compound matrices and ordinary differential equations, Rocky Mount. J. Math. 1990(20), 857-972. [15] M.Y. Li, J .S. Muldowney: Global stability for the SEIR in epidemiology. Math. Biosci. 1995(125), 155-164. [16] M.I. Li, H.I. Smith, L. Wang: Global dynamics of an SEIR epidemic model with vertical transmission. SIAM, J. Appi. Math. 2001(62), 58-69. [17] J. Zhang, Z.E. Ma: Global dynamics of an SEIR epidemic model with saturating contact rate. Math. Biosci. 2003(185), 15-32. [18] J. Zhang, J.Q. Li, Z.E. Ma: Global dynamics of an SEIR model with immigration of different compartments. Acta Mathematica Scientia Series B. 2006(26), No.3 551-567. [19] Z.E. Ma, J.P. Liu, J. Li: Stability analysis for differential infectivity epidemic models. Nonlinear Analysis: RWA, 2003(4), 841-856. [20] J. Li, Z. Ma, et aI.: Coexistence of pathagens in sexually transimitIed disease models. J. Math. BioI. 2003(47), 547-568. [21] K. Diftz, D.Schenzle: Proportionate mixing models for agedependent infection transmission. J. Math. BioI. 1985(22), 117-120. [22] H.R. Thieme, C. Castillo-Chavez: How may infection-agedependent infectivity affect the dynamics of H IV/AIDS? SIAM, J. Appi. Math. 1993(53), 1447-1479. [23] Y.C. Zhou, B.J. Song, Z.E. Ma: The global stability analysis for an SIS model with age and infection age structures. IMA Volume 126, C. Castillo-Chavez, S. Blower, eds., Springer-Verlag, 2001, 313-335.
34
Zhien Ma
[24] Y.C. Zhou, P. Fergola: Dynamics of a discrete age-structured SIS models. Discrete and Continuous Dynamical Systems Series B. 2004(4),843-852. [25] J.L. Ma, Z.E. Ma: Epidemic theshold conditions for seasonally forced SEIR models. Mathematical Biosciences and Engineering, 2006(3), 161-172. [26] RM. Anderson, RM. May, eds.: Population biology of infectious diseases, Springar-Verlag, New York, 1982. [27] RM. Anderson, RM. May: The invasion, persistence, and spread of infectious diseases within animal and plant communities, Philos. Trans. R Soc. London 1986(B314), 533-570. [28] E. Venturino: Epidemics in predator-prey models: Disease in the prey, In mathematical population dynamics: Analysis of hetergeneity, one: Theory of epidemics, Wuerz publishing, Canada, 1995. [29] Y.N. Xiao, L.S. Chen: Modelling and analysis of a predator-prey model with disease in the prey. Math Biosci. 2001(170), 59-82. [30] RG. Bowers, M. Begon: A host-pathogen model with free living infective stage, applicable to microbial pest control. J. Theor. BioI. 1991(148), 305-329. [31] M. Begon, RG Bowers: Host-host-pathogen models and microbial pest control:the effect of host self-regulation, J. Theor. BioI. 1995(169), 275-287. [32] L.T. Han, Z.E. Ma, H.M. Hethcote: Four predator-prey models with infectious diseases. Math. Comput. Modelling, 2001(34), 849-858. [33] L.T. Han, Z.E. Ma, T. Shi: An SIRS epidemic model of two competitive species. Math. Comput. Modelling, 2003(37), 87-108. [34] T.G. Hallem, C.E. Clam, G.S. Jordan: Effects of toxicants on populations: a qualitative approach, II. First order kinetics, J. Math. BioI. 1983(109), 411-429. [35] T.G. Hallam, Z.E. Ma: Persistense in population models with demographic fluctuations. J. Math. BioI. 1986(24), 327-339. [36] Z.E. Ma, B.J. Sang, T. Hallam: The threshold of sarvival for systems in a fluctuating environment. Bull. Math. BioI. 1989(51),311323. [37] Z.E. Ma, G.R Cui, W.D. Wang: Persistence and extinction of a population in a polluted environment. Math. Biosci. 1990(101), 597. [38] F. Wang, Z.E. Ma: Persistence and periodic orbits for an SIS model in a polluted environment. Comput. Math. Appl. 2004 (47),779-792.
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[39] K.L. Cooke, P.Van den Driessche: On zeros of some transcendental equation, Funkcial. Evac. 1986(29), 77-90. [40] E. Beretta, Y. Kuang: Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM, J. Math. Anal. 2002(23), 1144-1165. [41] J.Q. Li, Z.E. Ma: Ultimate stability of a type of characteristic equation with delay dependent parameters. J. Syst. Sci. and Complexity, 2006(19), 137-144. [42] J.Q. Li, Z.E. Ma: Stability switches in a class of characteristic equations with delay-dependent parameters. Nonlinear Anal. RWA, 2004(5), 389-408. [43] J. Lou, Z.E. Ma, Y.M. Shao, L.T. Han: Modelling the interaction of T-cells, Antigen presenting cells, and HIV-1 in Vivo. Comput. Math. Appl. 2004(48),9-33. [44] J. Lou, Z.E. Ma, Y.M. Shao: HIV-1 population dynamics in Vivo: implications for pathogenesis, effect of cytokine and therapy strategy. J. Biol. Syst. 2004(12), 315-333. [45] J. Lou, Z.E. Ma, Y.M. Shao, L.T. Han: The impact of the CD8+ cell non-cytotoxic antiviral response (CNAR) and cytotoxic T lymphocyte (CTL) activity in a cell-to-cell spread model for HIV-1 with a time delay. J. Biol. Syst. 2004(12), 73-90. [46] F. Wang, Z.E. Ma: A competition model of HI V with recombination effect. Math. Comput. Modelling, 2003(38), 1051-1065. [47] WHO and CDC, Available from http:/www.moh.gov.cn/was40/ detail ?record= 14&channelid=8085&searchword=%B 7%B5 %E4 % D2%DF%C3%E7. [48] MHC, Available from http://www.moh.gov.cn/zhg-/xgxx/fzzsjs/ 1200306030008.htm. [49] J. Zhang, J. Lou, Z.E. Ma, J.H. Wu: A compartmental model for the analysis of SARS transmission patterns and outbreak control nieasures in China. Appl. Math. Comput. 2005(162),909-924. [50] Y.C. Zhou, Z.E. Ma, F. Brauer: A discrete epidemic model for SARS transmission and control in China. Math. Comput. Modelling, 2004(40), 1491-1506.
36
Modeling SARS, West Nile Virus, Pandemic Influenza and Other Emerging Infectious Diseases: A Canadian Team's Adventure* Fred Brauer Department of Mathematics, University of British Columbia Vancouver, BC V6T lZ2, Canada E-mail:
[email protected]
Jianhong Wu Department of Mathematics and Statistics, York University Toronto, Ontario, M3J lP3, Canada E-mail:
[email protected] Abstract Shortly after the 2002-03 Severe Acute Respiratory Syndrome (SARS) outbreak, a Canadian team on modeling communicable diseases was established and an adventure of interdisciplinary research involving close interaction and collaboration between modelers, epidemiologists, and public health policy makers started. This article provides a review ofthe history of this team, its collective efforts and long-term goal in modeling emergingjreemmerging communicable diseases of critical importance to Canada and the international community.
1
Introd uction
Canada has a fine reputation for excellence in the general area of mathematical biology, and takes pride in its an admirable pool of world class modelers in life sciences. Therefore, it should not be very surprising that when an urgent call for collective efforts in modeling the transmission *Research was partially supported by Natural Sciences and Engineering Research Council of Canada, Social Sciences and Humanity Research Council of Canada, Mathematics for Information Technology and Complex Systems, and Canada Research Chairs Program. This work was also supported by a Canada-China Thematic Program on Disease Modeling, funded by the Canadian Federal Network of Centres of Excellence program and the International Research Development Centre.
Modeling SARS, West Nile Virus, Pandemic Influenza and . . .
37
potential and transmission dynamics was made, shortly after the 200203 Severe Acute Respiratory Syndrome (SARS) outbreak hit its largest city Toronto, 18 scientists responded almost simultaneously and acted immediately. Virtually within a couple of days after the call from Mathematics for Information Technology and Complex Systems (MITACS, a center under the Canadian federal program Network of Centers of Excellence) (www.mitacs.ca) , the team was established and an intensive interdisciplinary research project was launched. This MITACS team therefore started as a pilot project with a primary mandate to utilize mathematical models and computer simulations to make a useful contribution to the management of the SARS outbreak in Toronto, and more specifically, the Great Toronto Area (GTA). However, from the very beginning, the team has had a long-term goal of building a focused national group for analyzing, modeling and predicting transmission dynamics and spread patterns of infectious diseases that, in collaboration and coordination with other Canadian and international research centers, can provide the much needed national capacity for rigorous analyses and defensible decision-making in the domain of public health. The team has been making steady progress towards its long-term goal, which is partially reflected by its scientific contributions, outreach activities, interdisciplinary research programs, international collaborative efforts, promotion of mathematical research of infectious diseases, training programs including a series of summer schools and intensive courses, and the recent creation of the Center for Disease Modeling (CDM). This paper intends to provide a brief summary of the aforementioned progress. We shall, in Section 2, discuss the team's scientific contributions for the understanding of the transmission dynamics and spatial spread patterns of specific diseases such as SARS, West Nile virus and pandemic influenza. We shall then, in Section 3, describe the team's networking activities, its interaction with some Canadian public health institutes such as Ontario Ministry of Health and Long-team Care and Public Health Agency of Canada (PHAC). We shall, in Section 4, present some details of the team's efforts at developing a series of Summer School and intensive courses aimed at training the younger generation in a highly interdisciplinary setting, and at providing public health people with enough understanding of the modeling process to consult and communicate with modelers when they encounter something that lends itself to modeling. This section will also describe some of the team's international collaborative efforts including the on-going Canada-China collaboration, and the activities being planned at the newly created Center for Disease Modeling. In the final section, we provide further comments from the experience/lessons we have gained from the past.
38
2
Fred Brauer, Jianhong Wu
Progresses on specific diseases: scientific contributions
The spread of a communicable disease involves characteristics of the agent, the host and the environment in which transmissions take place. The goal of mathematical epidemiology - the art and science of modeling communicable diseases - in relation to public health, is to evaluate the agent-host-environment interface and efforts to alter the interface through intervention to our advantage, be they preventive or therapeutic in nature. Mathematical epidemiology has a long history, but in recent years researchers in this area have developed more complex and biologically relevant models that have become important for influencing the design of control programs. Some of these models have been developed for new diseases, some include new treatments, some involve evolutionary aspects and some consider new patterns of social behavior and travel. Recent mathematical approaches include deterministic compartmental models, stochastic models, network models and Markov chain Monte Carlo models. These techniques are often complemented with computer simulations, which use demographic and disease incidence data. Nevertheless, there remain many challenging problems in the understanding of disease transmission and spread and solutions of these problems often require well-coordinated team efforts and interdiscipli:g.ary collaboration. The MITACS team (http://www.liam.yorku.ca/research/MADI/) does have quite broad expertise in epidemiology, virology, statistical analysis, health information, public health policy, mathematical modeling and scientific computation. This diversified and complementary expertise has put the team in an excellent position to address some of these challenging problems, as illustrated by some of the samples presented in this section for some specific diseases of great current interest. The team's expertise and research program have been expanding to cover the area of mathematical immunology, and it is hoped that this growing strength will enable the team to elucidate the complex interplay between infectious pathogens and the immune system, drug therapy or vaccine interventions, and other key contributors to microbial pathogenesis. Most team members have their own active research programs giving a strong foundation for sustainable growth of the team's scientific advancement. Depending on the issues under consideration and the required expertise of individuals, various subsets of team's members have been developing (sub-) projects requiring collective efforts. The remaining part of this section provides a few samples of this self-organized collaboration within the team and its external collaborators. The choice of the three diseases reflects our desire to develop a variety of templates
Modeling SARS, West Nile Virus, Pandemic Influenza and· . .
39
for mathematical models to fit different types of disease dynamics and management. More specifically, we note that the 2002-03 SARS epidemic and the anticipated influenza pandemic have been catalysts for a great deal of mathematical modeling of epidemics. SARS models were the first to include isolation and quarantine, influenza models the first to introduce asymptomatic stages and antiviral treatment with implications for drug resistance. West Nile virus models were the first to incorporate the ecology, spatial dispersal and biological/physiological structures of hosts. These models have broader applicability, much in agreement with our belief that it would be worthwhile to work on general disease transmission models that include features relevant to current and possible future outbreaks of diseases.
2.1
SARS
Severe acute respiratory syndrome (SARS), a new, highly contagious, viral disease, emerged in China late in 2002 and quickly spread to 32 countries and regions causing in excess of 774 deaths and 8098 infections worldwide. In the absence of a rapid diagnostic test, therapy or vaccine, isolation of individuals diagnosed with SARS and quarantine of individuals feared exposed to SARS virus were used to control the spread of infection. In the study [1] we examined mathematically the impact of isolation and quarantine on the control of SARS during the outbreaks in Toronto, Hong Kong, Singapore and Beijing using a deterministic model that closely mimics the data for cumulative infected cases and SARSrelated deaths in the first three regions. However, the models did not fit the data for Beijing until mid-April, when China started to report data more accurately. The results reveal that achieving a reduction in the contact rate between susceptible and diseased individuals by isolating the latter is a critically important strategy that can control SARS outbreaks with or without quarantine. An optimal isolation program entails timely implementation under stringent hygienic precautions defined by a critical threshold value. Values below this threshold lead to control, but those above are associated with the incidence of new community outbreaks or nosocomial infections, a known cause for the spread of SARS in each region. Allocation of resources to implement optimal isolation is more effective than to implement sub-optimal isolation and quarantine together. This is of course a well-known observation now in the medical and public health community, but this model-based analysis and conclusion would have been a great contribution to crisis management had it been developed and made available to and accepted by those involved in the decision making. This supports the idea that developing general
40
Fred Brauer, Jianhong Wu
templates which can be adapted to model new disease outbreaks would be useful in order to be able to provide analysis quickly. One of the salient features of the outbreak of SARS in the Greater Toronto Area (GTA) was the role of the hospital in transmission. Of 144 early patients, 111 (77%) were exposed to SARS in the hospital setting; of these, 73 patients (51%) were health care workers, including nurses, respiratory therapists, physicians, radiology and electrocardiogram technicians, housekeepers, clerical staff, security personnel, paramedics, and research assistants. The high risk of transmission within the health care setting has significant impact on the conduct of public-health interventions in the SARS epidemic, and potentially for other emerging respiratory diseases. To examine the SARS outbreak in GTA, we developed in [2] a compartmental model dividing the entire population into classes of susceptibles, exposed, infectives, hospitalized and removed, and subclasses representing the general public and those in the hospital setting including health care workers and patients (HCWP). The model reflects the extremely intense exposure of HCWP to infected individuals prior to awareness of SARS by the medical community; their heightened risk continued until adequate precautions were fully operant in hospitals. The analysis shows that the secondary infection induced by a hospitalized patient for the HCWP (Ro ~ 4.5) is much larger than the secondary infection induced by an average infective for the general public (Ro ~ 1.6) during the first two weeks of the SARS outbreak in GTA. These secondary infection rates were later reduced when hospital infection control procedures and community-wide quarantine measures were introduced. Our considerations are also highly relevant to the spread of other respiratory infections, including pneumonic plague or other emerging respiratory infections, for which hospitals are amplifiers of diseases that involve the general public, and our models should be useful to address issues related to their control. The epidemiology of SARS has been complex with rapid spread in some areas (e.g. Beijing) but not others, (e.g. Shanghai) despite introduction of the causative agent, as well as the control of the epidemic in many localities without continuous community spread. These patterns are more consistent with secondary infection in the health-care setting being much greater than in the general community, with an overall Ro > 1. The ability of relatively modest quarantine measures to decrease Ro to a value less than 1 in the general community suggests a pathogen that is not well-adapted to human hosts, and implies strong selection for transmission. Thus, stringent control measures in hospital settings-the major amplifiers of transmission-are needed to minimize the risk for pandemic spread of SARS. This work appeared after the 2002-03 SARS outbreak, but the model formulation and the stratification strategy developed in this work to address the importance of protecting health care workers in order to maintain a critical health
Modeling SARS, West Nile Virus, Pandemic Influenza and···
41
care capacity for emergency management during an outbreak of pandemic have been utilized later (see the subsection on pandemic influenza later) by a small group of the team to discuss the significance of the prophylactic use of antiviral drugs for health care workers in a potential influenza pandemic. See also the final section for some relevant comments. As is clearly illustrated by the management of the 2002-03 SARS outbreak, the isolation and treatment of symptomatic individuals, coupled with the quarantining of individuals that have a high risk of having been infected, constitute two commonly used epidemic control measures. Although isolation is probably always a desirable public health measure, quarantine is more controversial. Mass quarantine can inflict significant social, psychological, and economic costs without resulting in the detection of many infected individuals. In the paper [3], we used probabilistic models to determine the conditions under which quarantine is expected to be useful. The results demonstrate that the number of infections averted (per initially infected individual) through the use of quarantine is expected to be very low provided that isolation is effective, but it increases abruptly and at an accelerating rate as the effectiveness of isolation diminishes. When isolation is ineffective, the use of quarantine will be most beneficial when there is significant asymptomatic transmission and if the asymptomatic period is neither very long nor very short. This paper gives a list of diseases for which quarantine can be effective, and provides explicit conditions under which quarantine should not be exercised. In the article [4], Day noted that preemptive quarantine through contact-tracing effectively controls emerging infectious diseases, but occasionally this quarantine fails and infected persons are released. The probability of quarantine failure is typically estimated from diseasespecific data, and this article derived a simple, exact estimate of the failure rate, that does not depend on disease-specific parameters. During flu-season, respiratory infections can cause non-specific influenza like illnesses (ILls) in up to one-half of the general population. If a future SARS outbreak were to coincide with flu season, it would become exceptionally difficult to rapidly and accurately distinguish SARS from other ILls, given the non-specific clinical presentation of SARS and the current lack of a widely available, rapid diagnostic test. In the study [5], we constructed a deterministic compartmental model to examine the potential impact of preemptive mass influenza vaccination on SARS containment during a hypothetical SARS outbreak coinciding with peak flu season. Our model was developed based upon the events of the 2003 SARS outbreak in Toronto. The relationship of different vaccination rates for influenza and the corresponding required quarantine rates for individuals who are exposed to SARS was analyzed and simulated un-
42
Fred Brauer, Jianhong Wu
der different assumptions. The study revealed that a campaign of mass influenza vaccination prior to the onset of flu season could aid the containment of a future SARS outbreak by decreasing the total number of persons with ILls presenting to the health-care system, and consequently decreasing nosocomial transmission of SARS in persons under investigation for the disease.
2.2
Pandemic influenza
Since 2003, the team has been developing models and model-based analysis for the spread of influenza, in the hope that our research can have some impacts on the public health policy for the preparedness of a potential pandemic influenza. Much of the team's efforts started from the workshop "Mathematical Modeling, Implications for Pandemic Influenza Preparedness" sponsored by the Public Health Agency of Canada, the British Columbia Centre for Disease Control, MITACS and the Pacific Institute of Mathematical Sciences (PIMS), held in Vancouver, March 30-31, 2005. The key objective of the workshop was to explore the application of modeling techniques to pandemic influenza planning and the workshop provided a forum for the mutual exchange of information between the influenza, public health and modeling areas. The potential for modeling to assist the Canadian Pandemic Influenza Committee in designing and prioritizing pandemic influenza preparedness and response strategies was discussed, and the need and interest were also explored in an interdisciplinary working group comprised of expertise in influenza biology, public health and epidemiology, epidemic models, and operations research. It seemed to some members of our team that some of the non-modelers in the Vancouver workshop had the attitude that model means a network black box, and our influenza work described below grew out of our reaction that it should be possible to get analogous results from simpler deterministic models. The dialogue started in this workshop continued over the last few years, including the follow-up workshops in Toronto (March 2006), Ottawa (January 2007) and Montreal (March 2007), and a workshop on spatial spread of pandemic is planned to be held in Vancouver early 2008. Some of the key issues identified through these meetings were addressed in our team's work published in peer-reviewed journals and preliminary versions of these studies were made available to PHAC and other relevant agencies. More specifically, our work on the role of prophylaxis of health care workers in order to reduce the disease mortality of the general population was based on a contract from Ontario Ministry of Health and Long-Term Care, and the work on drug resistance was delivered as a report of a short-term contract from PHAC. The Ottawa
Modeling SARS, West Nile Virus, Pandemic Influenza and . . .
43
workshop proceedings have been published in [6]. Influenza pandemics are caused by introduction into human population of a novel, virulent influenza A virus strain, likely generated by a genetic shift, against which a large segment of population lacks immunity. The pandemic viral strains typically cause heightened morbidity and mortality, and tend to drive previously circulating influenza virus strains to extinction. In order to prepare for future outbreaks of diseases for which model parameters are unknown, we suggest that simple models which address key features with relatively few parameters are an appropriate starting point. This was illustrated in the work by a subgroup of the team [7]. The model framework was then used by an extended group involving our collaborators in PHAC, University Health Network, and National Research Council (NRC) of Canada to address a few issues of critical importance to provincial and federal governments' pandemic plans. Stochastic simulations of network models have become the standard approach to studying epidemics. We believe that one should use simple models if there is little reliable data and use more complicated models only when there is more data. We showed in [7] that many of the predictions of these models can also be obtained from simple classical deterministic compartmental models. We suggest that simple models may be a better way to plan for a threatening pandemic with location and parameters as yet unknown, reserving more detailed network models for disease outbreaks already underway in localities where the social networks are well identified. We formulated compartmental models to describe outbreaks of influenza and attempt to manage a disease outbreak by vaccination or antiviral treatment. The models give an important prediction that may not have been noticed in other models, namely that the number of doses of antiviral treatment required is extremely sensitive to the number of initial infectives. This suggests that the actual number of doses needed cannot be estimated with any degree of reliability. The model is applicable to pre-epidemic vaccination, such as annual vaccination programs in anticipation of an ordinary influenza outbreak with limited drift, and as a combination of treatment both before and during an epidemic. A more general theory, for the final size of an epidemic, arising from [7] was developed in [8], and this general theory was later used in [9] to study in details issues related to vaccination and antiviral treatment. The use of antiviral drugs has been recognized as the primary public health strategy for mitigating the severity of a new influenza pandemic strain. However, the success of this strategy requires the prompt onset of therapy within 48 hours of the appearance of clinical symptoms. We showed in [10] that this requirement may be captured by a compartmental model that monitors the density of infected individuals in terms of the
Fred Brauer, Jianhong Wu
44
time elapsed since the onset of symptoms. We showed that such a model can be expressed by a system of delay differential equations with both discrete and distributed delays. The model was analyzed to derive the criterion for disease control based on two critical factors: (i) the profile of treatment rate; and (ii) the level of treatment as a function of time lag in commencing therapy. Simulations were also performed to illustrate the feasible region of disease control. Our findings show that, due to uncertainty in the attack rate of a pandemic strain, initiating therapy immediately upon diagnosis can significantly increase the likelihood of disease control and substantially reduce the required community-level of treatment. This suggests that reliable diagnostic methods for influenza cases should be rapidly implemented within an antiviral treatment strategy. The strategy for use of antiviral drugs is still under debate. We developed in [11] a compartmental model based on the study of the non-socomial transmission of SARS and the simple model for pandemic influenza, and we used this model to evaluate the impact of prophylaxis of healthcare workers (HCWs) through a mathematical model that considers attack rates in a range of 25% rv 35% in the general population and 25% rv 50% among HCWs. Our simulations and uncertainty analysis using the demographics of the province of Ontario, Canada show that increasing prophylaxis coverage of HCWs has little impact on reducing the reproduction number of disease transmission and may not prevent the occurrence of an outbreak if expected. However, it does enable a high level of treatment, which substantially reduces morbidity and mortality in the population as a whole. Therefore, we suggest that prophylaxis of HCWs should be considered an important part of public health efforts for minimizing influenza pandemic burden and its socio-economic disruption. This suggestion may have some political implication; see the final section for a brief discussion. A critical limitation to the use of these drugs is the evolution of highly transmissible drug-resistant viral mutants. In [12], we developed a mathematical model to evaluate the potential impact of an antiviral treatment strategy on the emergence of drug-resistance and containment of a pandemic. The results show that elimination of the wild-type strain depends crucially on both the early onset of treatment in indexed cases and the population level of treatment. Given the likely delay of 0.5 rv 1 day in seeking healthcare and therefore initiating therapy, the findings indicate that a single strategy of antiviral treatment will be unsuccessful at controlling the spread of disease if the reproduction number of the wild-type strain RO' exceeds 1.4. We demonstrated the possible occurrence of a self-sustaining epidemic of resistant strain, in terms of its transmission fitness relative to the wild-type, and the reproduction number Considering reproduction numbers estimated for the past three
Ro.
Modeling SARS, West Nile Virus, Pandemic Influenza and···
45
pandemics, the findings suggest that an uncontrollable pandemic is likely to occur if resistant viruses with a relative transmission fitness above 0.4 emerge. While an antiviral strategy is crucial for containing a pandemic, its effectiveness depends critically on timely and strategic use of drugs. While resistant strains may initially emerge with compromised viral fitness, mutations that largely compensate for this impaired fitness can arise. Understanding the extent to which these mutations affect the spread of disease in the population can have important implications for developing pandemic plans. By employing a deterministic mathematical model, we investigated in the work [13] possible scenarios of the emergence of population-wide resistance in the presence of antiviral drugs. The results show that if treatment level (the fraction of clinical infections receiving treatment) is maintained constant during the course of the outbreak, there is an optimal level that minimizes the final size of the pandemic. However, aggressive treatment above the optimal level can substantially promote the spread of highly transmissible resistant mutants and increase the total number of infections. We demonstrated that resistant outbreaks can occur more readily when the spread of disease is further delayed by applying other curtailing measures, even if treatment levels are kept modest. However, by changing treatment levels over the course of the pandemic, it is possible to reduce the final size of the pandemic below the minimum achieved at the optimal constant level. This reduction can occur with low treatment levels during the early stages of the pandemic, followed by a sharp increase in drug-use before the virus becomes widely spread. Our findings suggest that an adaptive antiviral strategy with conservative initial treatment levels, followed by a timely increase in the scale of drug-use can minimize the final size of a pandemic, while preventing large outbreaks of resistant infections. These findings may not necessarily agree with intuitions from non-modelers, and thus further critical examination of the models become necessary.
2.3
West Nile Virus
We started the work on modeling West Nile virus spread almost immediately after the establishment of the MITACS team, in close collaboration with the Foodborne, Waterborne and Zoonotic Infections Division of PHAC, with assistance and support from the Vector-Borne Disease Unit of the Infectious Diseases Branch at the Ontario Ministry of Health and Long Term Care. The early results were presented in the workshop "Mathematical Modeling and Prediction of Infectious Disease in Populations" associated with the symposium "Enhancing Information and Methods for Health System Planning and Research" sponsored by a number of Ontario health and health informatics research organizations including OHI
46
Fred Brauer, Jianhong Wu
UP, ICES, CIHF and OMHLTC, in January of 2004. Our research foci have been carefully guided by the central goal of incorporating the predictive power of mathematical modeling into a national surveillance system, and this has been achieved via close consultation with PHAC. Progress has been reported to PHAC in the formats of contract reports [14, 15], and multiple symposia have been organized to discuss the progress and to make plans for the future. An example of such an event is the MITACSjPHAC Mini-Symposium on West Nile Virus (http://www.liam.yorku.ca/research/MADI/link_events~ecember2007.htm ) that was held at York University on December 6, 2007. The MITACS team and the Foodborne, Waterborne and Zoonotic Infections Division of PHAC coordinated a strategic planning meeting June 3-5, 2007, to prepare for the 2008 International Forum on Climate Changes and Waterborne and Vector-borne Diseases to be held in Nanjing, China. Since its incursion into North America in 1999, West Nile virus (WNV) has spread rapidly across the continent resulting in numerous human infections and deaths. In the establishment phase of this disease, owing to the absence of an effective diagnostic test and therapeutic treatment against WNV, public health officials have focussed on the use of preventive measures in an attempt to halt the spread of WNV in humans. In [16), we used mathematical modeling and analysis to assess two main anti-WNV preventive strategies, namely: mosquito reduction strategies and personal protection. We proposed a single-season ordinary differential equation model for the transmission dynamics of WNV in a mosquito-bird-human community, with birds as reservoir hosts and culicine mosquitoes as vectors. The model was shown to exhibit two equilibria; namely the disease-free equilibrium and a unique endemic equilibrium. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number, as a threshold quantity R o, which depends solely on parameters associated with the mosquito-bird cycle, is less than unity. The public health implication of this is that WNV can be eradicated from the mosquito-bird cycle (and, consequently, from the human population) if the adopted mosquito reduction strategy (or strategies) can make Ro < 1. On the other hand, it was shown, using a novel and robust technique that is based on the theory of monotone dynamical systems coupled with a regular perturbation argument and a Liapunov function, that if Ro > 1, then the unique endemic equilibrium is globally stable for small WNV-induced avian mortality. Thus, in this case, WNV persists in the mosquito-bird population. Our recent study [17] shows that backward bifurcations are possible, and thus the system may exhibit coexistence of two stable equilibria and whether there is a disease outbreak in a particular year depends on the condition of the mosquito and bird infection at the beginning of the year.
Modeling SARS, West Nile Virus, Pandemic Influenza and···
47
Motivated by the purpose of evaluating the relative effectiveness of mosquito control involving larvicides and insecticide sprays, the work [18] derived a very general mathematical model to assess the effectiveness of culling as a tool to eradicate vector-borne diseases. The model, focused on the culling strategies determined by the stages during the development of the vector, becomes either a system of autonomous delay differential equations with impulses (in the case where the adult vector is subject to culling) or a system of non-autonomous delay differential equations where the time-varying coefficients are determined by the culling times and rates (in the case where only the immature vector is subject to cUlling). Sufficient conditions were derived to ensure eradication of the disease, and simulations were provided to compare the effectiveness of larvicides and insecticide sprays for the control of West Nile virus. We showed that eradication of vector-borne diseases is possible by culling the vector at either the immature or the mature phase, even though the size of the vector is oscillating and above a certain level. This work also indicates that the timing and frequency of culling is important, inappropriate choice may be counterproductive. Further extensions of the model to consider a combination of larvicides and insecticide spray are considered in [19]. The spatial spread patterns and speed of transmission are obviously of paramount importance in terms of prevention and control. In [20], a model was developed to understand the jump/discontinuous spatial spread patterns in the establishment phase of WNV, as shown in the 2000-03 Health Canada map of dead birds submitted for WNV diagnosis by health region. This discontinuous spatial spread seems to be the consequence of the combination of the local interaction and spatial diffusion of birds and mosquitoes and long-range dispersal of birds, and this motivated the use of patchy models instead of the reaction-diffusion model. The model was also used to see how the interaction of the ecology of birds and mosquitoes, the epidemiology of bird-mosquito cycles, and the diffusion and immigration patterns of birds affect the long-term and transient transmission of the diseases within the whole region consisting of multiple patches. We did so by calculating the basic reproduction number of the region as a function of the basic reproduction number of each patch, the spatial dispersal rates and patterns of birds, and the spatial scale of the birds' flying range in comparison with the mosquitoes' flying range. The patchy model developed in [20] is based on the spatially homogeneous single season ordinary differential equations model in [16] for the local interaction of birds and mosquitoes within a patch, and linear dispersal among patches was used. We then used the average distance a female mosquito can traverse during its lifetime as a measure for the partition of the region under consideration, and hence the number of patches that a bird can fly during the life span of a female mosquito becomes a
48
Fred Brauer, Jianhong Wu
natural and important parameter in the modeling and for the analysis and simulations. We also calculated the reproduction number Ro and studied how this number is affected by the direction-selective dispersal pattern of birds. Our analysis and the simulations illustrated that this direction-selective dispersal of birds decreases the reproduction number and slows the spatial spread of WNV. Our simulations seem to indicate that the jump spatial spread of WNV arises if the birds' long-range dispersal dominates the nearest neighborhood interaction and diffusion of mosquitoes and birds. This work focused on the one-dimensional patch model, which can only be regarded as a theoretical approximation of the West Nile virus landscape in Canada by connecting Ontario with British Columbia with a straight line and ignoring the transmission heterogeneities along other directions. Better understanding of the West Nile virus spread in the Canadian medical landscape can be achieved only by extending this work to a two-dimensional model and by incorporating more spatiotemporal heterogeneities. This extension is currently being addressed in collaboration with geosimulation experts in order to provide real-time detailed simulation for future WNV spread in Ontario and Quebec. In many vector-borne diseases such as WNV it is the mosquito that carries the virus, but ticks and fleas can also be responsible. The diseases can be spread to humans, birds, and other animals. Much has been done in terms of modeling and analysis of the transmission dynamics and spatial spread of vector borne diseases. However, one important biological aspect of the hosts-the stage structure-seems to have received little attention, although structured population models have been intensively studied in the context of population dynamics and spatial ecology, and the interaction of stage-structure with spatial dispersal has been receiving considerable attention in association with the theoretical development of the so-called reaction-diffusion equations with nonlocal delayed feedback. The developmental stages of hosts have a profound impact on the transmission dynamics of vector borne diseases. In the case of West Nile virus the transmission cycle involves both mosquitoes and birds, the crow species being particularly important. Nestling crows are crows that have hatched but are helpless and stay in the nest, receiving more-or-less continuous care from the mother for up to two weeks and less continuous care thereafter. Fledgling crows are old enough to have left the nest (they leave it after about five weeks), but they cannot fly very well. After three or four months these fledglings will be old enough to obtain all of their food by themselves. As these facts demonstrate, the maturation stages of adult birds, fledglings, and nestlings are all very different from a biological and an epidemiological perspective, and a realistic model needs to take these different stages into account. For example, in comparison with grown birds, the nestlings and fledglings
Modeling SARS, West Nile Virus, Pandemic Influenza and···
49
have much higher disease-induced death rate, much poorer ability to avoid being bitten by mosquitoes, and much less spatial mobility. In [21], we derived from a structured population model a system of delay differential equations describing the interaction of five subpopulations, namely susceptible and infected adult and juvenile reservoirs and infected adult vectors, for a vector borne disease with particular reference to West Nile virus, and we also incorporated spatial movements by considering the analogue reaction-diffusion systems with nonlocal delayed terms. Specific conditions for the disease eradication and sharp conditions for the local stability of the disease-free equilibrium are obtained using comparison techniques coupled with the spectral theory of monotone linear semiflows. A formal calculation of the minimal wave speed for the traveling waves was given and compared with data in the literature relating to the observed speed of spread of West Nile virus across North America. Further details about the spatial spread of diseases involving animal hosts can be found in the survey articles [22, 23, 24].
3
Other progress and networking activities
The members had been working on mathematical epidemiology very actively prior to the establishment of the MITACS team, and thus the team could make rapid progress in its effort to modeling SARS outbreak and assessing various outbreak management strategies. It was therefore timely and natural that the team organized the MITACS-PIMS-Health Canada Meeting on SARS in The Banff International Research Station (BIRS) in September of 2003 (http://www.birs.ca/workshops/2003/ 03w2315/). BIRS has been playing such an important role to facilitate the team's research and outreach activities that it deserves a special note of thanks. This joint Canada-US-Mexico initiative "provides an environment for creative interaction and the exchange of ideas, knowledge, and methods within the Mathematical Sciences, and with related sciences and industry. BIRS is located on the site of the world-renowned Banff Centre in Alberta. It has its own building (Corbett Hall) and facilities which allow mathematical scientists a secluded environment, complete with accommodation and board, and the necessary facilities, for uninterrupted research activities in a variety of formats, all in a magnificent mountain setting." (http://www.birs.ca/). In subsequent years, the MITACS team has been taking full advantage of this unique facility to launch and carry out its "uninterrupted research activities" in collaboration with a number of major organizations, as described in other parts of this article. The MITACS-PIMS-Health Canada meeting consisted of invited pre-
50
Fred Brauer, Jianhong Wu
sentations and round-table discussions in a broad range of areas. Issues addressed include • Mechanism of viral pathogenesis, development of resistance and therapeutics - their importance in management and control of viral diseases and the role of mathematical approach in understanding these issues; • What do we know about SARS now, in case of a recurrence? model contact rates influenced by events and distinguish between fitting data and underlying rules; • What can we say about disease outbreaks in general for other possible diseases - analysis of early and incomplete data, constant inflows of infectives, inclusion of health care workers in the model, and modeling of quarantine and isolation; • Estimation of incubation distributions during infectious disease outbreaks; • Possible implications of asymptomatic SARS infections - including asmptomatic non-infectious; asymptomatic infectious and a mixture of the two; • Key events via back-calculation, heterogeneity and super-spreading; • How qualitative analysis helps policymakers, legal issues; • How the SARS epidemic has influenced people's longer term research goals, if it has; • Stochastic vs. deterministic modeling; • How do we sustain the collaborative research considering the geographical distances? These discussions have had substantial influence in directing the team's work on SARS modeling and analysis, and its subsequent studies of other diseases and interactions with public health policy decision makers. It was clear from the team's progress over the last few years that purpose of this workshop - "to bring together international leaders and active researchers working in the areas related to the modeling, simulations and analysis of the transmission dynamics of SARS and other infectious diseases, to further the fruitful interplay among mathematical, statistical, epidemiological sciences and operations research, in order to speed up the process of finding effective tests and prevention and control measures" , was fully achieved. Incidentally, this meeting coincided with a meeting of the provincial/territorial and federal ministers of health in the Canadian east cost that, to our knowledge, was organized to discuss how to increase the country's capacity for managing future emerging infectious diseases. For this reason, our meeting received unprecedented media attention and we did not miss this opportunity to call for the public's attention and support to the basic research of infectious diseases and the importance to
Modeling SARS, West Nile Virus, Pandemic Influenza and···
51
build the national capacity for rigorous analyses and defensible decisionmaking in the domain of public health. The team's work has since received considerable media attention (see for more details at the webpage http://www.liam.yorku.ca/research/MADI/) . We are very pleased to witness the rapid establishment of the Public Health Agency of Canada, with which we have been enjoying fruitful collaboration although improving such a collaboration remains as a rewarding but challenging task ahead. Immediately after the creation of the MITACS team in April of 2003, the Canadian SARS Research Consortium (CSRC) was established in June of 2003, under the leadership of the Canadian Institutes for Health Research (CIHR) and its Institute of Infection and Immunity. MITACS was a member of the CRSC Management Group. We were invited to attend the two meetings coordinated by the Consortium and to give invited presentations on behalf of the team. Our team's progress was also included in the Summary of the SARS Research Performed By MITACS, as part of an appendix in the Canadian SARS Research Consortium Report (http://www.cihr-irsc.gc.ca/e/27342.html). In the past several years, we have organized various workshops in coordination with our partner organizations, notably Public Health Agency of Canada. Through close contact and communication, our partner organizations first identified key issues that are of critical importance for public health, and then we discussed what can and should be done through mathematical modeling. We identified expertise of the team, and then brought the team members together with carefully selected international authorities for brainstorming. Past workshops usually ended with a joint recommendation to both PHAC and MITACS that, in turn, leads to future joint projects. This had significant impact on the team's focus and work on pandemic influenza and WNV. Also, it was agreed that the 2007 PHAC-MITACS symposium on Modeling Sexually Transmitted and Blood-Borne Infections (http://www.birs.ca/birspages.php?task =displayevent&eventid=07w2156) should be expanded to a series of annual events. We are discussing with relevant agencies the accommodation of such an annual meeting.
4 4.1
Summer school/intensive course & CDM Summer school/intensive course, seminar series
The team fully realizes that whether its interdisciplinary research can be sustained relies heavily on the training of a highly qualified younger generation, and therefore considerable efforts have been devoted to the development of a summer school series. This started with the successful
52
Fred Brauer, Jianhong Wu
2004 Summer Program on Modeling Infectious Diseases at BIRS sponsored jointly by MITACS-PIMS-MSRI (http://www.birs.ca/birspages. php? task=displayevent&evenUd=04ss101) , followed by the 2006 MITACS-FIELDS-PIMS Summer School on Mathematical Modeling of Infectious Diseases (http://www.liam.yorku.ca/sc06/). Two future schools are already being planned: the 2008 Summer School on Mathematical Modeling ofInfectious Diseases at the University of Alberta (http://www. math. ualberta.ca/ rvirl/ summer ..school/mpssmjndex.html), and the 2009 Summer School on The Mathematics of Invasions in Ecology and Epidemiology in BIRS. These Summer Schools are designed to emphasize the modeling process and the interaction between epidemiology and mathematics, in order to provide a language and a framework for the students to think and formulate infectious diseases problems mathematically; and to show how particular modeling approaches work for issues of critical importance to the public health, and how these approaches are complementary to each other. In parallel, the team is also developing an intensive short course targeted at public health people, with the goal of giving public health people enough understanding of the modeling process to consult and communicate with modelers when they encounter something that lends itself to modeling. The team was invited by the US Center for Disease Control and Prevention to deliver such a short course in its head office at Atlanta in August of 2007. The idea of the Summer School series came out from the discussions of the 2003 MITACS-PIMS Health Canada Meeting on SARS that called for actions to further the fruitful interplay among mathematical, statistical, and epidemiological sciences, and to provide effective training for graduate students and junior researchers in the collaborative research in infectious diseases based on mathematical modeling and qualitative analysis. The 2004 program consisted of a Summer School (for graduate students and beginning postdoctoral fellows), followed by a Research Conference. Admission to the 2004 Summer School was quite competitive since the maximum capacity of the BIRS lecture room was 43. Although several students came from the medical sciences (Health Canada and St. Michael Hospital, for example), most were from graduate programs in mathematical and statistical sciences since the required level of mathematical and statistical background was high. It was suggested during the 2004 summer course that future courses should also be targeted at students and research scientists in the medical community, and this suggestion was adopted for the 2006 School. Group projects were an important part of the 2004 Summer School (and continued to be so for the 2006 School). Students were divided into eight teams working on one of five projects: HIV / AIDS, SARS, Cholera,
Modeling SARS, West Nile Virus, Pandemic Influenza and···
53
TB, Malaria. Since students ranged from recent PhD.'s to recent BA's, an effort was made to have each team contain a mixture of experienced and less experienced students and a mixture of students from mathematical, statistical and medical backgrounds. Instructors made themselves available in the afternoons for advice and help on the projects. The students blended well, and everyone seemed to have benefited significantly from their participation. The group presentations were extremely impressive, and all teams managed very well (within five days) to put together their proposed models, some qualitative analysis, computer simulations, epidemiological background and applications. Two of the teams were later invited to give their presentations in the Research Conference after the school, and one student project team published their work [25]. The 2006 School differed from the 2004 Program in several respects: the 2006 School had over 70 students with a much broader range of backgrounds to promote the interdisciplinary collaboration; the course consisted of not only the regular lectures introducing the basic concepts and techniques but also public lectures delivered by active leading researchers about the critical role of mathematics in public health policy formation and implementation, as well as some carefully selected lectures from the MITACS annual meeting (this summer school was attached to the combined MITACS and CAIMS(Canadian Industrial and Applied Mathematical Society)-annual meeting), bringing the students to the frontier of research in the subject area. We expected to admit 50 students for the 2006 School, but the response was so overwhelming and there were so many qualified applicants that we decided to admit 72 students, among whom were 11 epidemiologists and research scientists, 27 students from graduate programs in mathematics and statistics and the reminder from life sciences (biology, public health, epidemiology). Many scientists have been involved in the organization of these summer schools, these include Fred Brauer (University of British Columbia), Kamran Khan (Toronto/St. Michael Hospital), Mark Lewis and Michael Li (University of Alberta), Sabrina Plitt (Public Health Agency of Canada), Babak Pourbohloul (British Columbia CDC), Pauline ven den Driessche (University of Victoria), James Watmough (University of New Brunswick), Jianhong Wu (York University), Ping Yan (Public Health Agency of Canada), and Huaiping Zhu(York Uniersity). Julien Arino (University of Manitoba) and Lin Wang (University of New brunswick) acted as program assistants while they held their respective postdoctoral fellowships at McMaster University and University of British Columbia, and they are now faculty members. Lectures at these Schools were delivered by Linda Allen (Texas Tech), Chris Bauch (University of Guelph), Fred Brauer (University of British Columbia), Carlos Castillo-Chavez (Arizona State University), Troy Day (Queen's University), David Earn (McMaster University), John Glasser
54
Fred Brauer, Jianhong Wu
(CDC), Kamran Khan (St. Michael's Hospital), Jia Li (University of Alabama-Huntsville), Jha Prabhat (University of Toronto ), Robert Smith (University of Ottawa), Beni Sahai (Cadham Provincial Laboratory in Winnipeg), Pauline van den Driessche (University of Victoria), James Watmough (University of New Brunswick), Jianhong Wu (York University), Ping Yan (Public Health Agency of Canada) and Huaiping Zhu (York University). The focus of the 2004 and 2006 Schools was on deterministic models, but there were a substantial number of lectures to demonstrate how and when stochastic and network models should be used, how parameters in the deterministic models are identified and estimated using statistical methods; and how simple deterministic models should be further fine tuned in order to deal with disease transmission heterogeneity and spatio-temporal spread patterns. The lecture notes of these schools will be published [26]. The 2006 Summer School started with a short introduction to the background material in mathematics and epidemiology. This was a great challenge to the organizers and instructors due to the background diversity of students. These introductory sessions were well received by students who also suggested there were many places we could improve: providing lecture notes much earlier, having more sessions targeted to different groups of students with varying maturity in mathematics and epidemiology. Fred Brauer has prepared some lecture notes on calculus, matrices, and ordinary differential equations [27] for these introductory sessions. These notes, along with a couple of selected chapters from the book [28], provide the necessary backgrounds in calculus, matrices, statistics and probability. In addition to the summer schools, the team has also been coordinating with its partners various meetings, workshops, symposia and special sessions associated with annual events of major mathematical and public health organizations. Encouraging and supporting participation of involved students and postdoctoral fellows to these activities has been the top priority. Current effort includes a proposal to CAlMS for a minisymposium attached to the 2008 Canada-France Congress. The minisymposium is entitled "Models for Transmission of Communicable Diseases" , and a strong team of young researchers has been provisionally assembled. These speakers come from many different institutions (Universities and Research Institutions), and several are (former or current) students or PDFs of our project team. Finally, we started a seminar series in September of 2004, that remains active, with the most recent session held at York University on October 5, 2007 (Catherine Beauchemin, Ryerson University: Influenza Dynamics In-host; Beni Sahai, Cadham Provincial Laboratory: The Biology of Influenza Infection; Robert Smith, The University of Ot-
Modeling SARS, West Nile Virus, Pandemic Influenza and···
55
tawa: Perspectives on the Basic Reproductive Ratio). Normally, there are two speakers in each seminar (one emphasizing modeling and the other addressing epidemiology and public health concerns). Some seminars are jointly organized with other organizations to encourage multiinstitutional cooperation. For example, a seminar on mathematical immunology will be held in March of 2008, jointly sponsored by the Center for Disease Modeling and the Fields Institute Center for Mathematical Medicine. Maintaining such a series is quite useful, as it provides a platform for the team to interact with other leading researchers in focused areas.
4.2
International collaboration
The team has been coordinating with other leading scientists a few highprofile international events including the Workshop on Mathematics for Global Public Health (Tempe, USA, March 2007)(http://mtbi.asu.edu /workshop07/), the Canada-China Joint Program (a summer school followed by a workshop, Xian, China, May 2006) (http://www.liam.yorku.ca /research/MADI/finaLschedule-wu.pdf), and the Workshop on Mathematical Modeling of Infectious Diseases and a meeting with Taiwan CDC in June of 2006. Members of the team continue to integrate their linkage to the MITACS team into their international collaborations. For example, we are organizing an Africa-Canada-China session on Disease Modeling, in association with the Annual Meeting of the International Society of Mathematical Biology (The Fields Institute, summer of 2008). The 2003 SARS outbreak has raised globally the awareness of the importance of mathematical modeling and model-based simulations / analysis for the prevention and control of major outbreaks of emerging and/ or reemerging infectious diseases. As a consequence of this global awareness, the two SARS affected countries Canada and China established their national teams. In China, its National Science Foundation has recognized the importance of mathematical modeling of infectious diseases as a national strategic priority and started funding a similar "national strategic project" led by Professor Tatsien Li (Fudan University) and Professor Zhien Ma (Xian Jiaotong University). There have been collaborative efforts between the two teams at the individual researcher level (notably the long-term friendship and collaboration with Professor Zhien Ma), and there have been some successful stories about the collaborations at the team level as well [29, 30, 31, 32, 33, 34, 35]. One specific example of initial success is the Canada-China Joint 2006 Summer Program on Infectious Diseases Modeling, consisting of a research workshop and a summer school, hosted by Xian Jiaotong University. The Canada-China collaboration on disease modeling between the two teams
Fred Brauer, Jianhong Wu
56
has reached its current status of success, largely due to an exchange program launched in 2003 when The Mathematical Centre of the Chinese Ministry of Education (MCME) sent six scientists to participate in various MITACS projects. One of these visitors was Professor Yicang Zhou from Xian Jiaotong University, who has since been playing a very active role for the Canada-China collaboration in the subject area. The establishment of two national teams with similar scientific goals and major funding and the initial success of collaboration lead very naturally to a strong desire and great feasibility for a mutually fruitful and sustainable collaboration. Therefore, it was quite natural that the Canada-China Thematic Program on Disease Modeling was included in the MITACS's International Initiative in Mathematical Modeling of Complex Systems (I2 M 2CS) which was proposed by MITACS as part of the NCE pilot International Partnership Initiative (IPI). The detailed plan for such a program was finalized in the Canada-China Meeting on Mathematical Epidemiology held in Beijing University, May of 2007. A small planning meeting was also held in Nanjing (June of 2007), in collaboration with several Chinese agencies in health, food safety and environment, to work out some details for the 2008 Canada-China Forum for Modeling the Impact of Environmental Changes on Vector-borne Diseases. In addition, to take advantage of the presence of a large number of Chinese and Canadian participants to the 4th International Conference of Mathematical Biology, we organized a Canada-China session on Mathematics for Disease Dynamics and Health Research in Wuyishan (May 29-June 1, 2007), and the success of this special session led naturally to the proposed Africa-Canada-China session aforementioned. The 2008 Canada-China Thematic Program consists of multiple workshops, joint projects on major diseases of importance to both Canada and China, and exchange of students and research scientists.
4.3
The future
We have selected three particular diseases of current interest to illustrate the challenges and excitements that the team has been experiencing in its interdisciplinary research adventure. There are certainly other diseases and issues that this team has been studying, in close collaboration with our federal/provincial government and industrial partners. For example, in the PHAC-MITACS Symposium on Modeling Sexually Transmitted and Blood-borne Diseases (Banff, August of 2007), we identified three sub-projects of critical importance to the Canadian public health. A web site is being set up in the National Microbiology Laboratory secured facility for all participants to exchange information and ideas about these three sub-projects. Each sub-project has a co-leader from the MITACS team, and part of our current efforts is influenced by this development.
Modeling SARS, West Nile Virus, Pandemic Influenza and· . .
57
Our plans include building our expertise in modeling a wide range of diseases which will enable us to develop interdisciplinary and international collaborations in the future. For example, much of what the team will learn in studying West Nile Virus will also be applicable to malaria. Although malaria is not a big problem in Canada, it is considered a major threat worldwide by the World Health Organization. It is one of the diseases of great concern in China and a target of the Canada-China joint effort, and is certainly a concern in Africa. The experience from our study of the spatial spread of West Nile Virus is certainly helpful in our current effort to understand the spatial spread of H5Nl-avian influenza with some preliminary results reported in [36]. These plans are certainly ambitious, but they are feasible because of the large and diverse group of scientists involved. The team's current research spans the spectrum from detailed models of specific diseases, to more abstract frameworks, to the development of mathematical techniques. There are a number of subprojects that will be the focus of individual researchers or research groups, but all of these form part of a larger structure. These subprojects involve the prediction and control of influenza, sexually transmitted diseases (Chlamydia, HIV / AIDS, human papillomavirus), tuberculosis, West Nile Virus and antibiotic-resistant bacteria epidemics. They address issues identified as being of critical importance requiring immediate attention, through the aforementioned workshops and other forms of consultation with our partners in medical community and public health agencies. There are also subprojects involving the development of models that are less disease-specific. Much of this modeling is still motivated by particular diseases, but the focus ranges from specific questions about certain diseases, to questions about general epidemiological processes and mathematical foundations. Specific issues being addressed include: modeling behavioral changes by both susceptibles in response to news of an epidemic and infectives through inability to continue normal activities or voluntary self-quarantine; modeling heterogeneous mixing [37, 38]; and modeling contact tracing. All of the research in these subprojects is being conducted with an eye towards developing a greater qualitative understanding of the effects of various phenomena, such as heterogeneity among individuals, explicit spatial population structure, contact networks, and stochasticity. Some on-going work also touches on issues in social sciences such as media impact on behavior changes during an outbreak of an emerging disease
[39]. Based on the progress of this team and as part of our effort to build a more permanent network, we created the MITACS Center for Disease Modeling (CDM) in the spring of 2007, with its head office at York University and with participating universities and research institutes across the country. It is planned that CDM shall have newsletters, a web dis-
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Fred Brauer, Jianhong Wu
cussion platform, a preprint series, and an annual report to document its activities and to promote interaction between modelers and their users. Much remains to be done in order to secure support from major national research institutes and funding agencies to find the best way to sustain this initiative, and to develop and enhance with our international collaborators strong international collaborative projects and support. It is hoped that CDM, with the seminar series, the newsletter and preprint series, the Summer School-Intensive Course series, multiple research projects and strong international collaboration, will grow into a center of activities in the fascinating interface of mathematical theory and public health.
5
Final remarks
We have been asked on various occasions to talk about what we have achieved using mathematical modeling, that cannot be done otherwise. We have also been asked to report the impact of our work on public health policy development, and to document the influence of our research and networking activities on the training of the next generation. We hope this paper provides partial answers, although we also hope the paper can raise a number of new questions and raise the public awareness of the importance of mathematical modeling to assist public health decision formation and implementation. We conclude with a few remarks, based on the lessons learned from the past. • Our goals working on general disease transmission models includes the purpose of influencing public health policy makers to look at the predictions of models before a crisis, and at the assessment of different control strategies before their implementation. The team can be activated quickly when a crisis calls for assistance from mathematical modelers, if and only if its scientific activities and interdisciplinary research are maintained before the crisis occurs. • The amount of nosocomial transmission of SARS in Tornto caused many health care workers to feel that they had been put at extreme risk, and this has caused an insistence that they be better protected if there is an influenza pandemic. This motivated our work on the prophylactic use of antivirals as described earlier. This also suggests that prophylactic use of antivirals may be a political necessity independent of what models say. This may also be an even greater consideration in other countries, where the preparedness policy includes protecting police and other guardians of public order.
Modeling SARS, West Nile Virus, Pandemic Influenza and· . .
59
• Our compartmental models for pandemic influenza suggest that both the number of doses of treatment and the number of disease cases are very sensitive to the number of initial infectives (if the control reproduction number is close to or less than 1). This suggests that both the number of doses needed and the number of cases to be expected can not be predicted reliably, but the relative predictions of different management strategies are significant. It also suggests, as other models have said, that beginning control measures quickly is vital. • The model predictions for dealing with the evolution of drug resistance seem to run counter to the intuition of non-mathematicians. This is an important reason for looking carefully at such models further, and for further interaction and consultation with medical experts and public health people. • One of the students at the 2004 school in Banff as well as a participant in the 2006 school was Jane Heffernan, who is now a team member and a faculty member at York University. The 2004 Summer School benefited a great deal from the assistance of Julien Arino, a postdoctoral fellow and now a faculty member at the University of Winnipeg and a team member. Lin Wang was a student at the 2004 school and a postdoctoral fellow providing a great deal of assistance at the 2006 school. He now has a faculty position at the University of New Brunswick and is a team member. Perhaps, these are indications that the Summer School series has indeed achieved something. We note also that some of the students from the medical community and public health agencies have now collaborating with the team in a wide variety of activities including the coordination of the 2008 Summer School. • We have spoken of experience gained at the schools (things that we did right) and lessons learned (things that we need to improve). The schools not only provide opportunities for the students, but also have been learning experiences for us, both in improving what we do and in thinking about different target audiences, and the future of the subject areas.
Acknowledgement This paper is based partially on some of the publications and group discussions of the (past and current) members of the MITACS team: Sten Ardal (Ontario Ministry of Health and Long-Term Carea), Julien Arino (University of Manitoba), Jacques Belair (University of Montreal), Martin Blaser (New York University Medical School), Fred Brauer (Univer-
60
Fred Brauer, Jianhong Wu
sity of British Columbia), Troy Day (Queen's University), Charmaine Dean (Simon Fraser University), David N. Fisman (Hospital for Sick Children and Ontario Public Health Laboratories Branch), Michael Gardam (University Health Network), Abba Gumel (University of Manitoba), Jane Heffernan (York University), Zachary Jacobson (Health Canada), Kamran Khan (St. Michael's Hospital), Michael Li (University of Alberta), Neal Madras (York University), Babak Pourbohloul (British Columbia CDC), Marcia Rioux (York Institute for Health Studies,York University), Shigui Ruan (University of Miami), Beni M. Sahai (Cadham Provincial Laboratory), Robert Smith (University of Ottawa), Pauline van den Driessche (University of Victoria), Ling Wang (University of New Brunswick), James Watmough (University of New Brunswick), Glenn Webb (Vanderbilt University), Jianhong Wu (York University, team leader), Huaiping Zhu (York University), and Ping Yan (Public Health Agency of Canada). We would also like to acknowledge support and assistance from Natural Sciences and Engineering Research Council of Canada(NSERC), Mathematics for Information Technology and Complex Systems(MITACS), the Canada Research Chairs Program(CRC), International Development Research Center (IDRC), Ontario Ministry of Health and Long-term Care(MOHLTC), and Public Health Agency of Canada(PHAC).
References [1] A. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, D. Gabrielson, C. Bowman, M. E. Alexander, S. Ardal, J. Wu and B. Sahai, Modeling strategies for controlling SARS outbreaks in Toronto, Hong Kong, Singapore and Beijing, Proc. Royal Soc. London (B), (2004), 271, 2223-2232. [2] G. Webb, M. Blaser, H. Zhu, S. Ardal and J. Wu, Critical role of nosocomial transmission in the Toronto SARS outbreak, Math. Biosci. Eng., 1(2004), 1-13. [3] T. Day, A. Park, N. Madras, A. Gumel and J. Wu, When is quarantine a useful control strategy for emerging infectious diseases, Amer. J. Epidemiology, 163:5 (2006), 479-485. [4] T. Day, Predicting quarantine failure rates, Emerging Infectious Diseases, 10 (2004), 487-488. [5] K. Khan, J. Wu, Q. Zeng and H.Zhu, The utility of preemptive mass influenza vaccination in controlling a SARS outbreak during flu season, Math. Biosci. Eng., in press.
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[6] J. Wu, P. Yan and C. Archibald, Modeling the evolution of drug resistance in the presence of antiviral drugs, BioMed. Public Health, accepted. [7] J. Arino, F. Brauer, P. van den Driessche, J. Watmough and J. Wu, Simple models for containment of a pandemic, J. Royal Soc. London (Interface), 3:8 (2006), 453-458. [8] J. Arino, F. Brauer, P. van den Driessche, J. Watmough & J. Wu, The basic reproduction number and a final size relation for general epidemic models, Math. Biosci. Eng. 4 (2007), 159-175. [9] J. Arino, F. Brauer, P. van den Driessche, J. Watmough & J. Wu, A model for influenza with vaccination and antiviral treatment, J. Theoretical Biology, in revision. [10] M. Alexander, G. Rost, S. Moghadas and J. Wu, A delay differential model for pandemic influenza with antiviral treatment, Bulletin of Mathematical Biology, in press. [11] M. Gardam, D. Liang, S. Moghadas, J. Wu, Q. Zeng and H. Zhu, The impact of prophylaxis of healthcare workers on influenza pandemic burden, J. Royal Soci. (Interface), 2207 Published on line, doi: 10.1098 jrsif.2006.0204. [12] M. Alexander, C. Bowman1, Z. Feng, M. Gardam, S. Moghadas, G. Rost, J. Wu, and P. Yan, Emergence of drug resistance: implications for antiviral control of pandemic influenza, Proc. Royal Soc. London (B), 274 (2007), 1675-1684. [13] S. Moghadas, C. Bowman, G. Rost and J. Wu, Population-wide emergence of antiviral resistance during pandemic influenza, PLoS, in revision. [14] R. Liu, J. Wu and H. Zhu, Birds ecology and surveillance system of West Nile virus, a report for Public Health Agency of Canada, 2006. [15] R. Liu, J. Wu and H. Zhu, A patchy model of West Nile virus: applications to the Peel Region (Ontario), a report for Publc Health Agency of Canada, 2006. [16] C. Bowman, A. Gumel, P. Van den Driessche, J. Wu and H. Zhu, A mathematical models for assessing control strategies against West Nile virus, Bulletin of Mathematical Biology, 67(2005), 1107-1133. [17] J. Jiang, Z. Qiu, J. Wu and H. Zhu, Threshold conditions for West Nile virus outbreaks, Bulletin of Mathematical Biology, in revision. [18] S. Gourley, R. Liu and J. Wu, Eradicating vector-borne diseases via age-structured culling, J. Mathematical Biology, 54:3(2007), 309335.
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[19] X. Hu, Y. Liu and J. Wu, An impulsive system for the West Nile virus eradictiona using a combination of larvicides and insecticide sprays, in progess. [20] R. Liu, J. Shuai, J. Wu, and H. Zhu, Modeling spatial spread of West Nile virus and impact of directional dispersal of birds, Math. Biosci. Eng., 3:1 (2006), 145-160. [21] S. Gourley, R. Liu and J. Wu, Some vector borne diseases with structured host populations: extinction and spatial spread, SIAM J. Appl. Math., 67(2007), 408-432. [22] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine", eds. by Y. Takeuchi, K. Sato and Y. Iwasa, Springer-Verlag, New York, 2007, pp. 97-122. [23] S. Gourley, R. Liu and J. Wu, Spatiotemporal patterns of disease spread: interaction of physiological structures, spatial movements, disease progression and human intervention, in "Structured Population Models in Biology and Epidemiolo91l' , eds. by Pierre Magal and Shigui Ruan, in press. [24] S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in progress. [25] B. Rapatski, P. Klepac, S. Dueck, M. X. Liu and L. 1. Weiss, Mathematical epidemiology of HIV / AIDS in Cuba during the period 19862000, Math. Biosci. Eng., 3(3): (2006) 545-556. [26] F. Brauer, P. van den Driessche and J. Wu, Lecture Notes in Mathematical Epidemiology, in progress. [27] F. Brauer, Some mathematical background for mathematical epidemiology, CDM Preprint, 2008. [28] S. P. Otto and T. Day, A Biologist's Guide to Mathematical Modeling, Princeton Unviersity Press, 2007. [29] F. Brauer, Y. Zhou and Z. Ma, A discrete epidemic model for SARS transmission and control in China, Math. Compo Modeling, 40(2004), 1491-1506. [30] J. Zhang, J. Lou, Z. Ma and J. Wu, A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China, Appl. Math. f3 Comp., 162 (2005), 909-924. [31] J. Li, Y. Zhou, J. Wu and Z. Ma, Complex dynamics of a simple epidemic model with a nonlinear incidence rate, Discrete and Continuous Dynamical Systems (B), 8:1(2007),161-173.
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[32] Y. Zhou, K. Khan, Z. Feng and J. Wu, Projection of Canadian tuberculosis incidence with increasing immigration trends, J. Theoretical Biology, in revision. [33] Y. Zhou, Y. Shao, Y. Ruan, J. Xu, Z. Ma, C. Mei and J. Wu, Modeling and prediction of HIV in China using transmission rates structured by infection ages, Math. Biosci. Eng., in press. [34] L. Liu, Y. Zhou and J. Wu, Global dynamics in a TB model incorporating case detection and two treatment stages, submitted. [35] J. Liu, Y. Zhou and J. Wu, A delay differential equation for disease spread via transport-related infection, Mathematical Biosciences, in revision. [36] R. Liu, V.R.S.K. Duvvuri and J. Wu, Spread pattern formation of H5N1-avian influenca and its implications for control strategies, Mathematical Modelling of Natural Phenomena, in press. [37] F. Brauer, Epidemic models with treatment and heterogeneous mixing, in progress. [38] F. Brauer and J. Watmough, Age of infection models with heterogeneous mixing, in progress. [39] R. Liu, J. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, J. Computational f3 Mathematical Methods in Medicine, 8:3(2007), 153-164.
64
Diseases in Metapopulations* Julien Arino Department of Mathematics, University of Manitoba Winnipeg, Manitoba R3T 2N2, Canada E-mail:
[email protected]
Abstract Metapopulation models consist of graphs, with systems of differential equations in each vertex. This modeling paradigm is appropriate for the description of the spatio-temporal spread of infectious diseases. In this paper, I present the setting of these models, and some of the mathematical techniques that can be used to study them. I conclude with a brief review of some models using this approach.
1
Foreword -
Notations
These lecture notes attempt to give a relatively exhaustive overview of methodological aspects of ordinary differential equations metapopulation models in the context of the spatial spread of diseases. They are based on work carried out with Pauline van den Driessche (in particular [5, 6, 7, 8]) and extensions of this work, and the work of all the authors cited. It is assumed that basic mathematical epidemiology is known. A certain number of reference works can be consulted, if such is not the case. Some of the most significative are the books of Anderson and May [3], Diekmann and Heesterbeek [21], Brauer and Castillo-Chavez (14] and Thieme [59]. Hethcote also gave a good review that focuses on vaccination aspects [30]. There are also reference works concerning specific diseases. The book of Hethcote and Yorke on gonorrhea [32] and the one of Busenberg and Cooke on vertically transmitted diseases [15] are but two examples. See also the papers in [17, 18, 27, 35, 43]. We adopt the convention that roman letters represent demographic parameters, whereas Greek letters denote disease related parameters. Notations have been adjusted, where possible, to abide to this rule. The SEIRS model, and its sub cases (SI, SIS, SEI, SEIS, SIR and SIRS, to ·Partly supported by MITACS and NSERC.
Diseases in Metapopulations
65
cite the most commonly used), will appear throughout this document, it is therefore detailed here with the parameters used in the manuscript. The flow diagram of the model is as follows:
13(N)
The SEIRS system then takes the form
+ IIR - if> - dS, E' = if> - (e + d)E, I' = eE - b + d + <5)1, S'
=
8(N)
R'
= "(I -
(II
+ d)R.
(l.la) (l.lb) (l.lc) (l.ld)
It is assumed that there is no vertical transmission of the disease, so that all birth occurs into the susceptible class, at the rate 8(N) > 0, where N = S + E + I + R. Individuals in all epidemiological classes are subject to natural death, at the per capita rate d. If the birth rate is supposed equal to the death rate, as is done frequently, then the letter d is used for both. The force of infection is denoted if>. It describes the rate of apparition of new infections. The most commonly used forms are mass action incidence if> = f3SI, and proportional (or standard) incidence
if>=f3
SI N'
where f3 is the transmission parameter. When a generic form is needed, it will be assumed that the force of infection can be written as
if>
=
f3(N)SI,
where the function f3 operates a scaling of the contacts by the population size. It is assumed throughout that f3 is a nonnegative nonincreasing function of N. Note that f3 might depend additionally on individual components of N, such as S and I, although this is not indicated for clarity of notations. Upon infection, individuals move to the exposed (or latent) phase, where they are not yet infectious. The time of sojourn in the E class is exponentially distributed with mean lie, giving the rate of movement out of the E class e. From the E class, individuals move to the infective
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Julien Arino
(I) class, where they can infect susceptible individuals. When infective, individuals are subject to additional death due to infection, at the rate 8. The average duration of infection is Ih, after which time individuals move to the recovered (R) class. In the recovered class, individuals are immune to the disease. They lose immunity after a mean time period of l/v. The epidemic parameters are assumed to be nonnegative, with limiting cases giving simpler models. For example, if a disease confers permanent immunity, then v = 0 and an SEIR model results. If a disease has a very short latent period that can be ignored, then E ---; 00 (an SIRS model); and if in addition the period of immunity is so short that it can be ignored, then v ---; 00 and an SIS model results. Because of space limitations, and of the focus that is put here on mathematical aspects, I will assume that the readers know the reasons that lead to the use of a metapopulation-type framework in models of the spread of an infectious disease. If this is not the case, I recommend reading the lengthy introduction in [5], which provides some explanations as well as references. This document is organized as follows. Section 2 details the general framework of metapopulations, and in particular, graph theoretic aspects. Disease models set in the context of metapopulations are presented in Section 3. Specific questions are addressed, and steps of a general method to study these problems are outlined using three specific models. Equipped with this knowledge, the reader should then refer to Section 4 to see what the current state of the art is on the topic.
2
Metapopulations
This section deals with metapopulation dynamics in the most general setting, i.e., when seen as large systems of differential equations coupled together within a graph. We give here the graph theoretic and dynamical systems context in which metapopulation models are formulated.
2.1
Introduction
The subject of metapopulations dynamics is relatively recent in the mathematical biology field, although it has been used in ecology for a longer time. A reference work on mathematical models is the book of Levin, Powell and Steele [39], whereas Hanski and Gilpin [28] give a more ecological account of the subject. The simplest type of metapopulation models derives from the same type of models that led to discrete cellular automatons. In this setting,
Diseases in Metapopulations
67
often referred to as patch occupancy models, a patch (generally, a domain in space) is occupied or unoccupied by individuals of a given species. Typically, one considers the evolution of the number of occupied patches in a network, where the occupancy of a given patch depends on the occupancy of neighboring or connected patches. This type of system will not be discussed here. The type of systems that will be discussed here can be defined, loosely, as follows.
Practical definition: A metapopulation model involves explicit movements of the individuals between distinct locations. Movement can correspond to an actual physical movement of individuals, but can also represent the evolution of a trait. To summarize, in the context of this document, a metapopulation is a graph with vertices (in metapopulation terminology, patches) containing of a certain number of subpopulations, linked by migration as arcs, with explicit, non trivial dynamics for the subpopulations in each patch. To construct such models, several components must be defined, that are detailed in the remainder of this section.
2.2
The connection graph
Suppose that there are p patches. The set of patches is denoted P, with p = IPI. Each patch pEP contains a certain number of species belonging to a common set S of species. We denote s = lSI the total number of species in the system. Note that at this point in our exposition, "species" is employed in a loose sense: two different epidemiological states represent two species. Each patch is a vertex in a graph 9. The edges of 9 represent the possibility for a given species to move between two patches; as a consequence, any two patches are connected by a maximum of s edges. The edges are then given an orientation (they are arcs, in graph terminology), to take into account that movement is not always symmetric. Thus, the graph is a multi-digraph 9 = (P, A), where A is the set of arcs, i.e., an ordered multiset of pairs of elements of P. Any two vertices X, YEP are connected by at most s arcs from X to Y and at most s arcs from Y to X. The formalism of graphs is helpful to characterize some of the properties of metapopulation models, and so a few further definitions are given, in which X, YEP are patches.
Direct access.
Define the binary relation RS by
Julien Arino
68
RS(X, Y) if, for species s E S, there exists an arc A E A between X and Y. In this case, we say that species s has direct access to patch Y from patch X. We write R(X, Y), and say that patch X has direct access to patch Y or that patch Y can be accessed directly from patch X, if there exists s E S such that RS(X, Y). We write R(X, Y) and say that patch X has full direct access to patch Y if RS(X, Y) for all s E S. The converse properties are also defined: species s E S has no direct access to patch Y from patch X if RS(X, Y) does not hold, which we write not RS(X, Y). Patch X has no direct access to patch Y if there is no s E S such that RS(X, Y), i.e., Vs E S, not RS(X, Y). For a given patch X, define
Px---->
=
{Y
E
P: RS(X, Y)}
and
Px---->
=
{Y E P : 3s E S such that RS(X, Y)},
the sets of patches that can be directly accessed from patch X, and
and
P---->x
=
{Y
E
P: 3s
E
S such that RS(Y, X)},
the sets of patches that have direct access to patch X. Connection matrix For a given species s E S, a connection matrix can be associated to the multi-digraph g. Choosing an ordering PI, ... ,Pp for the elements of P, the (j, i) entry of the p x p-matrix Cs is one if R S (Pi, Pj ) and zero otherwise, that is, if Pi has no direct access to Pj . Note that this gives the transpose matrix of the adjacency matrix obtained with the usual convention in graph theory that entry (i, j) be 1 if Pi has direct access to P j . For convenience, the ordering of the patches is generally assumed the same for all species. Indirect access. A given species s E S has indirect access to patch Y from patch X if, for species s E S, there exists a path from X to Y in 9 but species s does not have direct access from X to Y. In other words, there exists X I E P such that
but
Diseases in Metapopulations
69
Indirect access can be defined on longer chains, by assuming that there exists Xl"'" Xn E P, with n ~ p, such that
but
For problems involving disease propagation, the notion of speciesindependent indirect access from one patch to another is also very important. Patch X has species-independent indirect access to patch Y if there exists two species Sl and 82 in S and a patch Xl E P such that
with not R(X, Y). As for indirect access, species-independent indirect access can also be defined on longer chains. For disease models, indirect access (in all forms) is particularly relevant for animals. If space is discretized in patches, humans typically have direct access from one patch to another, although some exceptions do occur if, for example, patches cover a very small surface area or in the case of some political restrictions of travel. Migrating animals, on the other hand, will typically follow a route involving sequences of patches that are connected two-by-two. The following graph illustrates the importance of species-independent indirect access. Suppose the top graph shows the connections for species A, while the bottom graph represents connections for species B.
o
species A species B
Then, despite the absence of species-specific connection between patches 1 and 3, there exists a link between patches 1 and 3. If a disease is transmitted between species A and B, this means that the disease can move to patch 3 from patch 1. For a given species, indirect access can be read in the connection matrix Cs . Indeed, entries of C; give the paths of length exactly 2 in 9 for species s, and by induction, entries of C: give the paths of length exactly k in g.
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Julien Arino
Access is the combination of direct and indirect access. Species s E S in patch X has access to patch Y if species s has direct or indirect access to patch Y from patch X, and patch X has access to patch Y if it has direct or indirect access to patch Y from patch X. Two patches X and Yare connected if X can be accessed from Y and/or Y can be accessed from X. For a given patch X, the sets P~ --->, P~x and P ---+x of patches that species s in X has access to, X has access to, for which species s has access to X and that have access to patch X, respectively, are defined as were the related sets for direct access, but considering the more notion of access.
(left) and P-;3 (right) for an example conFigure 2.1: The sets nection graph. Patches directly connected to 3 appear in darker gray, indirectly connected patches appear in lighter gray. Patches with no access to 3 or that cannot be accessed from 3 are white.
Symmetric multi-digraph. The multi-digraph 9 is symmetric for if species s if for all X, YEP, RS (X, Y) implies RS (Y, X), that the binary relation R S is symmetric. It is fully symmetric if, for all YEP, R(X, Y) implies R(Y, X). Note that this implies that the associated connection matrices are symmetric. Movement is similar for all species if, in the multi-digraph g, the existence of an s E S such that RS(X, Y) implies that RS(X, Y) for all s E S. ""T.·rn,'lUIV connected multi-digraph. The last property we consider is connectedness. For a given species s, the strongly connected components (or strong components, for short) are such that, for all
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patches X, Y in a strong component, species s in X has access to Y. The multi-digraph is strongly connected for species s if all patches belong to the same connected component. Strong connectedness is equivalent to irreducibility of the connection matrix Cs , which is defined as follows. The matrix A is irreducible if for all i,j = 1, ... ,p, there exists k such that j > 0, where j is the (i, j)-entry in A k. A matrix that is not irreducible is reducible. A characterization of reducible matrices that is useful is the following. A matrix A is reducible if there exists a permutation matrix P such that p T AP is block triangular,
a1
a1
pTAP=
(
~~~ A~2 Anl An2
with every block Aii square and either irreducible or a 1 x 1 null matrix (an irreducible matrix A has block (1,1) equal to A).
2.3
Dynamics in the vertices
The dynamics of the system combines the dynamics in each patch resulting from the interactions of the various species that are present, with an operator describing the movements of individuals between the patches. The models that are considered in the rest of this document are time autonomous with linear movement operators, so the exposition is now specialized to this case. Let Nf be the number of individuals of species s E S in patch pEP at time t, Ns = (N;, ... , Nf)T be the vector of distribution of the individuals of a given species s among the different patches and NP = (Ni, ... , Nn T be the vector of composition of the population of a given patch p. There are several ways of describing the evolution of the populations. The most obvious is to write the evolution of each individual component ofthe system; for all s = 1, ... ,s and p = 1, ... ,p,
(2.1) with If : IRP --+ IR a function describing the dynamics within patch p of individuals of species s. This function might involve all individuals that are present in the patch, regardless of their species, hence its dependence on NP; we suppose that there are no between-patch interactions, though, so that If only involves individuals from patch p. The term 2:f=l m~iN; describes the inflow of individuals of species s into patch p, from all
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72
patches in P --->po The term - 2:~=1 mipNf is the outflow of individuals of species s towards all patches in P p--->' Note that it is assumed here that mii = 0 for all s. Another way to deal with linear movement operators is to suppose that p
mii
=-
I: mji' j=1
jf-i
which allows to write (2.1) in the form ftNf = ff(NP) + 2:;=1 m~iN~. For the clarity of the exposition, we use here the convention mii = O. Although a painstaking approach, the use of the form (2.1) is sometimes required to establish properties such as the positive invariance of the positive orthant under the flow of the system. It is also the most straightforward way to formulate models. However, because of the notational burden, vector notations are often used. The most common of these vector notations (and the only one we detail here) proceeds species by species, using a vector equation for each successive species. For all s = 1, ... ,s,
(2.2) with fP : JR.P --+ JR." and Ms a j5 x j5 matrix representing the movement terms. For a given species s, it takes the form,
-
2:~=1 m k1 m s21
(
~f2 ",",P
...
·· ·
mfil
mfp m2p
s
- L..Jk=l m k2 •..
..
.
-
mfi2
) (2.3)
2:~~1 m kp
Note that the matrix Ms combines the connection matrix deduced from the graph of patches, and a description of the intensity of the connections. The connection matrix of the graph is thus easily reconstructed from Ms by setting diagonal terms in Ms to zero, and nonzero offdiagonal elements to 1. Throughout this text, the following notations and names are used for matrices. Let A, B be two n x n-matrices, with entries (aij). Then
• A ~ 0 is a nonnegative matrix if aij ~ 0 for all i, j, and A ~ B if A-B ~ O. • A > 0 is a positive matrix if additionally, there exists i, j such that aij > 0, and A > B if A - B > O. • A
»
0 is strongly positive if
A-B» O.
aij
> 0 for all i, j, and A
»
B if
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73
The same notions and names are used for vectors. Note that this terminology is not standard. Many authors in linear algebra say that a matrix A» 0 is positive, and use no specific name for a matrix A > o. Here, it is however important to distinguish between, for example, a vector that is positive and one that is strongly positive.
2.3.1
Properties of the movement matrix
Matrix Ms is used extensively in the remainder of this document. Its main properties are summarized in the following result.
Theorem 2.1. Consider a species s E S. Then (-Ms) is a singular Mmatrix. All its eigenvalues have nonpositive real parts. 0 is an eigenvalue of M s , and one of the eigenvectors associated to the eigenvalue 0 is the vector n~ = (1, ... ,1). In the case that Ms is irreducible, then 0 is an eigenvalue with multiplicity 1, n~ is (to a multiple) the only strongly positive eigenvector associated to M s , and all other eigenvalues have negative real parts.
Proof. (-Ms) has the Z-sign pattern, that is, has nonnegative diagonal and nonpositive offdiagonal entries. Each column sum of Ms is zero, i.e., n~(-Ms) = 0 for all s E S, where the 1 x j5 vector n~ = (1, ... ,1). (The index on nT is dropped in the sequel if unambiguous.) It follows that (-Ms) is a singular M-matrix; see, e.g., [12] or [23, Theorem 5.11]. Because the column sums of Ms are all zero, Gershgorin's circle theorem [41, 61] implies that all eigenvalues ,X of Ms have nonpositive real parts. Indeed, defining j5
d= t=l, .max_Lmki' ... ,p k=l
all Gershgoring disks are contained in the disk centered at -d and with radius d. Also, from the singularity of M s , 0 is an eigenvalue. Since nT Ms = 0, it also follows that nT is a (left) eigenvector of Ms associated to the eigenvalue O. To show that the eigenvector nT is, to a multiple, the only strongly positive (left) eigenvector of M s , we proceed as follows. The matrix
Ms+dI is nonnegative, and therefore, the Perron-Frobenius theorem for nonnegative matrices ([12, Theorem 2.1.4], [61, Theorem C.2]) implies that the spectral radius
p(Ms
+ (1)
=
max{I,X1 : ,X E Sp(Ms
+ dI)},
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where Sp(A) is the spectrum of the matrix A, is an eigenvalue associated to a strongly positive eigenvector v. Another conclusion of [12, Theorem 2.1.4] is that any other eigenvalue .x E Sp(Ms + df) such that l.xl = p(Ms + df) is also simple, and that any other nonnegative eigenvector of Ms + df is a multiple of v. Using a left eigenvector, we have
vT (Ms + df)
=
p(Ms + df)v T
T
»
T
0 unique to a multiple. Since nT (Ms + af) = n Ms + Jn = anT, it follows that p(Ms + (1) = d and vT = nT is the eigenvector associated to the spectral radius d of Ms + df. Now note that the spectra of Ms and Ms + df are translated of which implies that nT is the only strongly positive (left) eigenvector of Ms, and is associated to the eigenvalue O. In the case that Ms is irreducible, Ms + df is also irreducible (the irreducibility of Ms is not affected by modifying its diagonal entries; think of the associated connection graph). The Perron-Frobenius theorem in the irreducible case can be used (see, e.g., [61, Theorem C.1] or [56, Theorem I]), implying that, additionally to the formerly given properties, the spectral radius p(Ms + df) of Ms + df is positive, is an eigenvalue of multiplicity one, and is such that for all other .x E Sp(Ms + (1),
for vT
a,
l.xl < p(Ms + df). As a consequence, since the spectra of Ms and Ms+af are translated of 0 is the dominant eigenvalue of Ms and is of multiplicity one, and all other eigenvalues of Ms have negative real parts. 0
a,
Note that for matrix Ms, the difference between the reducible and the irreducible case is not as important as it is in general, because the nature of Ms implies that the conclusions we can draw in the reducible case are stronger than they typically are. Also, in the irreducible case, a shorter proof that 0 is the spectral radius is given by using [12, Theorem 2.2.35] with the zero column sum property, but the reasoning used seemed worth explaining in detail here. Similarly, the Perron-Frobenius theorem has been formulated directly for matrices such as Ms, which are called essentially nonnegative (or essentially positive in the irreducible case), but it seemed of interest to show how to obtain these results here. Lastly, all diagonal entries of Ms + df are positive, except the entry corresponding to in Ms which is zero, the matrix Ms is primitive, from [12, Corollary 2.4.8]. (In the case that Ms has only d on the diagonal, then it suffices to consider Ms + (d + e)1, for e > 0, instead of Ms + df to obtain positive terms on the main diagonal.) Therefore, the strongest version of the Perron-Frobenius theorem could also have been used, which applies to primitive matrices. But, again, because of
a
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75
the nature of M s , the conclusions drawn are not stronger than those obtained in the irreducible case.
2.3.2
Case of an irreducible movement matrix
The following theorem describes the dynamics for one species, in the case where there is no internal patch dynamics or that this dynamics simplifies, and that the movement matrix is irreducible. It is used frequently in Section 4.
Theorem 2.2. For a given species s E S, suppose that the movement matrix Ms is irreducible, and that the within-patch dynamics simplifies, that is, limt-+oo fP(NP(t)) = O. Then the migration component of (2.2) satisfies lim Ns(t) = N; » o. t-+oo
Proof. Since limt-+oo fP(NP(t)) = 0, we can suppose that fP(NP) = O. The existence part is adapted from [4]. Remark that system (2.2) with fP(NP) = 0 is overdetermined, in the sense that the total populaT tion n Ns = LpEP NJ: is constant. To find the equilibrium value N;, the system MsNs = 0
must be solved, with Ms a singular matrix. This is achieved by considering the augmented system of p + 1 equations in p unknowns,
(£J N.. ~
(T) ,
(2.4)
where NO = L:=1 NJ:(O). All column sums of the last p rows are zero, thus the second equation (for example) can be eliminated. Now perform column operations Cr f - Cr - C1 for r = 2, ... , P on the determinant of the resulting coefficient matrix, reducing it to the p - 1 determinant det(M(1) + Td, where M(1) denotes matrix Ms with its first row and column deleted, thus
- L:~: 1 mq2 m23 M(1)
=
(
mp2
......
mp3 ...
and T1 = m1nJ-1 = [-m21,"" -m p1]T[1, ... , 1], where m1 is the vector formed from the first column of Ms by omitting the first entry.
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By [12, M 35 , p.137] since mpq ~ 0, -M(l) is a nonsingular M-matrix (it has the Z-sign pattern and n~_l (-M(l)) ~ 0 and is not the zero vector by the assumption that Ms is irreducible). Thus det(-M(l)) > 0 and so detM(l) has sign (_l)p+l. Since Tl has rank 1, it follows from the linearity of the determinant subject to rank 1 perturbations, see, e.g., [46, Corollary 4.2], that det(M(l)+Tl) = det M(l)(l + nJ-l M(l)-lml). As -M(l) is an M-matrix, (-M(l)-l) ~ 0, thus M(l)-l ~ o. But ml ~ 0, thus 1 + n~_lM(l)-lml i~ positive and so det(M(l) + Td has the sign of det M(l), namely (_l)p+l. By Cramer's Rule,
detM(l)NO Nl
=
-de-t("""M-(.,.-'l-=--')+---=T'-:-d
Similarly by deleting the (p
+ l)st
detM(p)NO Np = det(M(p) + Tp)
equation in (2.4), NO
=
0
1 + n~_l(M(p))-lmp > ,
where Tp = mpn~_l = [-ml p, ... , -mp-l,p, -mp+l,p, ... , -mpp]Tn~_l for p = 1, ... ,p. Here mp is the vector formed from the pth column of M by omitting the pth entry. Thus given a value of NO, there is a unique positive solution Np = N; for p = 1, ... ,p. We now consider the stability of N*. 0
2.3.3
Case of a reducible movement matrix
In most of Section 3, it is assumed that the movement matrices are irreducible, for each species. It is however possible to assume that the movement matrices are reducible, but this changes the approach: results such as Theorem 2.2, which could be obtained in full generality in the irreducible situation, have to be treated on a case by case basis, depending on the precise nature of the movement matrices. Rather than attempt a systematic treatment of the reducible case, which would require a rather lengthy development, I discuss here the main differences with the irreducible situation, by presenting the possible cases for 3 patches, which can easily be extended to cover the general case. Generically, with 3 patches, the reducible situation corresponds to one of the following graphs (operating a relabelling of patches if need be). There can be no graph with only one strong connected component, as it corresponds to the irreducible case. There are two graphs with only isolated strong components: graphs ~h and {h:
77
Diseases in Metapopulations
®
®
Theorem 2.2 can be applied to the study of each of the isolated strong components in {h and {h. So, in practice, this is an irreducible configuration, which corresponds to the reduced form of Ms being block diagonal, that is, direct sum of irreducible blocks. In the case where the system does not separate, there can be two or three strong components. First, in the case of 2 strong components, we have the following two graphs, 03 and 04:
with associated movement matrices
M3
=
[-r:~1 -r:~2 ~23] o
0
-m23
and
M4
=
[-r:~1 -(m17~ m32) ~]. 0
m32
0
The remaining cases, graphs 05, 06, 07, 08 and 09, have 3 strong components:
Associated to these graphs are the movement matrices
M5 =
-(m 21 + m3d 0 0] [-m21 0 0] m21 -m32 0 , M6 = m21 -m32 0 , [ m31 m32 0 0 m32 0
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78
° ° °°0] , M8
[-(m 21 + m31) m21 m31
-m31
-m32 [ m31 m32
=
°°00]° 0
and
Mg
-m 21 =
[
m21
o
°. °° 00] 0
The main distinction between the cases is not the number of strong components, but rather the number of sinks and/or isolated strong components in the decomposition into strongly connected components. If, by abuse of language, we call sink a patch that is a sink, or a strong component that is a sink in the condensed graph, or an isolated strong component, then we observe that the multiplicity of the dominant eigenvalue 0 is equal to the number of sinks in the graph. This is summarized in the following table Case
# Sinks
91 92 93 94 95 96 97 98 9g
3 2 1 1 1 1 1 2 2
Eigenvalues
0,0,0
+ m21) 0, -m23, -(m12 + m2I) 0,0, -(m12
0,A2,A3
+ m3I) 0, -m21, -m32 0, -m31, -m32 0,0, -(m21 + m3I)
0, -m32, -(m21
0,0, -m21
where A2 and A3 are two negative eigenvalues with slightly more complicated expressions than those presented in the table. Studying (2.1), it is clear that the population vanishes in a source that is reduced to one patch. Intuition indicates that this is also the case for sources not reduced to a single patch. Solving, as in the irreducible case, the augmented system
with N(O) = N 1(0) + N 2(0) + N 3(0), confirms this intuition. Cases and 92 have equilibria given by the following table. Case
N 1 (0) m12(N1(0) + N 2(0)) m12 + m21
N 2 (0)
m21(N1(0) + N 2(0)) m12 +m21
91
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79
In ~he case of 93, patch 3 is a source and the strong component {I, 2} is a smk, and the equilibrium is given by
N*1
Case
N*2
N*3
o Cases 94 to 97, all the individuals migrate to patch 3, which is the only sink in the graphs, and equilibria take the form Case
N*1
o a a a
Finally,
N*3
o a o
N 1(O) N 1(O) N 1(O) N 1(O)
a
+ N 2 (O) + N3(O) + N 2(O) + N3(O) + N 2(O) + N3(O) + N 2(O) + N3(O)
98 and 99 have two sinks, and equilibria given by Case
N*1
o
o
N*2 N 2(O)
+
N 1 (O)
N*3
ffi21Nl 0
+ ffi31 + N 2 (O)
ffi21
Note that this gives a different interpretation of the discussion in [12, pp. 38-45], where a different vocabulary is used.
3
Methodological aspects
A certain number of objectives or steps can be isolated, when studying a metapopulation disease model. They are not very different from the steps that are carried out when lower dimensional systems are studied, but the high dimensionality of the systems makes them somewhat specific. This programme typically should involve at least the following steps, to take place once a satisfactory model is formulated. 1. Establish the well-posedness of the system.
2. Study the existence of disease free equilibria. 3. Compute a reproduction number for the system, and consider the local asymptotic stability of the disease free equilibria.
The steps above provide a basic understanding of the behavior of metapopulation disease models. Additionally to these steps, other questions worth addressing are the following. 4. If the disease free equilibrium is unique, prove that it is globally asymptotically stable when Ro < 1.
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5. Step 2 above addresses the existence of an equilibrium without disease for the whole system. Of importance in the context of epidemic spread is also the possibility for the system to have mixed equilibria, with some patches disease free and others with disease. 6. The expression obtained for no is typically very complicated, so obtaining some bounds on the value of no can be helpful. This programme is illustrated in the following sections, with examples chosen from the author's work.
3.1
The models under consideration
The models used here are formulated in a very general setting. Three types of models are considered, that are SEIRS-type models of the form of system (1.1), set in a patch setting. The population in each patch is divided into compartments of susceptible, exposed (latent), infective and recovered individuals with the number in each compartment denoted by S.(t), E.(t), I.(t) and R.(t), respectively. The symbol. is used to indicate that a variable or parameter might have one or several indices. The following system summarizes their common aspects, and will be used when genericity is needed. d
dtS. = B(N.) -1>. - d.S. d
dtE. = 1>. - (c. d
d/. = c.E. -
+ v.R. + 0 s (S.),
+ d.)E. + 0 E (E.),
b. + d. + b.)I. + 0
d dt R. = ,,(.1. - (v.
+ d.)R. + 0
R
(3.1a) (3.1b)
I
(I.),
(R.).
(3.1c) (3.1d)
The operators OX (X.), for X E {S, E, I, R}, are the movement operators; they potentially involve all components of a given state X. As discussed in Section 2.3, it is assumed that they are linear. Also, unless otherwise stated, it is assumed throughout that for a given state/species combination X E {S, E, I, R}, OX induces a graph that is strongly connected for that state/species combination. Finally, note that since travel is instantaneous, there is no change of epidemiological status during travel. We consider three types of models. The first is a particular case of the second, but it is useful for illustrating the type of computations needed without overburdenning the reader with notations. The other two serve the converse purpose of showing that even though they are complex, the computations can be carried out.
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3.1.1
81
Simple SEIRS
The model here is for disease transmission in one species, but allows for movement rates to depend on disease status. Within each patch conditions are assumed to be homogeneous. The total number of individuals in patch p = 1, ... ,p is Np(t) = Sp(t) + Ep(t) + Ip(t) + Rp(t). The rates of movement of individuals between patches are assumed to depend on disease status, and individuals do not change disease status during movement. Let mgq, m:q, m~q, m:q denote the rate of movement from patch q to patch p of susceptible, exposed, infective, recovered individuals, respectively, where mgp = m:p = m~p = m:p = O. This defines the nonnegative matrices MS = [mgq] , ME = [m:q] , MI = [m~q] and MR = [m~]. The movement matrices are deduced from these matrices by setting, for X E is, E,I, R},
M
X
= M X - diag
(nTMx).
Unless otherwise indicated, it is assumed that these matrices are irreducible. The above assumptions lead to a system of 4p ordinary differential equations (ODEs) describing the disease dynamics. For p = 1, ... ,p these equations are d j5 j5 dt Sp = Bp (Np) -
q=l d
j5
dtEp =
+ L m:qEq q=l
d
-Ip = cpEp - (')'p dt d
-Rp = "(pIp - (vp dt
q=l j5
L m!Ep,
j5
+ dp + 8p)Ip + L m~qIq q=l
j5
+ dp) Rp + L m~Rq q=l
(3.2b)
q=l
j5
L m~pIp, q=l
(3.2c)
j5
L m;;pRp q=l
(3.2d)
with initial conditions Sp(O) > 0 and Ep(O), Ip(O), Rp(O) ;;:: 0 such that
L:=l {Ep(O) + Ip(O)} > 0, so that there are initially infected individuals in the system. The force of infection
(3.3) with assumptions on (3p as in Section 1. In some specific cases, proportional incidence will be considered, with
Ii
(3.4)
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3.1.2
SEIRS for multiple species
The model here is the generalization of system (3.2) to s species (where we use the term species stricto sensu, not for epidemiological states). The dynamics for species s = 1, ... , s in patch p = 1, ... ,p is given by the following system of 4sp equations,
i5
i5
+ Lm~pqSsq - Lm~qpSsp, q=l
d dt Esp =
sp - (Esp
d
i5
q=l
q=l
(3.5b)
+ d sp + 6sp )Isp
i5
i5
+ Lm;pqIsq q=l
d
= 'YspIsp -
i5
+ dsp)Esp + Lm;;'qEsq - Lm~pEsp,
d/sP = EspEsp - (rsp
dt Rsp
(3.5a)
q=l
(vsp
(3.5c)
Lm;qpIsp, q=l i5
i5
q=l
q=l
+ dsp)Rsp + Lm!qRsq - Lm~pRsp.
(3.5d)
The parameters are defined similarly to those of system (3.2), but now the first subscript denotes the species; for example, l/'Ysp is the average period of infection for species s in patch p. Each species has its own movement matrices, for example M~ for infectious individuals of species s, obtained from the nonnegative matrix Ml = [m~pq] where m~pq denotes the rate of movement of an infective individual of species s from patch q to patch p, by setting
Denoting the total population of species s in patch p by Nsp Esp + Isp + R sp , the force of infection takes the form
=
Ssp
+
(3.6) in [4], but more generally, it can be assumed to take the form s
sp
=
L f3kp(Nkp)Ssph p, k=l
(3.7)
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83
with assumptions on (3sp as in Section 1. Thus, infection in patch p for susceptibles of species s involves contacts with infectives from all species that are present in the patch. This model is considered together with nonnegative initial conditions having, for all s = 1, ... , S and all p = 1, ... , p, Ssp(O) > 0 and Esp(O), Isp(O), Rsp(O) ~ 0 such that s
j5
L L {Esp(O) + Isp(O)} > O. s=lp=l
3.1.3
SEIRS model with residency patch
The main difference in the following model compared to (3.2) and (3.5) is that it tracks the place of residence of individuals: they can move between patches, but are always identified by the patch in which they were born. In this sense, this model is more adequate to describe the short term travels of humans, rather than migrations. Let Npq(t) be the number of residents of patch i who are present in patch j at time t, with Spq(t), Epq , Ipq(t) and Rpq(t) being the number that are susceptible, exposed, infective and recovered, respectively. The nonnegative matrix = [m~qr1gives the movement rates of susceptible individuals resident in patch p from patch r to patch q. Similarly = [m:qr ], = [m~qrl and = [m:qrl give these rates for exposed, infective and recovered individuals. From these matrices, the movement matrices for residents of patch p in epidemiological state X are constructed, using
M;
M£
Mff
M:
M;
For simplicity, denote j5
N; = LNpq,
(3.8)
q=l the population of individuals born in patch pEP (called the residents of patch p), and j5
N;= LNqp,
(3.9)
q=l the number of individuals currently in patch p. Birth is assumed to occur in the residency patch at rate Bpp(N;) and natural death occurs (in all disease states) in all patches at the rate dpq . For p patches, the model takes the following form for p, q = 1, ... ,p.
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84
!
Spq =Bpq (N;)
+ vpqRpq - dpqSpq -
15
+ L m~qTSpT T=l
d dt Epq =
15
L m~TqSpq, 15
+ dpq)Epq + L m:qTEpT T=l
d
d/pq =cpqEpq - (,pq
(3.lOa)
T=l
15
L m:TqEpq ,
(3.10b)
T=l
+ dpq + bpq)Ipq
15
+ L m~q,..IPT T=l
15
L m~Tqlpq,
(3.lOc)
T=l
d dt Rpq ="(pqlpq - (vpq
15
+ dpq)Rpq + L
15
m:qTRpT -
T=l
L m!qRpq. T=l
(3.lOd) The force of infection for residents of patch p that are susceptible and currently in patch q,
L {JTq(N~)SpqITq,
(3.11)
T=l
that is, describes the contacts, in patch q, between susceptible residents of patch p who are currently in patch q, and infective residents of all patches who are currently in patch q, related to the total current population in the patch. Assumptions on {JTq are as in Section 1. If standard incidence is used as in the previous models, giving (3.12) then {JPTq denotes the rate of transmission of the disease for a contact in patch q between a susceptible from patch p and an infective from patch r that results in disease transmission (it is the product of the proportion of such adequate contacts and the average number of such contacts). Note that this model is more complicated than the previous two. It consists in 4p2 equations. Also, where (3.2) and (3.5) have, respectively, at most 4 and 48 arcs in each direction between any two given patches, (3.10) has a maximum of 4p arcs between any two given patches. (Typically, p > 8). Here, the residents of each patch can be thought of as a species in the sense of Section 2.
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3.1.4
85
Types of movement matrices
In the context of the models considered here, movement similar for all states takes the following form.
Definition 3.1. Movement in system (3.2) is similar for all states if, for all p, q = 1, ... , p, there holds that either
or S
mpq
E I R = mpq = mpq = mpq = 0.
Movement in system (3.5) is similar for all states if, for all p, q and a given species s E S, there holds that either S
E
I
R
mspqmspqmspqmspq
= 1, ... ,p
>0
or S
mspq
E I R = mspq = mspq = mspq = 0.
If this is true for all states and all species, then movement in (3.5) is similar for all states and all species. Movement in system (3.10) is similar for all states if, for all p, q, r = 1, ... ,p, there holds that either
or S
mpqr
E I R = mpqr = mpqr = mpqr = 0.
Unless otherwise specified, it is assumed throughout that movement is similar for all epidemiological states. Results shown for similar movement rates are of course valid in the particular case of identical movement. When movement is similar for all states, we drop the superscript indicating the disease state in
x -x x -x x -sx X -sx Pp-+' P p-+' P -+p' P -+p' p;-+, p p-+' P"....,p and P-+p and thus denote by -
-
-8
-8
Pp-+, Pp-+, P -+p, P -+p, P;-+, P p-+, P"....,p and P-+ p the sets of patches that can be accessed directly from p, can be accessed from p, have access to p directly, have access to p, can be accessed directly from p by species s, can be accessed from p by species s, to which species s has direct access from p, and to which species s has access from p, respectively.
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3.2
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Well-posedness
A system is well-posed if its solutions exist for all positive times, are unique, exhibit sensitive dependence on parameters and initial data, and are bounded. For systems describing populations, an additional condition of well-posedness is that solutions remain nonnegative for nonnegative initial conditions, as negative population values have no interpretable meaning. To study the well-posedness and in particular, the boundedness of solutions, it is convenient to consider the demographic component, that is, the total population in each patch or in the whole system. 3.2.1
Existence and uniqueness of solutions
With systems such as (3.1), the existence and uniqueness of solutions, as well as continuous dependence on parameters and initial data, is assured by a proper choice of birth and force of infection functions, since the other processes are described by constant or linear terms. In the cases treated later, the birth function is either constant or a linear combination of state variables. There may exist problems at the origin, if the force of infection is not defined there, but this problem is solved by mollifying the functions if need be. Therefore, it is assumed from now on that solutions exist and are unique, and depend continuously on initial data and parameters. 3.2.2
Nonnegativity and/or positivity of solutions
Take for example (3.2). To show the positive invariance of the positive 4orthant under the flow of system (3.2), it suffices to show that each of the faces of the positive orthant cannot be crossed, that is, that the vector field points inward on these faces. Assume that initially, all variables are nonnegative. Setting Sp = 0 in (3.2a) gives
R.':
d
dt Sp = Bp(Np) + vpRp
P
+L
m;'qSq ~ 0,
q=l
implying that Sp = 0 cannot be crossed from positive to negative Sp. Similarly, setting Ep = in (3.2b) gives
°
d dt Ep
P
=
IPp +
L m:qEq ~ 0, q=l
and Ep = 0 cannot be crossed. The same argument shows that neither Ip nor Rp can be crossed. Hence solutions remain nonnegative for nonnegative initial conditions.
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The result can be in fact strengthened, by noting that for positive
N p, Bp(Np) > 0, and for q E P~p, m~q > 0, implying that dSp/dt > 0, in turn implying that for positive initial conditions, Sp remains positive. The same type of reasoning can be applied to systems (3.5) and (3.10). To summarize, in all cases above, it has been assumed that initial conditions are such that S.(O) > 0 and E.(O), I.(O), R.(O) ? 0, with L.{E.(O) + I.(O)} > 0 (so that there are exposed and/or infectives at initial time). This implies that S. > 0 for all times. The following generic result can be stated. Theorem 3.2. If a given movement rate m. > 0, then the linked S. > 0 provided that S.(O) > O. 3.2.3
Boundedness of solutions
Establishing that solutions are bounded can be more difficult, and requires that the behavior of the total population in each patch or the total population in the system be studied. We make the following hypotheses on the birth functions. For system (3.5), we assume that (3.13)
with the species index s dropped in the case of (3.2), while for system (3.10), we suppose that if p = q, if pol q,
(3.14)
where for each p, one of the following two conditions holds:
HI A. > 0, d. > b•. H2 A. = 0, d. = b•. Theorem 3.3. Provided that the birth functions satisfy HI or H2, solutions to system (3.2), (3.5) and (3.10) are bounded.
Proof. Consider system (3.5). When the behavior of the demographic component is considered, the fact that movement depends on the state implies that the dynamics of the total population Nsp in patch p for species s takes the form
(3.15)
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which cannot be evaluated independently of the dynamics of (3.5). However, the total population in the system for species s, N s , satisfies d
j5
-Ns
dt
= L (Bsp(Nsp ) - dspNsp - bspIsp) p=1
+
t( p=1
L
(t m~qXSq t m~pxsp)) -
XE{S,E,I,R}
q=l
q=l
j5
=
L
(Bsp(Nsp ) - dspNsp - bspIsp)
p=l
L
+
(t (t m~qXSq t m~pxsp)) -
XE{S,E,I,R}
p=1
q=1
q=1
j5
= L (Bsp(Nsp ) - dspNsp - bspIsp) .
(3.16)
p=1 The last equality results from the fact that for each state X E {S, E, I, R}, the sums j5
j5
j5
j5
L L m~qXSq - L L m~pXsp
p=lq=l
p=lq=l
cancel. This is readily established by noticing that in these sums, each term appears exactly once with a positive sign and once with a negative sign. That solutions to (3.2) are bounded is deduced directly from the boundedness of solutions to (3.5) in the case s = l. Similarly, in the case of system (3.10), the variation of the number N; of residents in patch p is given by
:t
N;
= Bpp(N;) -
t
(dpqNpq - bpqlpq)
q=l
As in the case of (3.16) for system (3.5), the movement terms cancel when considering the total population for system (3.10), N = L: P=1 NT, which implies that N satisfies the equation P p (3.18)
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Therefore the total population for system (3.5) is nonincreasing if p
p
p=l
p=l
p
p
p=l
p=l
L Bsp(Nsp ) ~ L (dspNsp - 6spIsp) , or increasing if
L Bsp(Nsp ) > L (dspNsp -
6sp I sp) ,
and that of system (3.10) depends on the sign of
If HI holds, as was assumed in [50], then the population in patch p is bounded above by max{Apj(dp - bp),N;(O)}. If H2 holds, as was assumed in [6, 7] with dp = d, then the resident population of patch i is constant with N; = N;(O), and if this is true for all patches, then the total population in the system is constant. 0
3.3
Behavior of the demographic component
Consider system (3.2). In the case of mild diseases it may be reasonable to assume that movement rates are independent of disease status, thus M S = ME = MI = MR =: M. In this case, the behavior of the demographic component can be linked to the behavior of the underlying metapopulation model without disease. The total population Np in patch p, evolves following the equation d
dt Np
= Bp(Np) - dpNp - 6plp +
P
P
q=l
q=l
L mpqNq - L mqpNp.
(3.19)
First, consider a particular case, with assumptions made in the multispecies model of [4]. Theorem 3.4. Suppose that in system (3.2), there is no disease induced death (6 p = 0 for all p), movement is identical for all epidemiological states, that in each patch, birth compensates natural death, that is, Bp(Np) = dpNp, and that the movement matrix is irreducible. Then for every patch p = 1, ... ,p, there holds
lim Np(t) = N; > O.
t---+oo
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Proof. Under the assumptions of the theorem, the system is linear and de couples from the epidemic variables, and takes the form, for p = 1, ... ,p, d P P dt Np = mpqNq mqpNp, q=l q=l
L
L
o
the result follows from Theorem 2.2.
The same type of result holds for systems (3.5) and (3.10), that are given here without proof. Theorem 3.5. Suppose that, in system (3.5), there is no disease induced death, movement is identical for all epidemiological states and that in each patch, birth compensates natural death, that is, Bsp(Nsp ) = dspNsp . Then the movement model is given, for all s = 1, ... , s and all p = 1, ... ,p, by
(3.20)
and there holds Theorem 3.6. Suppose that, in system (3.10), there is no disease induced death, movement is identical for all epidemiological states and yields an irreducible movement matrix, and that in each patch, birth compensates natural death, that is,
t
B pq (Npr ) = {d pq r=l N pr , 0, Then for every patch p holds
if p = q, if p
=1= q.
= 1, ... ,p and subpopulation q = 1, ... ,p, there
Theorems 3.4, 3.5 and 3.6 establish that if movement is identical for all classes and there is no disease induced mortality, then the behavior of the total population is independent of the disease characteristics. In the general case, the demography is not independent of the disease, and it is not possible to characterize the behavior of the former independently of the latter. It is clear, however, that the convergence of N can still be established, provided that I converges to some value 1*. For example, for system (3.2), it is established in Section 3.6 that I ----. 0 when the basic reproduction number, no, is less than 1, and this is done with no assumption on N.
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3.4
Existence of a disease free equilibrium (DFE)
The metapopulation model is at equilibrium if the time derivatives are zero. In the case of system (3.2), patch p is at a disease free equilibrium (DFE) if Ep = Ip = 0, and the p-patch model is at a DFE if Ep = Ip = 0 for all p = 1, ... ,po System (3.5) is at a DFE if Esp = Isp = 0 for all s = 1, ... , s and all p = 1, ... ,po System (3.10) is at a DFE if Epq = Ipq = 0 for all p,q = 1, ... ,po At this point, the objective is to find the DFE for the p-patch model. In Section 3.7, the existence of mixed equilibria, with some patches at the DFE and others at an endemic equilibrium, is considered. First, it must be shown that at a DFE, R. = o. We have the following result.
Theorem 3.7. Suppose that, in system (3.2), Ep p = 1, ... ,po Then for all p = 1, ... ,p,
=
Ip
=
0 for all
lim Rp(t) = O.
t-+oo
Suppose that, in system (3.5), Esp = Isp = 0 for all p = 1, ... ,p and all species s = 1, ... ,5. Then, for all p = 1, ... ,p and all s = 1, ... ,5, there holds, lim Rsp(t) = O. t-+oo
Suppose that, in system (3.10), there holds that Epq = Ipq for all p, q = 1, ... ,p. Then, for allp,q= 1, ... ,p,
lim Rpq(t)
t-+oo
=
o.
Proof. Substituting Ip = 0 in (3.2d), there holds that at the DFE, using the vector form of the equation,
From Theorem 2.1, the matrix (_MR) is a singular M-matrix. It follows that MR - diag (ZIp + dp ) is nonsingular, and at a DFE, R = O. To show the result for (3.5), it suffices to proceed species by species, while for (3.10), proceeding resident patch by resident patch leads to the same result. 0 Thus at a DFE, system (3.2) is such that, for allp and satisfies ji
13p (Np) - dpNp +
= 1, ... ,p, Sp =
ji
L mf,qNq - L m~pNp = o. q=l
Np
q=l
(3.21 )
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S; N;,
= which Assume that (3.21) has a solution that gives the DFE is unique. This is certainly true if Bp (Np) = dpNp (i.e., birth rate equal to the death rate) and bp = 0 (i.e., no disease related death) giving a constant total population as in [4]. It is also true if Bp (Np) = bp as assumed in [49, 50]. At the DFE, system (3.5) takes the form Bsp(Nsp ) - dspSsp +
15
15
q=l
q=l
L m~pqSSq - L m~qpSsp =
0,
(3.22)
O.
(3.23)
while (3.10) takes the form 15
Bpq(N;) - dpqSpq
15
+L
m;qrSpr -
r=l
L m;rqSpq = r=l
More generally, letting d = (d 1 , ... , d15 )T, then if B (S) -=Jproblem (3.21) can be written as a fixed point problem,
(d I -
M
S
) -1
d I,
the
B(S) = S.
Since -Ms is a singular M-matrix, J I _Ms is a nonsingular M-matrix. As a consequence, (d 1- M S )-l is a nonnegative matrix that leaves the positive cone IR+. invariant. If B has the required property, then the contraction mapping principle can be used and there is a unique solution to the fixed point problem. Otherwise, provided B is a continuous mapping such that the total population is bounded, fixed point results ensure that there exist solutions to the problem, although uniqueness is not guaranteed.
3.5
Reproduction number and local stability of DFE
In this part, we assume that a DFE exists. Linear stability of the disease free equilibrium can be investigated by using the next generation matrix [21,60]. Note that, in general, depends on the demographic, disease and mobility parameters.
no
3.5.1
Simple SEIRS
To derive the basic reproduction number in the most general context, system (3.2) with a generic force of infection (3.3) is first considered. Using the notation of [60], and ordering the infected variables as E 1 , ... ,E15 , h, ... ,I15 , form the vectors
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93
representing new infections into the infected classes E I , ... , Ep,h, ... , I p , and
V=-
+ L;=l mffJEj - L;=l mfpEp + d l + (h)h + L;=l mLlj - L;=l mJI h
-(€p + dp)Ep
€IEI - hI
representing other flows within and out of the infected classes E I , ... ,Ep , h, ... ,Ip (note that V has a minus sign). The matrix of new infections F and the matrix of transfer between compartments V are then the Jacobian matrices obtained by differentiating F and V with respect to the infected variables, evaluated at the disease free equilibrium (DFE). Note that 8iP p 8 ( ) 8E = 8E (3p Np SpIp, p
p
and therefore it follows that 8iP pj8Ep = 0 at the DFE. Therefore, the matrices F and V are given in partitioned form by an dV - [ Vll 0] - -Y21 Y22
(3.24)
with
p
Fl2
. (8iP = dmg 81 I p
)
'
DFE
and
+ diag (€i + di ), V21 = diag (€i) , V22 = _MI + diag hi + di + bi).
Vll = _ME
Matrices Vll and V22 are P x P irreducible M-matrices [12] and thus have positive inverses. The next generation matrix
has spectral radius, denoted by p, given by
As shown in [60], the Jacobian matrix of the infected compartments at the DFE, which is given by F - V, has all eigenvalues with negative real
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parts if and only if p (FV-1) < 1. Note that the conditions of application of Theorem 2 in [60] are satisfied, and in particular, condition (A5) holds if B is such that the demographic component without disease converges. This is summarized in the following theorem.
Theorem 3.8. Define the basic reproduction number (3.2) with force of infection (3.3) by
no = p (FV-1)
no
for system
= P (F12 V;2 1V21 V111) ,
(3.25)
with matrices F and V defined by (3.24). Then the DFE is locally asymptotically stable if no < 1, and unstable if no > 1. With force of infection (3.4), i.e., using standard incidence,
so the matrix V is unchanged, whereas F12 = diag ((31, ... , (3p) provided that Bp(Np) is such that = (as is the case for example if Bp(Np) = dpNp). In this case, no does not depend on the movement parameters, only on the within-patch parameters.
S; N;
3.5.2
SEIRS with multiple species
Suppose that the functions Bsp are such that for all s = 1, ... , sand = 1, ... ,p, limt-+oo Nsp(t) = N;p > O. The method for multiple species is essentially the same as for a single species. To determine the matrices F and V, order the state variables by species, then by patch, i.e.,
p
The vector F then takes the form
with sp zeros corresponding to the I variables. As for the single species case, there holds that, at the DFE, 8ipspj8Eij = 0, for all s, i = 1, ... , s and p,j = 1, ... ,p. Also, 8ipspj8Iij = 0 for all s,i = 1, ... ,s and p,j = 1, ... ,p, whenever p =I- i, since there are no contacts outside ofthe patch. Then the nonnegative matrix F takes the form
o ......................... -
o
0 I
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95
where ffiGp denotes the direct sum of the Gp's, with G p an s x s-matrix with (r, s) entry equal to
[GpJrs
=
I '
rp &
DFE
and representing the contacts between species rand s in patch p. The matrix V is the block matrix ·
.
All: ... :A
. lp :
.........................
··
..
o
..
. :', : : ... . .. .;... .... ... : ... : ~
',
V =
[!drZ·-]
· -1: AP:
...
.:A-- .:
, ,: PP:. .........................:........ : ....... : ....... . p
-E9c
: Bll: ... :B l : ........:....... ~ ..... p.. . .
..
..
.
:. . .. ,:....:..... : ..:... :...
p
p=1
: BpI: ... :Bpp
;
;
;
Matrix A is a block matrix, with each block Apq a s x s diagonal matrix. The (r, r) entry of App is equal to drp + crp + L:f=1 m~p' whereas for p -=1= q the (r, r) entry of Apq is -m;;'q. The (p, p) entry of Bpp is equal to drp + "(rp + L:f=1 m;lp, whereas for p -=1= q the (r, r) entry of Bpq is -m~pq. Finally, Cp is an s x s diagonal matrix with (r, r) entry equal to crp· Matrices G, A, Band Care sp x sp..matrices. Matrices A and B are nonsingular M-matrices since they have the Z-sign pattern and are diagonally dominant by columns [12, M 35 , p. 137J. Thus A-I and B- 1 are nonnegative. Due to the particular structure of F and V, the computation of p( FV- I ) is greatly simplified. Indeed, the inverse V-I of V keeps its block triangular structure
and it follows that
Thus
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Since GB-1CA -1 is a nonnegative matrix, its spectral radius is attained at the largest real eigenvalue. If Ro < 1, then the DFE is locally stable, whereas if Ro > 1, then the DFE is unstable [60, Theorem 2]. The following result has been proved. Theorem 3.9. For system (3.5) with
s species and fi patches, (3.26)
If Ro < 1, then the DFE is locally asymptotically stable, if Ro > 1 then the DFE is unstable. In the case of a force of infection with proportional incidence such as (3.6), the (1', s) entry of G p takes the form f3rspSrp/N;p' 3.5.3
SEIRS with residency patch
Order the infected variables (exposed and infectives) as
E 11 ,: .. , E 1p , E 21 ,.··, E 2p ,"" E pp , 111 , ... , Ir p, hI"'" 12p ,"" Ipp. Since
pq describes the infection of susceptible residents of patch p who are currently in patch q, there holds that 8pq/8Iij = 0 if q =I- j, for all i, j,p, q = 1, ... ,p, since contacts only involve individuals that are in the same patch. This gives the block matrix F, F =
[~~.~-]
where G is a p2 x p2 matrix having p2 blocks, with each block G pq a
p x p diagonal matrix of the form
,8P2 1 ' ... , 8pp I ) . 8Iq2 DFE 8Iqp DFE Also, V is a lower triangular block matrix,
G pq = diag (8 P l 8Iq1
I
DFE
o
p=l p
.
-EBcpi p=l
where each block A, Band C is p2 matrix with CPl
+ dp1 + ~t=l m:kl -m p21
[
E -m pp1
X
p
EBBp : p=l I
p2. For p
... '"
=
1, .. . p, Ap is a p x p
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97
is a f5 x p-matrix with I'P l
+ dpl + t=~=l m~kl -m p2l
[
.. . .. .
I -m ppl
and Cp is a f5 x f5 diagonal matrix with Cp = diag (cpl, ... , cpp). Since Ak and Bk have the Z-sign pattern and have all positive column sums, Ak and Bk are nonsingular M-matrices [12]. Note that
Therefore, the inverse of V is the nonnegative matrix
V- l
=
p=l
p
(3.27)
: p
ffi (A C- l B ) -1: ffi B- 1
'17 p=l
p
p
p
:'17
p
: p=l I
Since V-I is lower triangular by blocks, FV-l can be given by blocks. By [60, Theorem 2], the basic reproduction number for system (3.10) is
(3.28) and the following result holds.
Theorem 3.10. Let no be defined as in (3.28). If no < 1, then the DFE of (3.10) is locally asymptotically stable. If no > 1, then the DFE of (3.10) is unstable.
3.6
Global stability of the disease free equilibrium
In the case of proportional incidence, a comparison theorem argument can be used to show that if no < 1, then the DFE is globally asymptotically stable.
3.6.1
Simple SEIRS
In the case of system (3.2), the local asymptotic stability result for 1 is readily strengthened to a global result.
no <
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Theorem 3.11. Consider system (3.2) with standard incidence (3.4), and assume that the birth function Bp is such that (3.2) has a unique DFE. Let Ro be defined by (3.25). lfR o < 1, then the DFE is globally asymptotically stable. Proof. Since Sp :( N p, it follows that <]?p :( {3pNplp/Np equation (3.2b) gives the inequality
{3plp, and
For comparison, define a linear system given by (3.29) with equality and equation (3.2c), namely d dt Ep = {3plp - (cp
P
+ dp)Ep + L
P
m:qEq -
q=l
dt lp
q=l
P
d
= cpEp -
(rp
L m:pEp,
+ dp + op)lp + L m~qlq q=l
P
L m~plp. q=l
This linear system has coefficient matrix F - V, and so by the argument in the proof of Theorem 3.8, satisfies lim Ep = 0 and lim Ip = 0 for t----'t-OO
t---+oo
Ro = p(FV-l) < 1. Using a comparison theorem (e.g, [38, Theorem 1.5.4] or [57, Theorem 13.1]) and noting (3.29), it follows that these limits also hold for the nonlinear system (3.2b) and (3.2c). That lim Rp = 0 follows from Theorem 3.7, and lim Sp t-+oo
= S;
t-+oo
follows from (3.2a) and the
assumption that a unique DFE exists. Thus for Ro < 1, the disease free equilibrium is globally asymptotically stable and the disease dies
D
~.
Note that it is clear from this proof that any incidence function <]?p such that
would lead to the same conclusion.
3.6.2
SEIRS with multiple species
A comparison theorem argument can be used as in Theorem 3.11 to show that if < 1, then the DFE of the multiple species system (3.5) is globally asymptotically stable. Note, however, that the proofrequires here the use of asymptotically autonomous differential equations.
no
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Theorem 3.12. For system (3.5) with s species and p patches, birth rate of the form (3.13), no disease induced mortality and proportional incidence (3.6), define no as in (3.26). If no < 1, then the DFE is globally asymptotically stable. Proof. To establish the global stability of the DFE, consider the nonautonomous system consisting of (3.5b)-(3.5d), with (3.5b) written in the form
d ~ I jp dt Esp = ~ f3sjp(Nsp - Esp - Isp - Rsp) N j=1 lP P P - (dsp + €sp)Esp + m!,qEsq m~pEsp, q=1 q=1
L
(3.30)
L
in which Ssp has been replaced by Nsp - Esp - Isp - R sp , and Nsp is a solution of (3.20). Write this system as X' =
(3.31 )
f(t,x),
where x is the 3sp dimensional vector consisting of the Esp, Isp and Rsp. The DFE of (3.5) corresponds to the equilibrium x = 0 in (3.31). Since 6sp = 0, system (3.20) can be solved for Nsp(t) independently of the epidemic variables, and Theorem 3.5 implies that the time dependent functions Nsp(t) -'> N;p as t -'> 00. Substituting this large time limit value N;p for Nsp in (3.30) gives d Esp dt
* =~ ~ f3sjp (Nsp
I jp - Esp - Isp - Rsp ) N*
j=1
lP
n
- (dsp
+ €sp)Esp + L
n
m!,qESq -
q=1
(3.32)
L m~pEsp. q=1
Therefore, system (3.31) is asymptotically autonomous, with limit equation (3.33) x' = g(x). To show that 0 is a globally asymptotically stable equilibrium for the limit system (3.33), remark that the linear system
x'
=
Lx,
(3.34)
where x is the 3sp dimensional vector consisting ofthe Esp, Isp and R sp ,
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but with the equation for Esp taking the form
p
+L
p
m1fr,q E Sq -
q=l
(3.35)
L m~pEsp q=l
is such that g(x) ::; Lx for all x E lR.~n. In system (3.34), the equations for Esp and Isp do not involve R sp , and can thus be considered independently from the latter. Let x be the part of the vector x corresponding to the variables Esp and I sp , and L be the corresponding submatrix of L. The term (3sjp N;p/ Njp corresponds to the (s, j) entry of the matrix Gp used in Theorem 3.9, since Ssp ---t N;p under the current assumptions. (See the remark following Theorem 3.9.) Therefore, the method used in Section 3.5 to prove local stability can also be applied to study the stability of the x = 0 equilibrium of the subsystem x' = Lx, with L = F - V. Therefore, if no < 1, then the equilibrium x = 0 of the subsystem x' = Lx is stable. When x = 0, the conclusion of Theorem 3.7 holds, and limt--+ooRs(t) = 0, with Rs = (Rsl, ... ,Rsp)T. Thus the equilibrium Rs = 0 of this linear system in Rs is stable. As a consequence, the equilibrium x = 0 of (3.34) is stable when no < 1. Using a standard comparison theorem (see, e.g., [38, Theorem 1.5.4]), it follows that 0 is a globally asymptotically stable equilibrium of (3.33). For no < 1, the linear system (3.35) and (3.5c) has a unique equilibrium (the DFE) since its coefficient matrix F - V is nonsingular. The proof of global stability is completed using results on asymptotically autonomous equations; see, e.g., [58, Theorem 4.1] and [19]. D As in the simple SEIRS case (Theorem 3.11), any incidence function
O
I
Isp
DFE
would lead to the same conclusion. Also, the assumption that there is no disease induced mortality can be relaxed, provided that it can be shown that Nsp converges for all s, p.
3.6.3
SEIRS with residence patch
Theorem 3.13. For system (3.10) with p patches, birth rate Bpq such that limt--+oo Spq = N;q, and proportional incidence (3.12), define no as in (3.28). If no < 1, then the DFE is globally asymptotically stable.
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The proof proceeds exactly as the proof of Theorem 3.12, except that the incidence function takes the form
and thus we obtain an upper bound by noting that
As in the proof of Theorem 3.12, letting Spq --+ N;q we obtain the terms that appear in the matrix F used in Theorem 3.10 when considering proportional incidence (3.12). The same remark about incidence holds as in the simple and the multiple species cases above. Note however that the formulation of Theorem 3.13 assumes the convergence of S, which is obtained by assuming no disease induced mortality, but also with properly chosen birth functions.
3.7
Existence of mixed equilibria
A mixed equilibrium is an equilibrium for the whole system with some patches at a disease free equilibrium and others at an endemic equilibrium. The previous assumption of strongly connected movement graphs that is made in the rest of this chapter is here relaxed. To summarize the results established in this section, if movement is similar for all states, then the type of equilibria is fixed for each strongly connected component in the movement graph g. If movement is dissimilar, the situation is unresolved.
3.7.1
Model with classic movement
The situation is discussed in the case of the system with multiple species (3.5). Specialization to the case of a single species is trivial.
Theorem 3.14. Suppose that (3.5) with movement similar for all states is at an equilibrium. If a given patch p is at a DFE, then all patches that have an access to patch p for a given species s, i.e., patches q E P"--.p, are also at a DFE. Proof. Fix the species index at s. For simplicity suppose that p = 1, i.e., there is no disease in patch 1. Thus Esl = lsI = O. Then for p = 1, (3.5c) is
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Since Isr :? 0 and p
L mi1r I sr = L r=2
m;lr I sr
rEP~l
+
L
ms1r I sr
= 0,
rf/cP~l
it follows that Isr = 0 for r E P--->l. Similarly, setting p = 1 in (3.5b) and using P --->1, it follows that Esr = 0 for rEP--->1. Thus, all patches r with a direct access to patch 1 have no disease, i.e., are such that Esr = Isr = o. Now consider a patch r in P ---> 1. Using the same argument as previously, it follows that Esw = Isw = 0 for all w E P --->r' Patches that are in P --->r \ P --->1 have a length 2 access to patch 1. By induction, all patches in P --->1 are at the DFE if patch 1 is at the DFE. D Note that this result is independent ofthe nature ofthe birth function Bsp. Also, conclusions have not been derived on the nature of Rp for those patches that are at the DFE, nor has it been shown that Sp --7 N;. In fact, with a little additional work and using the same type of argument used in Theorem 3.7, it can be shown that limt--->oo Rk(t) = 0 for k E P--->p' In this case, Sp --7 N;, with the precise value of N; undetermined until a birth function has been chosen. Theorem 3.15. Suppose that (3.5) with movement similar for all states is at an equilibrium. If a given patch p is at an endemic equilibrium, then all patches that can be accessed from patch p for a given species s, i.e., patches q E P;--->, are also at an endemic equilibrium.
Proof. Fix the species index s. For simplicity suppose that p Es1 + Is1 > O. From (3.5b) and (3.5c) with q =I- 1, d 0= dt (Esq
+ Isq)
=
+ Isq)
=
1, i.e.,
- !,sqIsq
p
p
+L
msqr(Esr
+ Isr)
-
r=l
L msrq(Esq + Isq). r=l
Assume that Esq + Isq = 0 and msq1 > 0, i.e., patch 1 has access to patch q. Then the above equation reduces to p
0=
L msqr(Esr + I sr ), r=l
and implies that Es1 + Is1 = 0, giving a contradiction. Thus the disease in patch q is at an endemic equilibrium. The remainder of the proof follows as in the proof of Theorem 3.14. D
Diseases in Metapopulations
3.7.2
103
Model with residency patch
Theorem 3.16. Suppose that system (3.10) with movement similar for all states is at an equilibrium, and that a given patch p is at the DFE. Then all patches that can be accessed from patch p, and all patches that have an access to patch p, are at the DFE. Proof. Suppose for simplicity and without loss of generality that patch 1 is at the DFE, i.e., h1 = Ek1 = 0 for all k = 1, ... ,po Then consider (3.1Ob) and (3.10c) with i = 1. Since III = 0, it follows that
Recall that variables remain nonnegative. It follows that, since the system is at equilibrium, Elk = 0 for all patches in P!!.l' and Ilk = 0 for all patches k in P~l' Since movement is similar for all states, P!:l = P~l =: P ..... 1 . In summary, if patch 1 is at the DFE, then there holds that for all patches j in P ..... 1 , E 1j = hj = 0, i.e., there is no disease in visitors from patch 1 visiting patches that have direct access to patch 1. Consider now (3.10b) for one of these patches, i.e., for j E P ..... 1. There holds, for visitors to j from patch 1,
d
-E1"=
dt
J
J
Since the system is at an equilibrium, E 1j = 0, that is fi
0=
= Lf3rj(Nj)Slj Irj . r=l
It was shown (Theorem 3.2) that Spq > 0 for all positive times, provided that mgqr > O. Since the Spq are positive, there holds that for all j E P ..... 1 such that mfjk > 0, hj = O. Using the same type of argument that was used to show that R = 0 in the proof of Theorem 3.12, it follows that for these patches, there also holds that Ekj = O. To summarize, if patch 1 is at the DFE, then patches j E P ..... 1 are also at the DFE. By induction, all patches that are in P ..... 1 are at the DFE. D
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3.8
Bounds on Ro
The expression obtained for no in all three models is complicated. Therefore, it is useful to derive some bounds on the value of no. We do so in the most complicated case, that of the SEIRS model with residency patch. The results are also formulated in the simpler cases, but no proofs are provided. 3.8.1
SEIRS with residency patch
For convenience, write
Bib:?'Bib pr I BIjk := BIjk DFE' Theorem 3.17. Let V;;;l and v-;/ be the minimum and maximum column sums of the (2,1) block EB~=l (A pC;l Bp) in matrix V-I defined by (3.27). Then there holds that
P(
min
BibD ) ----..1k
i,j,k=l, ... ,p BIjk
D
V-I:::;;
m
B no : :; p (i,j,k=l, maxi b ----..1k ) ... ,p BIjk
V-I. M
(3.36)
Proof. We denote G( EB(A.C;l B.)-l) := G (EBk(A kCk 1 B k )-l) for simplicity. The (p, q) block of G (EB(A.C;l B.)-l) is G pq (A qC;l B q)-l for all p, q. As G pq is diagonal, multiplication with (A qC;l B q)-l amounts to multiplying row k = 1, ... ,p of (A qC;lBq)-l by the kth diagonal entry of G pq , that is, Bib~/BIqk. Let vk/(q) denote the (k, I) entry of (A qC;l Bq)-l, for k, I = 1, ... ,po Then a given block G pq (A qC;l B q)-l
takes the form
Bib~
-1
- - V -1
BIqp p
Bib~
-1
(q) ... >:II _ vpp (q) U qp
It follows that
Summing for p = 1, ... ,p gives the column sums in the qth block of
Diseases in Metapopulations
105
columns as [nTG(EB(A.C;lB.)-l)]
[q]
=
p a~D p a~D ) L a/:v;;}(q), ... , L a/kv;j(q) , ( p,k=l qk q p,k=l
where we denote
p
p
p
L:=LL
p,k=l
p=l k=l
in order to simplify notations. Thus, for the whole matrix,
(3.37) Define a~D
=
min ~ i,j,k=l, ... ,p aIjk
and a~D
max ~ i,j,k=l, ... ,p aIjk Then, for any column c in the
Defining v;;; 1 and v 1, ... ,p,
Ai
jth
block of columns, there holds
as in the theorem, it follows that for all c, j
=
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106 and thus
Using a standard result on the localization of the dominant eigenvalue of a nonnegative matrix (see, e.g., [42, Theorem 1.1]), which states that the dominant eigenvalue of a nonnegative matrix is bounded below and above by the minimum and maximum of its column sums, the result then follows. 0 Consider patch p isolated from the other patches (with movement rates into and out of the patch set to zero). In this case, all nonresident populations tend to zero. The basic reproduction number in patch p, Rg, is given by RP _ o - (cpp
C: pp
+ dpp)bpp + dpp ) .
8iP£' 8Ipp .
Corollary 3.18. Suppose that iPpq = iPq for all p, q = 1, ... ,p, i.e., infection occurs at the same rate for all individuals in a given patch q. Then min_Rg'::;Ro'::; max_Rg. p=I, ... ,p p=I, ... ,p
Proof. Note that Gpq represents infections in patch q of susceptibles from patch p. The assumptions of the corollary imply that G pq = G q for all p= 1, ... ,p. 0 In the particular case where parameters except the force of infection are equal in each patch, the bounds in Theorem 3.17 take an easier form. Theorem 3.19. Suppose that for system (3.10), parameters are the same in all patches, i.e., Cpq = c, 'Ypq = 'Y and dpq = d for all p, q = 1, ... ,po Then
(
. 8iP~) mm i,j,k
8Ijk
pc
-,-----='---- ~ d)(c d) ~
b+
+
Ro
(
~
~
8iP~) pc max - -,----':"",.--i,j,k 8Ijk b + d)(c: + d) .
(3.39)
Proof. Under the assumptions of the theorem, the following holds true: nT Ap
= (c + d)nT
=?
nT A C-1
=?
nT A
p
p
p
= c: + dnT c
c- 1 BP = (c + d)b + d) nT c· p
Diseases in Metapopulations
107
Consider an invertible matrix M such that nT M = enT. Then there holds that n™M-1 = en™-1, and thus n™-1 = lie. This implies that
nT(A a- 1 B )-1 = P
Substituting this value for gives the result.
P
€
(€ + d)(-r + d)"
2:::=1 V pc1 (j)
in (3.38), and using in (3.37) D
So, in the case that disease characteristics are identical for all individuals, and that transmission is identical for all individuals in a given patch, there are easily computable bounds for Ro. In particular, if Rg < 1 for all p = 1, ... ,p, then the DFE is locally asymptotically stable, or globally asymptotically stable if the stronger hypotheses needed for this are satisfied; if Rg > 1 for all p = 1, ... ,p, then the DFE is unstable. If, additionally,
3.9
Further problems
I mention briefly here other problems that could and should be considered. They are not detailed here, but references are provided in Section 4.
3.9.1
Existence of endemic equilibria
The existence of endemic equilibria, that is, equilibria with positive numbers of infectives, has barely been discussed here. In Section 3.7, it was established that endemic equilibria, if they exist, populate whole strongly connected components. However, no method was given to prove their existence. Numerical simulations seem to indicate that, for the systems presented here, there is a unique, globally asymptotically stable equilibrium point, when Ro > 1. Clearly, establishing properties of persistence of the system when Ro > 1 would be interesting steps in that direction.
3.9.2
Understand the effect of movement
Theorem 3.17 establishes that in the case of a relatively homogeneous system, the movement matrix plays a role only insofar as it determines the value at the DFE. If disease transmission is also homogeneous within each patch, then Corollary 3.18 proves that the situation is even more constrained. In particular, in that case, it is impossible, for example, for movement to stabilize an unstable situation, or to destabilize a stable situation. Indeed, consider a system consisting of two connected patches,
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and suppose that both are such that their ng < 1 when taken in isolation. If the conditions of Corollary 3.18 hold, then movement cannot change this situation. The same holds true if both patches are such that ng > 1: movement cannot stabilize such a situation. In a less restrictive setting, movement can either stabilize an unstable situation, or destabilize a stable one. This has been investigated, for example, in [1, 49, 50].
4 4.1
Diseases in metapopulations -
A review
Focus of the review
Our definition of metapopulations de facto excludes pure group models, that consider strict interactions between groups. Such models have been considered for about the same amount of time as metapopulation models. The first such models are due to Rushton and Mautner [47] and Haskey [29]. Other well known examples are due to Lajmanovich and Yorke [37], Hethcote [33], Hethcote and Thieme [31]. While these models are conceptually quite similar to metapopulation models, they make the assumption that there is no exchange of individuals between the subpopulations. Their analysis can be quite similar to the analysis of metapopulation models. Also, the focus is on deterministic models that have been mathematically analyzed. Simulation work as well as stochastic models will be evoked when they give insight into the mechanisms. Finally, we focus on time continuous models. There are some very interesting works that are formulated in discrete time (see, e.g., [2, 16]), but the theory is quite different.
4.2
Early works
Bartlett, 1956 The first work that we are aware of that uses a patch approach is due to Bartlett [11]. He considers the following model on two patches, 8~ = -«(31h
I~ 8~
I~
+ (32h)8 1 + b + ms(82 - 8 1 ),
= Uhh + (32h)81 - (d + p)J.lh + mI(I2 - II), = -«(31h + (32h)82 + b + ms(81 - 82 ), = «(31h + (3212)82 - (d + p)I2 + mI(h - h).
(4.1a) (4.1b) (4.1c) (4.1d)
The rate d + p incorporates the natural death rate d as well as the rate p of occurence of any other event leading to an individual leaving the infected class (disease specific death, recovery, etc.). b is the birth rate.
Diseases in Metapopulations
109
Note that this model is a hybrid of metapopulation and group models. Indeed, there is an exchange of individuals between patches through migration, but there is also cross patch infection. Baroyan and R vachev, late 60s, and directly related articles Following the work of Bartlett comes works by Baroyan, Rvachev and collaborators [9, 10]. They consider the spatial spread of influenza between cities in the Soviet Union. In their approach, a large geographic region (country) is partitioned into smaller sub-regions (cities). Migration and transportation between these cities are explicitly incorporated, and within a given city, transmission is modeled by a discrete deterministic compartmental SIR model. In [48], the parameters of the model are estimated using Hong Kong as a reference; the model is then used to simulate the spread of the Hong Kong influenza pandemic between 52 world cities. In [40], an epidemic threshold theorem is obtained. An SEIR version of this model was used recently [26]. Using the framework of Rvachev and Longini, Hyman and LaForce [34] formulate a multy-city transmission model for the spread of influenza between cities (patches) with the assumption that people continue to travel when they are infectious and there is no death due to influenza. Because influenza is more likely to spread in the winter than in the summer, they assume that the infection rate has a periodic component. In addition, they introduce a new disease state P in which people have partial immunity to the current strain of influenza. Thus they have an SIRPS model in which both susceptible and partially immune individuals can be infected, but this is more likely for susceptibles. A symmetric travel matrix M = [mij] with mij = mji is assumed, thus the population of each city remains constant. Their model for p cities is formulated as a 4p system of non autonomous ODEs. Epidemic parameters appropriate for influenza virus are used, in particular for strains of H3N2 in the 1996-2001 influenza seasons with an infectious period of l/ex = 4.1 days in all cities. Parameters modeling the number of adequate contacts per person per day and the seasonal change of infectivity are estimated by a least squares fit to data. The populations of the largest 33 cities in the US are taken from 2000 census data, and migration between cities is approximated by airline flight data. A sensitivity analysis reveals that 1/ ex, the average duration of infection, is the most important parameter.
4.3
Kermack-McKendrick-type models
The model known as the Kermack-McKendrick (KMK) model takes the form [36]
d
-8 = -f38I,
dt
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110
d -I dt d dtR
=
f3S1 - "II,
= "II,
that is, an SIR model without demography. The parameter "I represents here the rate of removal from the I class, it aggregating disease induced death and recovery from the disease. This system has the advantage that an explicit solution can be found (see, e.g., [13]). Several authors have used KMK-type models in a metapopulation context. Faddy, 1986 model
In a short note, Faddy [22] introduces a KMK-type SI n
S:
=
-Si ~f3jilj,
(4.2a)
j=l
n
I:
= Si L j=l
f3ji l j - 'Yili
+ ~ mij I j Ji-i
L mjih
(4.2b)
Ji-i
where "Ii represents the sum of all removals from the I class. As in the case of system (4.1), this system mixes group models with migration. An interesting remark made by Faddy is that there exists a sort of conservation law, since the quantity
does not change over time. The interest here is on the final size of the epidemic, for which an expression is obtained. In the spatially homogeneous case where Si(O) = S(O), a given value in all patches, mi = 2:Ji-i mij = m, f3ij = f3 and "Ii = "I, he obtains that
where Si( 00) is the final number of susceptibles remaining uninfected in patch i. Clancy, 1996 Clancy [20] introduces a Kermack-McKendrick type model on patches, but that describes the dynamics of a very simple epidemic with carriers, whose numbers is denoted C. The carriers are
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111
subject to specific removal (either through treatment of death), at a rate "(. The system includes a removed class that we do not show here as it bears no influence on the dynamics of the system. The latter takes the form (4.3a) n
C~ = -,,(Ci +
L m~Cj.
(4.3b)
j=l
He also introduces a corresponding stochastic version. The focus here is on the ultimate size of the epidemic; more precisely, estimates of Si as t --+ 00 are sought. Rodriguez and Torres-Sorando, 2001, consider in [44] a direct transmission model and a model of malaria. The malaria model tracks the evolution of the numbers Ii and Yi of infectious humans and infectious mosquitoes, respectively, on patch i. The total population of both species is assumed constant on each patch, and denoted Ni and M i , respectively, for humans and mosquitoes. Thus, the numbers of susceptibles are obtained by Si = Ni - Ii and Zi = Mi - Yi. The system takes the form II
= /3SiYi
- "(Ii
+ /3Si L
mij Yj,
(4.4a)
jf-i
~'
=
/3ZJi - d M Yi
+ /3Zi L
mjiIj.
(4.4b)
#i
/3 is the rate of transmission of the disease when a contact occurs. They consider the effect of different migration patterns and of the environment heterogeneity on the dynamics of the system, and in particular on the possibility of the disease becoming established. To study this, they consider the jacobian matrix at the DFE, and study the sign of the dominant eigenvalue. 4.4
Migration models
Wang and Mulone, 2003 Wang and Mulone [63] consider the following model in the case p = 2 patches,
S~ = di(Ni -
Si) - /3i S i ~. ,
j5
+ "(iIi + L m~Sj, j=l
(4.5a)
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112
(4.5b)
They establish a series of interesting results concerning the conditions under which the disease is persistent in the system. One particular conclusion that they draw is that, provided they are positive, the migration rates of susceptibles mr2, m~1 do not playa role in the permanence conditions. Wang and Zhao, 2004 form
Wang and Zhao [64] consider a model of the n
S~ = Bi(Ni)Ni - /liSi - (3i S Ji
+ fili + L m~Sj,
(4.6a)
j=1
n
I~
=
(3iSi1i - (/li
+ fi)Ii + L
mIjlj .
(4.6b)
j=1
With this more general birth function B i , even finding a disease free equilibrium is a difficult task. It is shown that, in this case, population movement can either intensify or reduce the spread of disease. Salmani and van den Driessche, 2006 Salmani and van den Driessche introduce in [50] a single species SEIRS model with status dependent movement and disease induced death, from which (3.2) is derived. In a first part, following the approach of [6, 7], they establish a basic reproduction number for the system, and as in [4], the global stability of the DFE when Ro < 1. They then proceed to a more detailed study of an SIS particular case in two patches. They establish that with different movement rates, different situations can prevail, with for example a global Ro < 1 and individual Rg in the patches less than 1 (it will be established in the next chapter that this cannot be the case when the movement rates are identical for all epidemiological states). Fulford, Roberts and Heesterbeek, 2002 The spread of bovine tuberculosis amongst the common brushtail possum in New Zealand, is modeled by Fulford et al [25]. Since only maturing possums (1 to 2 year old males) travel large distances, a two-age class metapopulation model is formulated, with juvenile and adult possums. As this disease is fatal, an SEI model is appropriate. In addition to horizontal transmission between both age-classes, pseudo-vertical transmission is included since juveniles may become infected by their mothers. Susceptible and exposed juveniles (but not infective juveniles) travel between patches as they
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113
mature. For p patches, the authors formulate a system of 6p ODEs to describe the disease dynamics. Using the next generation matrix method [21], the authors explicitly calculate Ro for p = 1 and for p = 2, and give the structures of the next generation matrices for p = 4 and three spatial topologies, namely a spider, chain and loop. The design of control strategies (culling) based on these three spatial topologies is considered. The critical culling rates (where Ro = 1) are calculated and the spatial aspects are shown to be important.
4.5
Model including residency patch
Sattenspiel and coauthors In [55], Sattenspiel and Simon introduced a model for the interaction between individuals in p neighborhoods, taking into account that some individuals only have contacts in their neighborhoods. Although this is a strict group model, since individuals do not move explicitly between neighborhoods, it is mentioned here because it is an obvious prequel to the models with residency patch. Also, it contains some interesting matrix-based analysis. Sattenspiel and Dietz [51] introduced a single species, multi-patch model that describes the travel of individuals, and keeps track of the patch where an individual is born and usually resides as well as the patch where an individual is at a given time. Hence this type of model describes human travel rather than migration. This framework was subsequently used numerically by Sattenspiel and others to describe various situations linked to the spread of influenza in the Canadian subartic [52], the effect of quarantining [53] and the influence of the mobility patterns [54]. Arino and van den Driessche We studied the model of [51] in [6, 7], giving some analytical results and calculating the basic reproduction number in the SIS [7] and SEIRS [6] cases, giving the first example of application of the method of [60] to such high dimensional models. These models have a unique DFE. Numerical simulations show that a change in travel rates can lead to a bifurcation at Ro = 1; thus travel can stabilize or destabilize the disease free equilibrium. This model is the basis for [8], which extends the model of [51] by allowing individuals to travel between two patches that are not their residency patch. The resulting model is system (3.10), analyzed in detail in Section 3. Ruan, Wang and Levin, 2006 Using the framework of [51]' Ruan, Wang and Levin [45] study the global spread of SARS. The system takes the form of an SEIRS model with an additional class for quarantined individuals, denoted Q, that do not travel. They study the existence of
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114
a DFE, and establish the basic reproduction number R o, deducing the local asymptotic stability of the DFE when Ro < 1 using the result of [60], as detailed in the SEIRS case in Section 3.5. The basic reproduction number depends explicitly on quarantining parameters. A particular case for two cities (Honk Kong and Toronto) is then considered.
4.6
New directions
To conclude this brief review of diseases in metapopulations, a few directions that appear promising to the author are now listed. It is hoped that, although necessarily biased by the author's opinions, this will encourage readers to study more in detail some of the aspects.
4.6.1
True patch heterogeneity
Metapopulations have been introduced in the context of epidemic diseases to take into account spatial heterogeneities. However, in all the models discussed so far, the spatial heterogeneity is not 'true'. Indeed, it is assumed that in the different patches, parameters are different, but that the incidence function is similar. The effect of the contact structure (the nature of the incidence function) on the dynamics is determinant. A model of Fromont, Pontier and Langlais [24] is the first we know that breaks this homogeneity. They consider a model appropriate for Feline Leukemia Virus among a population of domestic cats. There are p patches called farms or villages depending on the magnitude of the patch carrying capacity. Dispersal (which depends on disease state) can take place between any pair of patches or int%ut of non-specified populations surrounding the patches (representing transient feral males). Infected cats become either infectious or immune and remain so for life, thus the model is of SIR type, but a proportion of cats go directly from the susceptible to the immune state. A density dependent mortality function is assumed, as well as different incidence functions depending on the population density (mass action for cats on farms, standard incidence for cats in villages). The model consists of 3p ODEs and is analyzed for the case p = 2, taking data appropriate for the virus with one patch being a village and one patch being a farm, or both patches being farms. For a set of parameters such that in isolation the virus develops in the village but goes extinct on the farm, travel between the patches of either susceptible and immune cats or of infective cats can result in the virus persisting in both patches. Thus results show that, in general, spatial heterogeneity promotes disease persistence.
Diseases in Metapopulations 4.6.2
115
Models with infinite dimensional aspects
Wang, Fergola and Tenneriello, 2003 Wang, Fergola and Tennierello [62] study a model for the diffusion of innovation in a two patch environment. Although not strictly an epidemic model, the spread of innovation can easily be reread in terms of disease propagation. They first formulate the model in ODE and show that it has a globally asymptotically stable equilibrium (note that this is different from classical epidemic models, in the sense that there is no bifurcation from a disease free equilibrium to an endemic equilibrium). They then incorporate delay, in the form of product duration. Written in epidemiological terms, the model then takes the form d
dt 51 = III - (a1 + f31h)5 1 - d 15 1 + m2(52 + (1 - k 2)h) - m 15 1 + e- dIT1 (11 51(t - Td + f3 15 1(t - TdJ(t - Td), (4.7a) d
dt h
=
(a1
+ f31h)51 -
d1h - m1h
- e- d1T1 ('"Yl51 (t - Td d
dt 52
=
II2 - (a2
+ f32h)5 2 -
+ e-d2T2(1252(t d
dth
=
+ k2m2h
+ f3 15(t -
T1)J(t - Td),
d25 2 + m1 (51
+ (1 -
(4. 7b)
k 1)h) - m252
T2) + f32S2(t - T2)J(t - T2)),
(4.7c)
(a2 + f32lz)52 - d2lz - m2J2 + k1m1h - e-d2T2(1252(t - T2)
+ f3 25(t -
T2)J(t - T2)).
(4.7d)
In each patch i = 1,2, besides the usual parameters, IIi is the birth rate (a constant), ai is the intensity of advertisement for the products (this additional recruitment term is the main difference from classical epidemic models) and Ti is the duration of the product (that is, of infection) in each patch. The migration is slightly different from the other models seen so far, in the sense that individuals can change status when they move from one patch to the other: m1 is the rate of movement from 1 to 2, m2 is the rate of movement from 2 to 1, k1 is the fraction of infected from patch 1 that remain infected when moving to 2 and k2 is the fraction of infected from patch 2 that remain infected when moving to patch 1. Here again, there is a unique positive equilibrium, which is shown to be globally asymptotically stable under some conditions on parameter values. The paper concludes with a study of periodic solutions in the case where advertisement, i.e., ai, is periodic (in the delayed case). It is shown that there exists parameter values for which a periodic solution exists and is globally stable.
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116
Wang and Zhao, 2005 Wang and Zhao [65J formulated a model for an SIS model on patches, with a class of juveniles (denoted by J). On each patch p = 1, ... ,p,
(4.8a)
(4.8b)
(4.8c) where Ap = Sp + Ip (the population of adults), dt is the death rate of juveniles and d p is the death rate of adults, and Rp(t) is rate of recruitment of juveniles into the susceptible adult class. It is assumed that B(Ap) > 0 for Ap > 0, Bp continuously differentiable for Ai > 0 and B~(Ap) < 0 for all Ap > O. Recruitment into the adult class has to take into account that juveniles can be born in a given patch, and become adults in another patch. Let r be the age of recruitment into the adult class (assumed the same in each patch), and J(t, a) := (h(t, a), ... , Jp(t, a))T, with Jp(t, a) the number of juveniles in patch p at time t that are of age a. The recruitment R(t) then satisfies R(t) := (Rl (t), ... ,Rp(t) f = J(t, r). The age-space dynamics is described by
(at + oa)Jp(t, a) =
t,
mtkJk(t, a) -
(~m"£p + d~) Jp(t, a)
p
=
L m~kJk(t, a) -
d~ Jp(t, a),
k=l with J(t,O) = B(A(t)) := (Bl(Al(t))Al(t), ... ,Bp(Ap(t))Ap(t))T. Then, after some computations,
R(t) where
=
J(t,r)
-d{ + mil
~~l
CJ =
(
J m pl
=
exp(CJr)B(A(t - r)), C12
-d£ +m{2··· J m p2
Using R(t), the equations for S and I decouple from the equations for J, giving a system of 2p delay differential equations.
Diseases in Metapopulations
117
The authors then establish the existence of a unique disease free equilibrium under a certain number of assumptions. They then derive a basic reproduction number for the system, and consider the global stability of the disease free equilibrium, as well as the persistence of the system when this equilibrium is unstable, and the existence of an endemic equilibrium. The paper concludes with a study of a two patch particular case.
5
Cond usion
My aim here was to show that metapopulation models are usable in the context of epidemiology, to provide an extensive overview of the mathematical problems that arise when studying such models, and to illustrate some of the solutions that can be given to these problems. This was done through two classes of models that van den Driessche and I have considered, with a simple single population SEIRS also used to illustrate the most simple properties. I hope to have convinced the reader, at the cost of maybe a little too much detail, that the mathematical complications arising in these models can be dealt with, and that there is a pattern to these solutions that allows to envision a general theory of metapopulation models in epidemiology. This theory is barely sketched here.
References [1] L.J.S. Allen, B.M. Bolker, Y. Lou, and A.L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic patch model. Submitted. [2] L.J.S. Allen, D.A. Flores, RK. Ratnayake, and J.R Herbold. Discrete-time deterministic and stochastic models for the spread of rabies. Applied Mathematics and Computation, 132: 271-292, 2002. [3] RM. Anderson and RM. May. Oxford University Press, 1991.
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[6] J. Arino and P. van den Driessche. The basic reproduction number in a multi-city compartmental epidemic model. Lecture Notes in Control and Information Science, 294: 135-142, 2003. [7] J. Arino and P. van den Driessche. A multi-city epidemic model. Mathematical Population Studies, 10(3): 175-193,2003. [8] J. Arino and P. van den Driessche. Metapopulation epidemic models. A survey. Fields Institute Communications, 48: 1-13, 2006. [9] V. 0 Baroyan and L.A Rvachev. Deterministic epidemic models for a territory with a transport network. Kibernetica, 3: 67-73,1967. [10] V. O. Baroyan, L.A. Rvachev, U.V. Basilevsky, V.V. Ezmakov, K.D. Frank, M.A. Rvachev, and V.A. Shaskov. Computer modeling of influenza epidemics for the whole country (USSR). Adv. App. Prob., 3: 224-226, 1971. [11] M.S. Bartlett. Deterministic and stochastic models for recurrent epidemics. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, volume IV, pages 81-109. University of California Press, 1956. [12] A. Berman and R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences, volume 9 of Classics in Applied Mathematics. SIAM, 1994. [13] F. Brauer. The Kermack-McKendrick epidemic model revisited. Math. Biosci., 198: 119-131, 2005. [14] F. Brauer and C. Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology. Springer, 200l. [15] S. Busenberg and K.L. Cooke. Springer-Verlag, 1993.
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[16] C. Castillo-Chavez and Yakubu A.-A. Intraspecific competition, dispersal and disease dynamics in discrete-time patchy invironments. In C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and Yakubu A.-A., editors, Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, volume 125 of IMA Series on Mathematics and its Applications. Springer, 2002. [17] C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu, editors. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, volume 125 of IMA Series on Mathematics and its Applications. Springer, 200l. [18] C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu, editors. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory,
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123
Modeling the Start of a Disease Out break* Fred Brauer Department of Mathematics, University of British Columbia Vancouver, BC V6T lZ2, Canada E-mail:
[email protected]
1
Introduction
The Kermack-McKendrick compartmental epidemic model assumes that the sizes of the compartments are large enough that the mixing of members is homogeneous, or at least that there is homogeneous mixing in each subgroup if the population is stratified by activity levels. However, at the beginning of a disease outbreak, there are a very small number of infective individuals and the transmission of infection is a stochastic event depending on the pattern of contacts between members of the population; a description should take this pattern into account. It has often been observed in epidemics that there is a small number of "superspreaders" who transmit infection to many other members of the population, while most infectives do not transmit infections at all or transmit infections to very few others [17]. This suggests that homogeneous mixing at the beginning of an epidemic may not be a good approximation. The SARS epidemic of 2002-2003 spread much more slowly than would have been expected on the basis of the data on disease spread at the start of the epidemic. Early in the SARS epidemic of 2002-2003 it was estimated that Ro had a value between 2.2 and 3.6. At the beginning of an epidemic, the exponential rate of growth of the number of infectives is approximately (Ro - l)/a, where l/a is the generation time of the epidemic, estimated to be approximately 10 days for SARS. This would have predicted at least 30,000 cases of SARS in China during the first four months of the epidemic. In fact, there were fewer than 800 cases reported in this time. An explanation for this discrepancy is that the estimates were based on transmission data in hospitals and crowded apartment complexes. It was observed that there was intense activity in some locations and very little in others. This suggests that the actual reproduction number (averaged over the whole population) was much -This work has been supported by NSERC and MITACS.
124
Fred Brauer
lower, perhaps in the range 1.2-1.6, and that heterogeneous mixing was a very important aspect of the epidemic.
2
The network corresponding to a disease outbreak
Our approach will be to give a stochastic branching process description of the beginning of a disease outbreak to be applied so long as the number of infectives remains small, distinguishing a (minor) disease outbreak confined to this stage from a (major) epidemic which occurs if the number of infectives begins to grow at an exponential rate. Once an epidemic has started we may switch to a deterministic compartmental model, arguing that in a major epidemic contacts would tend to be more homogeneously distributed. However, if we continue to follow the network model we would obtain a somewhat different estimate of the final size of the epidemic. Simulations indicate that the assumption of homogeneous mixing in a compartmental model would lead to a higher estimate of the final size of the epidemic than the prediction of the network model. We describe the network of contacts between individuals by a graph with members of the population represented by vertices and with contacts between individuals represented by edges. The study of graphs originated with the abstract theory of Erdos and Renyi of the 1950's and 1960's [3, 4, 5], and has become important more recently in many areas, including social contacts, computer networks, as well as the spread of communicable diseases. We will think of networks as bi-directional, with disease transmission possible in either direction along an edge. An edge is a contact between vertices that can transmit infection. The number of edges of a graph at a vertex is called the degree of the vertex. The degree distribution of a graph is {pd, where Ph is the fraction of vertices having degree k. The degree distribution is fundamental in the description of the spread of disease. Initially, we proceed as if all contacts between an infective and a susceptible transmit infection, but we will not make this assumption when we study the course of a disease outbreak in Section 3. We think of a small number of infectives in a population of susceptibles large enough that in the initial stage we may neglect the decrease in the susceptible population. Our development begins along the lines of that of [7] and then develops along the lines of [6, 14, 16]. We assume that the infectives make contacts independently of one another and let Ph denote the probability that the number of contacts by a randomly chosen individual is exactly k, with L~o Ph = 1. In other words, {Ph}
Modeling the Start of a Disease Outbreak
125
is the degree distribution of the vertices of the graph corresponding to the population network. For convenience, we define the generating function 00
Go(z) =
2::>k Zk . k=O
Since L~o Pk = 1, this power series converges for 0 ~ be differentiated term by term. Thus
_ G~k)(O) k! '
Pk -
k
=
z ~ 1, and may
0,1,2, ....
It is easy to verify that the generating function has the properties
Go(O) = Po,
Go(1) = 1,
G~(z)
> 0,
G~(z)
> O.
< k >, is
The mean degree, which we denote by 00
< k >= :L kPk = G~(1). k=l
More generally, we define the moments 00
< k j >= :L kjPk,
j
= 1,2,···
,00.
k=l
When a disease is introduced into a network, we think of it as starting at a vertex (patient zero) who may transmit infection to every individual to whom this individual is connected, that is, along every edge of the graph from the vertex corresponding to this individual. For transmission of disease after the first generation we need to use the excess degree of a vertex. If we follow an edge to a vertex, the excess degree of this vertex is one less than the degree. We use the excess degree because infection can not be transmitted back along the edge whence it came. The probability of reaching a vertex of degree k, or excess degree (k - 1), by following a random edge is proportional to k, and thus the probability that a vertex at the end of a random edge has excess degree (k - 1) is a constant multiple of kPk with the constant chosen to make the sum over k of the probabilities equal to 1. Then the probability that a vertex has excess degree (k - 1) is kPk qk-l = < k >. This leads to a generating function G1(z) for the excess degree
Fred Brauer
126
and the mean excess degree, which we denote by 1
< ke >, is
00
2:
k (k -l)Pk < ke > = - k < > k=l 1
=
1
00
k=l
00
- - L k 2 Pk- - - L k p k
k=l
2
< k > _ 1 = G' (1).
3
1
Transmissibility
Contacts do not necessarily transmit infection. For each contact between individuals of whom one has been infected and the other is susceptible there is a probability that infection will actually be transmitted. This probability depends on such factors as the closeness of the contact, the infectivity of the member who has been infected, and the susceptibility of the susceptible member. We assume that there is a mean probability T, called the transmissibility, of transmission of infection. The transmissibility depends on the rate of contacts, the probability that a contact will transmit infection, the duration time of the infection, and the susceptibility. In this section, we will continue to assume that there is a network describing the contacts between members of the population whose degree distribution is given by the generating function Go (z), but we will assume in addition that there is a mean transmissibility T. If all contacts transmit infection, then T = 1. When disease begins in a network, it spreads to some of the vertices of the network. Edges that are infected during a disease outbreak are called occupied, and the size of the disease outbreak is the cluster of vertices connected to the initial vertex by a continuous chain of occupied edges. The probability that exactly m infections are transmitted by an infective vertex of degree k is
We define r 0 (z, T) be the generating function for the distribution of the number of occupied edges attached to a randomly chosen vertex, which is the same as the distribution of the infections transmitted by a randomly
Modeling the Start of a Disease Outbreak
127
chosen individual for any (fixed) transmissibility T. Then
r" (z, T)
~ %;, [~ p, (,~,) 1~'(1 - T),'-m)1zm
~
t, [t (!) p,
(zT)m(l - T),'-m)
1
(3, 1)
00
= LPk[zT + (1 -
T)]k
=
G o(l
+ (z -
l)T).
k=O
In this calculation we have used the binomial theorem to see that
Note that
ro(o, T) = G o(1- T),
ro(l, T) = G o(l) = 1,
r~(z, T) = TG~(l
+ (z -
l)T).
For secondary infections beyond the first generation we need the generating function rl(z,T) for the distribution of occupied edges leaving a vertex reached by following a randomly chosen edge. This is obtained from the excess degree distribution in the same way,
and rdO,T) = G I (l- T), r~ (z, T)
r l (l,T) =
= TG~ (1 + (z -
G I (l)
=
1,
l)T).
We let RI = r~ (1, T) = TG~ (1), the mean number of occupied edges. Although Ro = r~(l, T), the mean number of secondary cases infected by patient zero is the basic reproduction number as usually defined, the mean excess number of occupied edges is a more accurate description for the spread of disease and the threshold for an epidemic is determined by R I . If the incidence is mass action, the degree distribution is a Poisson distribution and Go(z) = GI(z), so that RI = Ro. For other degree distributions the values of RI and Ro could be quite different. Our next goal is to calculate the probability that the infection will die out and will not develop into a major epidemic. We begin by assuming that patient zero is a vertex of degree k. Suppose patient zero transmits infection to a vertex of degree j. We let Zn (T) denote the probability that
Fred Brauer
128
this infection dies out within the next n generations. The probability that there are m infections caused by a secondary vertex of degree j is
If there are m secondary infections coming from this vertex, for the infec-
tion to die out in n generations each of these secondary infections must die out within (n - 1) generations. The probability of this is Zn-l (T) for each secondary infection, and the probability that all secondary infections will die out in (n - 1) generations is [zn_l(T)]m. Thus the probability that all secondary infections from this vertex die out within (n - 1) generations is
Now zn(T) is the sum over j of these probabilities, weighted by the probability qj of j secondary infections. Thus
Since f1(z, T) is an increasing function of z, the sequence zn(T) is an increasing sequence and has a limit zoo(T), which is the probability that this infection will die out eventually. Then ZOO (T) is the limit as n ~ 00 of the solution of the difference equation
Thus Zoo (T) must be an equilibrium of this difference equation, that is, a solution of Z = f1(z, T). Let w be the smallest positive solution of Z = f1(z,T). Then, because f1(z,T) is an increasing function of z, k Z ~ f1(z,T) ~ f1(w,T) = w for 0 ~ Z ~ w. Since za ) = 0 < wand Z~~-;.l) (T) ~ w implies
it follows by induction that
z~k) (T) ~ w,
n
= 0,1, ...
,00.
k
= 1,2""
,00.
From this we deduce that
z~)(T) = w,
Modeling the Start of a Disease Outbreak
129
The equation r 1 (Z, T) = Z has a root Z = 1 since r 1 (1, T) = 1. Because the function r 1 (Z, T) - Z has a positive second derivative, its derivative r~ (z, T) - 1 is increasing and can have at most one zero. This implies that the equation r 1 (z, T) = z has at most two roots in 0 ~ z ~ 1. If Ro < 1 the function r1(z, T) - z has a negative first derivative r~(z,T) -1 ~ r~(l,T) -1
= TG~(l) -1 = Rl
-1 < 0
and the equation r1(z,T) = z has only one root, namely z = 1. On the other hand, if Rl > 1 the function r 1 (z, T) - z is positive for z = 0 and negative near z = 1 since it is zero at z = 1 and its derivative is positive for z < 1 and z near 1. Thus in this case the equation r 1 (z, T) = z has a second root ZOO (T) < 1. In either case, the limit z~:,l (T) is independent of k, and we denote it by zoo(T). The probability that the disease outbreak will die out eventually is the sum over k of the probabilities that the initial infection in a vertex of degree k will die out, weighted by the degree distribution {pd of the original infection, and this is 00
LPkZ~(T) = Go(zoo(T)). k=O
To summarize this analysis, we see that if Rl = TG~ (1) < 1 the probability that the infection will die out is 1. If Rl > 1 there is a solution ZOO (T) < 1 of
r1(z, T)
=
z,
and a probability 1 - ro(zoo(T), T) > 0 that the infection will persist, and will lead to an epidemic. However, there is a positive probability r 1 (zoo (T), T) that the infection will increase initially but will produce only a minor outbreak and will die out before triggering a major epidemic. Another interpretation of the basic reproduction number is that there is a critical transmissibility Tc defined by
In other words, the critical transmissibility is the transmissibility that makes the basic reproduction number equal to 1. If the mean transmissibility can be decreased below the critical transmissibility, then an epidemic can be prevented. The measures used to try to control an epidemic may include contact interventions, that is, measures affecting the network such as avoidance of public gatherings and rearrangement of the patterns of interaction between caregivers and patients in a hospital, and transmission interventions such as careful hand washing or face masks to decrease the probability that a contact will lead to disease transmission.
130
Fred Brauer
More sophisticated network analysis makes it possible to predict such quantities as the size of an epidemic, the probability that an individual will set off an epidemic, the risk for an individual of becoming infected, the probability that a cluster of infections will set off an epidemic small disease outbreak when the transmissibility is less than the critical transmissibility and how the probability of an epidemic depends on the degree of patient zero, the initial disease case [12, 14].
4
Some examples of contact networks
The above analysis assumes that there is a known generating function Go(z) or, equivalently, a degree distribution {Pk}. In studying a disease outbreak, we need to know the degree distribution of the network. If we know the degree distribution we can calculate the basic reproduction number and also the probability of an epidemic. What kinds of networks are observed in practice in social interactions? There are some standard examples. If contacts between members of the population are random, corresponding to the assumption of mass action in the transmission of disease, then the probabilities Pk are given by the Poisson distribution
e-cc k Pk =
-;;;!
for some constant c. To show this, we think of a probability of contact cD.t in a time interval D.t, and we let n
=
1 D.t.
Then the probability of k contacts in a time interval ,6.t is
where
(nk)=~n! k!(n - k)!
is the binomial coefficient. We rewrite this probability as n(n - l)(n - 2)··· (n - k nk
We let D.t
----+
0, or n
----+ 00.
+ 1) c k (1 -
Since
n(n - l)(n - 2)··· (n - k
+ 1)
~----'--.::.---'-:---'------'- ----+ nk
-;;,-)n
k!(l--;;'-)k·
1,
Modeling the Start of a Disease Outbreak and (1-
~r
->
e-
c
131
,
the limiting probability that there are k contacts is
Then the generating function is
L 00
Go(z)
= e- c
k
~! zk = e-ce cz =
ec(z-l) ,
k=O
and G~(z) = cec(z-l),
G~(l)
= c.
The generating function for the Poisson distribution is ec(z-l) and no TG~(l) = cT. We note also that for the Poisson distribution G1(z)
Go(z), n 1
=
= =
n ,.
The commonly observed situation that most infectives do not pass on infection but there are a few "superspreading events" [17] corresponds to a probability distribution that is quite different from a Poisson distribution, and could give a quite different probability that an epidemic will occur. For example, taking T = 1 for simplicity, if no = 2.5 the assumption of a Poisson distribution gives Zoo = 0.107 and Go(zoo) = 0.107, so that the probability of an epidemic is 0.893. The assumption that nine out of ten infectives do not transmit infection while the tenth transmits 25 infections gives
from which we see that the probability of an epidemic is 0.1. Another example, possibly more realistic, is to assume that a fraction (1 - p) of the population follows a Poisson distribution with constant r while the remaining fraction p consists of superspreaders each of whom makes L contacts. This would give a generating function
Go(z) = (1 - p)er(Z-l) so that no
+ pzL
= r(l - p) + pL, G1(z)
=
and
n1 =
r(l - p)er(z-l) + pLzL-l r(l-p)+pL ' r2(1 - p) + pL(L - 1) . r(l- p) + pL
Fred Brauer
132 For example, if r simulation gives
= 2.2, L = 10, P = 0.01, so that Ro = 2.278 numerical Rl
= 2.5,
Zoo
= 0.146,
so that the probability of an epidemic is 0.849. For network models, Rl is a better description of the spread of a disease out break than Ro. These examples demonstrate that the probability of a major epidemic depends strongly on the nature of the contact network. Simulations suggest that for a given value of the basic reproduction number the Poisson distribution is the one with the maximum probability of a major epidemic. It has been observed in many situations that there are a small number of long range conections in the graph, allowing rapid spread of infection. There is a high degree of clustering (some vertices with many edges) and there are short path lengths. Such a situation may arise if a disease is spread to a distant location by an air traveller. This type of network is called a small world network. Long range connections in a network can increase the likelihood of an epidemic dramatically. A third kind of network frequently observed is a scale free network. In a random network, the quantity Pk approaches zero very rapidly (exponentially) as k -+ 00. A scale free network has a "fatter tail", with Pk approaching zero as k -+ 00 but more slowly than in a random network. In an epidemic setting it corresponds to a situation in which there is an active core group but there are also "superspreaders" making many contacts. In a scale free network, Pk is proportional to k- a with a a constant. In practice, a is usually between 2 and 3. Often an exponential cutoff is introduced in applications of scale free networks in order to make G~(l) < 00 for every choice of a, so that Pk
= Ck-ae- k / e.
The constant C, chosen so that '2:.':=0 Pk = 1, can be expressed in terms of logarithmic integrals. These examples indicate that the probability of an epidemic depends strongly on the contact network at the beginning of a disease outbreak. The study of complex networks is a field which is developing very rapidly. Some basic references are [15, 18], and other references to particular kinds of networks include [1, 2, 13, 19]. Examination of the contact network in a disease outbreak situation may lead to an estimate of the probability distribution for the number of contacts [11, 12], and thus to a prediction of the course of the disease outbreak. A recent development in the study of networks in epidemic modeling is the construction of very detailed networks by observation of particular locations. The data that goes into such a network includes household sizes, age distributions, travel to schools, workplaces, and other public
Modeling the Start of a Disease Outbreak
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locations. The networks constructed are very complex but may offer a great deal of realism. However, it is very difficult to estimate how sensitive the predictions obtained from a model using such a complex network will be to small changes in the network. Nevertheless, simulations based on complicated networks are the primary models currently being used for developing strategies to cope with a potential influenza pandemic. This approach has been followed in [8,9, 10]. An alternative to simulations based on a very detailed network would be to analyze the behaviour of a model based on a simpler network, such as a random network or a scale-free network with parameters chosen to match the reproduction number corresponding to the detailed network. A truncated scale free network would have superspreaders and thus may be closer than a random network to what is often observed in actual epidemics.
5
Conclusions
We have described the beginning of a disease outbreak in terms of the degree distribution of a branching process, and have related this to a contact network. There is a developing theory of network epidemic models which is not confined to the early stages [12, 14]. This involves more complicated considerations, such as the way in which a contact network may change over the course of an epidemic. We have restricted our attention to the beginning of an epidemic in order not to have to examine these complications. Another extension would be to semi-directional networks with disease transmission in only one direction for some edges. For example, person A may go to a hospital only if infected, and may transmit infection to a health care worker B in hospital, but if A is not infected and never goes to the hospital to meet B, then A can not infect B. There are many aspects of network models for epidemics that have not yet been studied. While we have suggested using a deterministic compartmental model once an epidemic is underway, it may be reasonable to go beyond the simplest Kermack-McKendrick epidemic model. Heterogeneity of contact rates, age structure, and other aspects of an actual epidemic can be modeled. Ideally, for the initial stages of an epidemic we would like to use a network somewhere between the over-simplification of a random network and the extreme complication of an individual-based model.
References [lJ R. Albert, A.-L. Barabasi: Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47-97 (2002).
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[2] A.-L. Barabasi, R Albert: Emergence of scaling in random networks, Science 286, 509-512 (1999). [3] P. Erdos, A. Renyi: On random graphs, Publicationes Mathematicae 6, 290-297 (1959). [4] P. Erdos, A. Renyi: On the evolution of random graphs, Pub. Math. Inst. Hung. Acad. Science 5, 17-61 (1960). [5] P. Erdos, A. Renyi: On the strengths of connectedness of a random graph, Acta Math. Scientiae Hung. 12,261-267 (1961). [6] D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Network robustness and fragility: Percolation on random graphs, Phys. Rev. Letters 85,5468-5471 (2000). [7] O. Diekmann, J.A.P. Heesterbeek: Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester (2000). [8] N.M. Ferguson, D.A.T. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn, D.S. Burke: Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature 437, 209-214 (2005). [9] LM. Longini, M.E. Halloran, A. Nizam, Y. Yang: Containing pandemic influenza with antiviral agents, Am. J. Epidem. 159, 623633 (2004). [10] LM. Longini, A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D.A.T. Cummings, M.E. Halloran: Containing pandemic influenza at the source, Science 309, 1083-1087 (2005). [11] M.J. Keeling, K.T.D. Eames: Networks and epidemic models J. Roy. Soc. Interface 2, 295-307 (2006). [12] L.A. Meyers, B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, RC. Brunham: Network theory and SARS: predicting outbreak diversity. J. Theor. BioI. 232, 71-81 (2005). [13] A.L. Lloyd, RM. May: Epedemiology: How viruses spread among computers and people, Science 292,1316-1317 (2001). [14] M.E.J. Newman: The spread of epidemic disease on networks, Phys. Rev. E 66, 016128 (2002). [15] M.E.J. Newman: The structure and function of complex networks. SIAM Review 45, 167-256 (2003). [16] M.E.J. Newman, S.H. Strogatz, D.J. Watts: Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64, 026118 (2001). [17] S. Riley, C. Fraser, C.A. Donnelly, A.C. Ghani, L.J. Abu-Raddad, A.J. Hedley, G.M. Leung, L-M Ho, T-H Lam, T.Q. Thach, P.
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Chau, K-P Chan, S-V Lo, P-Y Leung, T. Tsang, W. Ho, K-H Lee, E.M.C. Lau, N.M. Ferguson, RM. Anderson: Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions, Science 300, 1961-1966 (2003). [18] S.H. Strogatz: Exploring complex networks. Nature 410, 268-276 (2001). [19] D.J. Watts, S.H. Strogatz: Collective dynamics of 'small world' networks, Nature 393, 440-442 (1998).
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Mathematical Techniques in the Evolutionary Epidemiology of Infectious Diseases Troy Day Departments of Mathematics/Statistics fj Biology, Queen's University Kingston, Ontario, K7L 3N6 Canada E-mail: [email protected]
Abstract I provide a brief introduction to two complimentary mathematical approaches for incorporating evolution into epidemiological models. These are referred to as the invasion-analysis technique and the Price-equation technique.
1
Introduction
The epidemiology of infectious diseases is a vibrant and growing area of research. Mathematics has come to play a central role in this field because it allows one to better understand disease dynamics and to assess the utility of different potential control measures. There have been many new developments and extensions of epidemiological models in recent years, including the development of models that account for pathogen evolution. In this chapter I provide a brief overview of two different ways in which the dynamics of pathogen evolution can be incorporated into epidemiological models. The need for incorporating evolution into epidemiological models of infectious disease stems, in part, from the high levels of genetic variation that are often generated through mutation and recombination during individual infections. These different genetic strains can have very different epidemiological characteristics, and therefore an accurate prediction of the epidemiological dynamics often cannot be made without accounting for this variation. Most epidemiological models follow the seminal work of Kermack and McKendrick [12] and are referred to as compartment models [l1J. In such models, the host population is divided into mutually exclusive classes (e.g., susceptible, infected, recovered, etc), and dynamical equa-
Mathematical Techniques in the Evolutionary Epidemiology· . .
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tions are developed to model the flow of individuals among the different classes. In this way, the dynamics of the pathogen itself are not explicitly modeled, but rather the epidemiological consequences at the level of the host population are tracked, and related to various aspects of infection at the level of the individual host (e.g., transmission rate between hosts, parasite induced mortality rate, recovery rate, etc). Most epidemiological models of pathogen evolution have been developed from compartment models, because the ultimate interest again is typically the epidemiological dynamics at the level of the host population. The challenge, therefore, has been to bridge the scale of pathogen replication and evolution within hosts (which is where all genetic variation is generated), to the scale of pathogen evolution and replication between hosts. Some strains of pathogens might be very good competitors within hosts but very poor at transmitting to new hosts and vice versa. In evolutionary terms, selection on the pathogen population acts at different levels of biological organization (i.e., within-host level and the between-host level). One of the key obstacles to developing a complete theory for the evolution of infectious diseases is this complexity of "multi-level selection" . To make any progress, some simplifications are necessary. Because this chapter is meant to be an introductory overview, I will make an extreme simplification and assume that selection acts only at the between-host level. In other words, I will suppose that any given infected host harbours only a single strain of pathogen at any given time. This strain type might change due to mutation, but I assume that if a mutant strain ever does arise within a host, it either dies out, or it displaces the original strain instantaneously [2, 18, 4]. Consequently, the evolution of the pathogen population occurs solely as a result of differences among strains in their ability to transmit from host-to-host, as well as differences in the mortality they induce and their susceptibility to clearance by host immunological responses. More realistic extensions of the techniques to be presented here have been developed [19, 18, 4, 16, 5], but this simple case is sufficient for introductory purposes.
2
Mathematical models of pathogen evolution
The two approaches for modeling pathogen evolution to be presented will be referred to as the 'Invasion-analysis' technique and the 'Priceequation' technique. The invasion-analysis approach is based on an assumption that evolutionary change is very slow relative to the timescale of the epidemiological dynamics. More specifically, it assumes that the epidemiological dynamics always reach their limiting behavior (typically
Troy Day
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c
assumed to be a point equilibrium) in between the appearance of successive mutations. As a result, there is only ever at most two-strains circulating in the population at any given time. The other key feature of this ~pproach is that it typically seeks to identify the endpoints of evolution only (as will be described below) but not to provide any information about the evolutionary dynamics that occur along the way. These simplifications allow for a relatively complete mathematical analysis of evolution in many contexts. The Price-equation technique is more complex but allows for any time scale of evolutionary change. It supposes that there are multiple strains present at any given time, and that mutations continually occur among these. It then tracks the simultaneous epidemiological and evolutionary dynamics. This is clearly a more realistic approach, but it comes with the drawback that rarely can a complete mathematical analysis be conducted. Rather, this approach is best suited towards providing qualitative insights into the epidemiological and evolutionary dynamics, although numerical analysis can also sometimes be used to gain quantitative insights as well. The above distinction is a bit artificial, and one can readily imagine constructing models with elements of both approaches. Nevertheless this distinction is useful because, in practice, most published models tend to use one or the other approach. In what follows I will use a very simple toy epidemiological model to highlight the main ingredients of each of these two approaches. To give the examples a concrete grounding in some epidemiological question of interest, I will focus on the issue of virulence evolution. This has been a question of considerable interest in evolutionary epidemiology, and is one for which both of the approaches have been used.
2.1
The underlying epidemiological model
Many pathogens infect their hosts without causing substantial damage (e.g., many rhinoviruses that cause the common cold) while others induce higher levels of mortality (e.g., flavivurses that cause dengue fever). One explanation for this variation in pathogen virulence is that the costs and benefits of pathogen-induced mortality vary among pathogens, and this has resulted in the evolution of different levels of virulence. I will work with a simple epidemiological model that is meant to explore this possibility, and to allow predictions of the level of virulence that we expect to evolve. Consider a simple 81 model, where S and I are the numbers of sus-
Mathematical Techniques in the Evolutionary Epidemiology. . .
139
ceptible and infected hosts respectively: dS -=edt
dI
II
B-S(31 ,
~
dt = S(31 -
(2.1)
(J.L + lI)I.
The parameter e is the immigration rate of susceptible hosts, J.L is the per capita background mortality rate of hosts, 1I is the increased mortality rate of hosts due to infection (the virulence), and (3 is the transmission rate (assuming mass-action transmission). This model has two equilibria, one in which the pathogen is absentE1 := (e / J.L, 0) and the other in which the pathogen is endemic E2 := ((J.L+1I)/(3, ~). The diseasefree equilibrium, E 1 , is always biologically feasibl:' whereas the endemic equilibrium, E 2 , is feasible if and only if Ro > 1 where Ro = !l.L. FurJ.LJ.L+v thermore, it can be shown (e.g., by the method of Lyapunov functions; [13]; Appendix) that, when E2 is feasible, it is globally asymptotically stable. Otherwise, El is globally asymptotically stable. It is this model that will be used in presenting the two techniques below.
!v -
2.2
Invasion analysis technique
The technique of invasion analysis a quite general approach for modeling evolution (see [20]) but here I will discuss it within the context of the above epidemiological model. Suppose that all pathogen strains can be characterized by their transmission rate, (3, as well as the level of virulence, 1I, that they induce. Thus, there is also a value of Ro specific to each strain. I will restrict attention to those strains having a value of Ro larger than one. The underlying logic of invasion analysis is as follows. Suppose that a single strain is currently present in the population, and that it has reached its endemic steady state of (2.1). Now imagine that a small number of individuals carrying a second strain are introduced into the population. An invasion analysis seeks to determine whether this new strain invades or dies out. More specifically, it seeks to determine if there is a strain that, once present at endemic levels, can resist invasion by all possible mutants that might arise. If so, it is reasoned that this is a plausible endpoint of evolution because, once here, no further evolutionary change can occur. Such strains are said to be evolutionarily stable (ES). To look for an ES strain for model (2.1) we need to augment this model to allow for a second strain. Using a subscript 'm' to denote the
140
Troy Day
mutant parameter values, we have
~~ = () dI dt
=
dIm dt
p,8 - 8(31 - 8(3mIm,
8(31 - (11 + v)I,
(2.2)
= 8(3m Im - (11 + vm)Im.
System (2.2) implicitly assumes that the only way in which the two strains interact is through competition for the infection of a common pool of susceptible hosts. As expected, one equilibrium of the augmented system (2.2) is S = fl+ V , i = JL!v -~, and im = 0, and it is the stability of this equilibrium tgat determines whether or not the mutant strain can invade. If and only if this equilibrium is asymptotically stable for all possible mutants is the resident strain ES. Model (2.2) is simple enough that a complete, global analysis is possible (Appendix). For most models, however, only a local analysis can be done and therefore I will focus on such local results here to illustrate the general approach. In this case we are looking for locally evolutionarily stable (LES) strains. A linear stability analysis of system (2.2) at the equilibrium S = flt, i = JL!v -~, im = 0 yields the following Jacobian matrix:
(
-I1,-i(3
-8(3
1(3
-11-v+8(3
o
0
(2.3a)
To appreciate its structure, rewrite matrix (2.3a) in a way that emphasizes its blocktriangular form: (2.3b)
where -0 := (0 0),
u :=
(-8(3m) 0 ' J mut := -11 - Vm
~ + 8(3m,
and J res
is given by J res := (-P,I'-(3i(3
- 8(3 ) -11- v +8(3 .
(2.4)
Thus, the eigenvalues of (2.3a) are simply the eigenvalues of the diagonal blocks, J res and J mut . The notation J res emphasizes the fact that, a linear stability analysis of system (2.1) at the endemic equilibrium E2 yields a Jacobian matrix
Mathematical Techniques in the Evolutionary Epidemiology·..
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that is exactly (2.4). Because we have assumed that the epidemiological dynamics reach a stable equilibrium when only the resident strain is present, we know that the eigenvalues of (2.4) have negative real parts (throughout I will focus only on hyperbolic equilibria). Therefore, the stability of the equilibrium when we introduce the mutant type is completely determined by the leading eigenvalue of the submatrix J mut . In this case, this submatrix is simply a single element and thus the eigenvalue is trivially equal to r := -p, - Vm + S(3m. Now, as mentioned, the resident strain is LES if and only if this equilibrium is locally asymptotically stable. Thus, we have
+ S(3m < 0, >~,
LES {::} -p, - Vm LES {::}; S
p,+vm
(2.5)
LES {::} R > R m , where LES is shorthand for the statement "The resident strain is locally evolutionarily stable", and where I have defined R = (3/(p, + v) and Rm = (3m/(P, + vm ). Thus, the resident strain is locally evolutionarily stable if and only if it has the largest value of (3d(p, + Vi) of all possible strains, i. These results can be shown to hold globally for this particular model as well (Appendix). This general approach has been used in a very wide variety of epidemiological models to characterize evolutionarily stable strains (see references in [5]). In the present case, we have seen that evolution maximizes a quantity (R in this case) and we can use this fact to elucidate important properties of pathogen evolution. In other more complex models, however, there need not be a simple maximization principle such as this. Rather, the inequality in (2.5) for more complex models often cannot be separated into terms solely involving mutant parameters on one side and resident parameters on the other. In this case, slightly more sophisticated analyses are required that are beyond the scope of this overview (interested readers should consult Otto and Day 2007). Let us now see how the above maximization principle can be used to understand pathogen evolution. To begin, we can immediately see that strains with very high transmission and very low virulence will be best (i.e., they have the largest value of R). For some pathogens, however, it is not possible for a strain to have a high transmission rate without also inducing a high mortality rate [6, 1, 15, 17, 14, 7]. The simplest way to account for this constraint is to suppose that transmission rate is an increasing function of virulence. In this case we can then seek the level of virulence that maximizes R = (3(v)/(p, + v). So long as the function (3(v) increases at a diminishing rate (i.e., d 2 (3/dv 2 < 0), Rwill be maximized at an intermediate level of virulence, v*. Furthermore,
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142
under these conditions it is easy to see that the ES level of virulence is an increasing function of host mortality rate, JL. In particular, v* must satisfy the following, first order necessary condition: d,6 dv
,6(v) JL
(2.6)
+v 2
Since d,6/dv decreases with increasing v (Le., d ,6/dv2 < 0), a simple implicit differentiation argument using (2.6) shows that that value of v satisfying (2.6) is larger, for larger values of JL. Thus, host species with high levels of natural mortality are predicted to harbour highly virulence pathogens.
2.3
Price equation technique
Price's equation has been widely used in evolutionary biology to model the dynamics of allele frequencies [22, 3, 8, 21]. Recently, however, it has been adapted to model the dynamics of the frequency of different pathogen strains in epidemiological models as well [4, 5]. To describe this approach let us return to the toy model (2.1) and first extend it to allow for n pathogen strains. We have dS
cit
= () -
dI· dt'
= S,6Ji - (JL + Vi) h
JLS - S L-i ,6ih (2.7) Vi E {I, 2, ... , n}.
System (2.7) consists of n+l equations; one for each type of infected host, and one for the dynamics of susceptible hosts. To maintain genetic variation in strains within the population, we now further extend system (2.7) to allow for the occurrence of mutation. Biologically, when mutations arise, they do so in a host that is already infected with some other genotype of pathogen. This then creates a host harbouring more than one type of pathogen. In order to maintain a simplified model in which hosts only ever contain a single pathogen, we therefore assume that such mutations either die out or supplant the original strain instantaneously. 'Mutation' in the model therefore really represents a change from one genotype of infection to another. Thus, as is common in many models of evolutionary epidemiology, we assume that a polymorphism is never maintained within a host [2, 19, 9, 10, 18]. Extending system (2.7) to incorporate this type of mutation, we have dS
cit = () - JLS - S L-i ,6i Ii,
~i = S,6i Ii -
(JL + Vi) Ii - rtIi
Vi,j E {1,2, ... ,n}.
+ rt L-j mjiIj,
(2.8)
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Here ry is the rate at which infections change genotype through mutation, and mji is the probability that, given such a change, an infection of genotype j changes to one of genotype i. System (2.8) completely specifies the dynamics of the different strain types. From an evolutionary standpoint, however, it is often more useful to change variables and to track these dynamics in terms of the frequencies of the different strain types. Defining qi := Id Ir as the frequency of strain type i, where Ir = L::i Ii, we have
dqi dt
-
dIddt dIT/dt Ir Ir = qi (Ti - r) - ryqi + ry L:: j mjiqj,
= ---qi---
(2.9)
where Ti = S(3i - f.l - Vi is referred to as the 'fitness' of strain i, and r := L::i qiTi is the mean fitness of all strains. Let's now step back for a moment and see what we have done. Given n strains of pathogen, we now have n - 1 equations for the dynamics of their frequencies (since L::i qi = 1). To completely specify system (2.8) in these new variables, however, we must also track the dynamics of the total number of infected individuals, as well as the number of susceptible individuals. Using (2.8) this gives
dS
cit = dIT
cit =
B - f.lS - S(3IT , S(3Ir - (f.l + v) Ir,
(2.10)
where x := L::i qiXi, and where I have made use of the fact that L::i mji = 1. System (2.10) provides the final two equations required to completely specify the dynamics (bringing the total number of equations back to n
+1). At this stage it might seem strange to employ this change of variables because it has resulted in a system of equations that is more complex than the original system (2.7). In fact, if we were primarily interested in the dynamics of the number of each strain type this would not be a useful change of variables. If, however, we are primarily interested in the evolution of some characteristic of the pathogen (e.g., its transmission rate or its virulence) then this change of variables does prove to be useful because we can now readily derive equations for the dynamics of the average value of these characteristics across all pathogens. Furthermore, this change of variables has also separated the epidemiological dynamics of the system (given by equations (2.10)) from the evolutionary dynamics of the system (given by equations (2.9)). As we will see, this is also a useful thing to do.
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To derive equations for the evolutionary dynamics of the mean level of virulence and transmission, we simply need to differentiate D = I:i qiVi and 13 = I:i qi(3i with respect to time. Doing so, and using (2.9) to simplify the result, we get [4]
(2.11a)
or (2.11b)
Here O'xy is the covariance between x and y across the pathogen strains that are circulating in the population, and xm := I:i,j Ximjiqj is the average value of trait x among all new mutations. Equations (2.11) are versions of Price's equation [22, 3, 8, 21] and each has a useful interpretation. The average trait value in the population changes as a result of two processes. First, the average trait value changes in a direction given by the sign of the covariance between the trait and fitness. For example, the average value of transmission is driven upward by the fact that strains with large values of transmission, (3, tend to have higher fitness (the term SO'{3{3), but it is also affected by the fact that strains with high virulence have lower fitness, and virulence might be genetically correlated with transmission across parasite strains (the term -a(3v)' Second, the average trait value changes in a direction governed by any mutational bias that might occur (e.g., the term -"1(13 -13m) for the dynamics of
13).
The variances and covariances in system (2.11), O'xy, will also change through time, and equations for these dynamics will typically depend on higher moments of the strain distribution. So in this sense, equations (2.11) cannot be immediately solved to obtain the evolutionary dynamics of the average values of v and (3. Nevertheless, system (2.11) can be used to gain some important insights into pathogen evolution without requiring a full solution. To see how, it is useful to write equations (2.11) in matrix notation:
(2.12)
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145
where G is the genetic (co )variance matrix and ( -1 S) T is termed the selection gradient. The product of G with the selection gradient in equation (2.12) describes the way in which natural selection changes the average level of virulence and transmission in the pathogen population. Natural selection always favours reduced virulence with a strength of -1. On the other hand, natural selection always favours an increased transmission rate with a strength that is proportional to the density of susceptible hosts, S. At equilibrium the force of mutation must balance the force of natural selection, as mediated through the genetic covariance structure of the pathogen population [4]. Interestingly, this formulation also separates the effects of the epidemiological dynamics on evolution (represented here by the selection gradient vector) and the genetic structure of the pathogen population (represented here by the genetic covariance matrix). In the invasion analysis approach, we were only able to infer properties of the endpoint of evolution, and therefore we needed to make an assumption of a tradeoff between virulence and transmission for such an endpoint to exist. Here, however, we can make predictions about the evolutionary dynamics regardless of whether such tradeoffs exist. Of course, we still do not expect there to be an intermediate equilibrium level of virulence unless, ultimately, some sort of constraint between ever increasing transmission and ever decreasing virulence occurs. In the Price-equation approach, this would be manifest as a positive covariance between the two traits. Let us now suppose such a constraint occurs, and return to the question of how natural host mortality rate affects virulence evolution using the Price equation approach. We saw previously (using an invasion analysis) that high mortality selects for the evolution of high virulence. An examination of equations (2.11) however, reveals that host mortality /1 does not enter directly into the evolutionary dynamics. Therefore, such mortality affects virulence evolution only if it indirectly affects either the genetics of the pathogen population, or the selection gradient (-1 S)T. We typically do not expect host mortality to significantly alter the genetics of the pathogen population, and therefore the only way in which host mortality affects virulence evolution is indirectly through the epidemiological dynamics. For example, if we assume that the epidemiological dynamics are always approximately at equilibrium (as in the invasion analysis), then from (2.10) we have S ~ (/1 + D) / jJ. Thus, higher host mortality rates lead, indirectly, to a higher number of susceptible hosts. This, in turn, increases the advantage of strains with higher transmission rate, and to the extent that transmission and virulence are positively correlated with one another, this leads to the evolution of higher virulence. Thus, the Price equation approach provides a more mechanistic picture
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of the factors that govern the evolution of pathogen populations. For example, if we were to test these predictions by experimentally elevating host mortality rate and measuring evolutionary changes in pathogen virulence, the invasion analysis would lead us to believe that higher virulence is always expected. The Price equation approach, however, reveals that this will only be true if our experimental manipulation allows this change in host mortality to indirectly feed back, through the epidemiological dynamics, to elevate the number of susceptible hosts (something that might well not occur in all experimental systems).
3
Discussion
This chapter has presented a brief overview of two different approaches to modeling evolutionary epidemiology. I have concentrated on a very simple toy epidemiological model that is not meant to represent the dynamics of any particular disease. Rather, it is simply meant as a tool to elucidate the similarities and differences of the two techniques. The interested reader should consult [4, 5] for more complex and realistic examples. The invasion analysis technique presented is somewhat restricted in its scope and does not provide complete information about the mechanistic details driving evolution in pathogen populations. It main advantage, however, is that it provides an analytically tractable approach for modeling pathogen evolution. The Price equation approach is more general in the sense that it makes fewer restrictive assumptions, and it also provides a more complete description of the mechanistic details of evolution. Its primary drawback, however, is that it rarely allows for a complete, analytical treatment of pathogen evolution. Rather, it is best at providing qualitative insights. As such these two approaches are best viewed as complimentary techniques.
Appendix Consider the following function
V(8, 1)
=8
-
8 In (818) + 1 -
j In (111),
(A1)
where Sand j are the equilibrium values of 8 and 1 at the endemic equilibrium, E 2 · Expression (A1) is a Lyapunov function for system (2.1) [13]. To see this, note that this function has a unique minimum in 8 and 1 at the equilibrium values Sand 1. Therefore, all we need to show is that dV/dt :::;; 0 along all trajectories of system (2.1), with
Mathematical Techniques in the Evolutionary Epidemiology··· equality holding only when S using equations (2.1) gives
d V dt
=
I
= 1.
dS + (1-!) ( S8) - ~
(1-~) S
=e
= 8,
dt
I
147
Differentiating (A1), and
dI dt (A2)
J-l S + J-l S - j3 S 1+ (J.l + 1/)1.
1-
A
A
Now, using the fact that j3Si = j38i~ = (e-J-l8)~ and e = J-l8+(J-l+I/)i, (A2) can be re-written as
dV dt
= e (2 -
§.) :;(
~S _ 8 '" 0,
(A3)
s.
with equality holding only when S = We can also use an extension of the Lyapunov function provided by [13] to demonstrate that the local results given in (2.5) of the text, hold globally as well. Consider the following function
V(S, I, 1m) = S -
8 In (SI8) + I
- i In (I I i)
+ 1m ,
(A4)
where Sand i are the equilibrium values of S and I at the endemic equilibrium, E 2 • Expression (A4) has a unique minimum in S, I, and 1m at (8, i, 0). The time derivative of (A4) along trajectories of system (2.2) is
(A5)
Now suppose that R > Rm. Expression (A5) is then clearly negative everywhere except when S = 8, 1m = 0, in which case it is zero. Thus, (A4) is a Lyapunov function for equilibrium (S, i, 0) of system (~.~) if R> Rm. If R < R m , then we already know that the equilibrium (S, I, 0) is unstable. Thus, result (2.5) of the text holds globally.
References [1] R.M. Anderson, R.M. May, Coevolution Of Hosts and Parasites, Parasitology, 85 (1982), 411-426.
Troy Day
148
[2] S. Bonhoeffer, M.A. Nowak, Mutation and the Evolution of Virulence, Proceedings of the Royal Society of London Series BBiological Sciences, 258 (1994), 133-140. [3] M. Bulmer, Theoretical evolutionary ecology, Sinauer Associates, Sunderland Massachusetts, 1994. [4] T. Day, S. Gandon, Insights from Price's equation into evolutionary epidemiology, in Feng, Dieckmann and Levin, eds., Disease Evolution: Models, Concepts, and Data Analysis, AMS, 2006. [5] T. Day, S.R Proulx, A general theory for the evolutionary dynamics of virulence, American Naturalist, 163 (2004), E40-E63. [6J D. Ebert, Virulence and Local Adaptation of a Horizontally Transmitted Parasite, Science, 265 (1994), 1084-1086. [7] D. Ebert, K.L. Mangin, The influence of host demography on the evolution of virulence of a microsporidian gut parasite, Evolution, 51 (1997),1828-1837. [8] S.A. Frank, George Price's Contributions to Evolutionary Genetics, Journal of Theoretical Biology, 175 (1995), 373-388. [9] S. Gandon, V.A.A. Jansen, M. van Baalen, Host life history and the evolution of parasite virulence, Evolution, 55 (2001), 1056-1062. [10] S. Gandon, M. van Baalen, V.A.A. Jansen, The evolution of parasite virulence, superinfection, and host resistance, American Naturalist, 159 (2002), 658-669.
[11] H.W. Hethcote, The mathematics of infectious diseases, Siam Review, 42 (2000), 599-653. [12] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics, part 1, Proceedings of the Royal Society of London, Series A (1927), 700-72l. [13J A. Korobeinikov, G.C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Applied Mathematics Letters (2002), 955-960. [14] M. Lipsitch, E.R Moxon, Virulence and transmissibility of pathogens: What is the relationship?, Trends in Microbiology, 5 (1997), 31-37. [15] M.J. Mackinnon, A.F. Read, Selection for high and low virulence in the malaria parasite Plasmodium chabaudi, Proceedings of the Royal Society of London Series B-Biological Sciences, 266 (1999), 741-748. [16] RM. May, M.A. Nowak, Coinfection and the Evolution of Parasite Virulence, Proceedings of the Royal Society of London Series BBiological Sciences, 261 (1995), 209-215.
Mathematical Techniques in the Evolutionary Epidemiology···
149
[17] S.L. Messenger, 1.J. Molineux, J.J. Bull, Virulence evolution in a virus obeys a trade-off, Proceedings Of the Royal Society Of London Series B-Biological Sciences, 266 (1999), 397-404. [18] J. Mosquera, F.R. Adler, Evolution of virulence: a unified framework for coinfection and superinfection, Journal of Theoretical Biology, 195 (1998), 293-313. [19] M.A. Nowak, R.M. May, Superinfection and the Evolution of Parasite Virulence, Proceedings of the Royal Society of London Series B-Biological Sciences, 255 (1994), 81-89. [20] S.P. Otto, T. Day, A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, Princeton University Press, Princeton, N.J., U.S.A., 2007. [21] G. Price, Selection and covariance, Nature, 227 (1970), 520-52l. [22] G. Price, Extension of covariance selection mathematics, Annals of Human Genetics, 35 (1972), 485-490.
150
The Uses of Epidemiological Models in the Study of Disease Control Zhilan Feng* Department of Mathematics, Purdue University West Lafayette, IN 47907, USA E-mail: [email protected]
Dashun Xu Department of Mathematics, Southern Illinois University Carbondale, IL 62901, USA E-mail: [email protected]
Haiyun Zhao Department of Mathematics, Purdue University West Lafayette, IN 47907, USA E-mail: [email protected]
Abstract Recently many mathematical models have been used to study the effectiveness of quarantine and isolation as control measures for the spread of infectious diseases. Most of the deterministic models have focused on the use of ordinary differential equation (ODE) models with the assumption of exponentially distributed disease stages. In this paper we demonstrate that some of these models may generate misleading predictions. We formulate a general integral equation model which assumes an arbitrarily distributed disease stage for both the latent and the infectious stages, and show that it reduces to the commonly used SEIR model (ODE) with quarantine and isolation when the stage distributions are exponential. The general model reduces to another ODE model when the disease stages are assumed to have a gamma distribution. The control reproductive number Rc for each model is calculated. The effect of control strategies of these models is compared in terms of both Rc and the final epidemic size. We demonstrate that the two ODE models can produce conflict pre-This author is partly supported by NSF DMS-0314575.
The Uses of Epidemiological Models in the Study of . . .
151
dictions regarding the effectiveness of disease control strategies via quarantine and isolation.
1
Introduction
Recently, many mathematical models have been used to investigate how to more effectively control emerging and reemerging infectious diseases such as SARS and smallpox via various disease control measures including vaccination, quarantine, and isolation (see, for example, Chowell, et al., 2003; Lipsitch, et al., 2003; McLean, et al., 2005; Riley, et al., 2003). Most of the studies have chosen to use simple deterministic ODE models of the SIR or S E I R type or their variations. While such simple models can very often provide many important insights into the disease transmission dynamics and have been used to address other important biological questions, the underlying assumptions used to obtain the simplicity of these model need to be examined carefully in order to make sure that the model predictions are reliable. One of the most commonly used simplifying assumptions is the exponential distribution assumption (EDA) on disease stages. In Feng et al. (2006), the appropriateness of using simple SEIR type of models to assess disease control strategies is investigated. They considered several epidemiological models with quarantine and isolation, but only a special case is considered b = 0). In this paper, we study a more general case in which early quarantine is included b> 0). The most basic and simple model is an extension of the standard SEIR compartmental model with the inclusion of quarantine (Q) and isolation classes, which is an ODE model (see (2.1)). This model uses the EDA for both the latent and infectious stages. Another ODE model with a gamma distribution for the disease stages is also considered (see (2.2)). The gamma distribution assumption (GDA) in general provides a more accurate description of the epidemiological process than the exponential distribution assumption. Hence, in many cases the gamma distribution model (GDM) is more realistic than the exponential distribution model (EDM). On the other hand, the GDM is more complex than the EDM and the mathematical analysis is more difficult. We demonstrate that the two models may produce conflict predictions, suggesting that the simple EDM may not be appropriate to use for assessing disease control programs. Following the approach of Feng et al. (2006) we also consider a general integral equation model by using arbitrarily distributed disease stages. It is shown that the general model reduces to the EDM or the GDM when the corresponding disease stage distribution is used. The results show that similar conclusions obtained in Feng et al. (2006) still hold for the case of"( > o.
152
2
Zhilan Feng, Dashun Xu, Haiyun Zhao
Models with quarantine and isolation
In the standard S E I R model, the total population is divided into four epidemiological classes: the susceptible (S), exposed (E, individuals who are infected but not yet infectious), infectious (I, individuals who are capable of transmitting the disease), and removed (R). Under the assumption that the latent and infectious stages are exponential distributed the corresponding ODE model is
S' = p,N - {3Sf:t - p,S, E' = {3Sf:t - (a1 + p,)E, I' = alE - (51 + p,)I, R' = 511 - p,R.
(2.1)
" I " denotes the derivative with respect to time t. {3 is the transmission coefficient, a1 is the rate at which a latent individual becomes infectious, 51 is the recover rate and p, is the natural death rate. No disease-induced death is considered. All involved parameters are nonnegative constants, and all variables and parameters are listed in Table 1. Many extensions ofthe simple model (2.1) have been used to address important biological questions. One of such examples is obtained by incorporating quarantine and isolation as below:
S' = p,N - ).,(t)S - p,S + rSQ, So = (1 - b)).,(t)S - rSQ, E' = (1 - "()b).,(t)S - (X + a1 + p,)E, Q' = "(b).,(t)S + XE - (a2 + p,)Q, l' = alE - (> + 51 + p,)I, H' = a2Q + >1 - (52 + p,)H, R' = 511 + 52H - p,R, where
).,(t)
=
(3 [I(t)
+
(1; P)H(t)].
(2.2)
(2.3)
).,(t) denote the force of infection (a specific form is given below). In this model it is assumed that a fraction b of contacts (susceptible individuals who have had contacts with an infectious person) is actually infected, and that the other fraction (1- b) of contacts remains susceptible who will be quarantined (SQ) and will return to the S class at a rate r (see, e.g., Lipsitch, 2003). Among the infected individuals (b).,(t)S) a fraction "( will be quarantined (Q) at the early stage of infection (i.e., there is a rate, "(b).,(t)S, from S to Q directly). The fraction (1 - "()
The Uses of Epidemiological Models in the Study of . . .
153
of the exposed individuals who are not quarantined at the beginning of infection will be quarantined at a constant rate X throughout the latent period. The non-quarantined and quarantined (exposed) individuals will progress to the infectious stage at constant rates Q:1 and Q:2 respectively (the relationship between Q:1 and Q:2 will be discussed later). Infectious individuals will be isolated (H) at a rate ¢> and individuals in the H class will recover at a rate (h (the relationship between 81 and 82 will be discussed later). p E [0,1] is the fraction of reduction in the transmission rate of isolated individuals with p = 1, P = 0, and 0 < p < 1 representing a completely effective, completely ineffective, and partially effective isolation, respectively. It is known that the ordinary differential equation model (2.2) implicitly assumes the exponential distribution for the latent and infectious stages. More precisely, the exponential functions pE(S) = e- a1S and PJ(s) = e-
1
1
00
TE
=
o
PE(s)ds
= -, Q:1
TJ
roo
= Jo
Similarly, the mean sojourn times in the Q
1
= 81 '
(2.4)
classes are TQ
=~
pJ(s)ds
~nd H
Q:2
and TH
=
1 8 , respectively. The memory-less property of the exponential 2
distribution implies that TH Q:2
=
= TJ and TQ = T E , which is equivalent to
Q:1
=:
Q:,
82 = 81 =: 8
(see Feng et al. 2006 for more detailed explanations). It has been argued that the exponential distribution assumption, while providing a good approximation of the process in many examples and capable of making the model very simple, may not be appropriate in some cases. Non-exponential distributions have been previously considered in epidemiological models (see, for example, Hethcote and TUdor, 1980; Hethcote, et al., 1981; Lloyd, 200la, 200lb; Plant and Wilson, 1986; Taylor and Karlin, 1998; Feng, et al., 2001; and Feng and Thieme, 200la, 2001b). However, none of these studies focuses on the evaluation of disease intervention policies. It has been observed that models with more realistic assumptions on disease stages may produce outcomes that are inconsistent with that of models with the EDA. The appropriateness of the EDA in models with quarantine and isolation is examined in Feng et al. (2006). Their results are obtained for only a special case when there is no early quarantine b = 0). In this paper, we extend their results by including the more general case when
154
Zhilan Feng, Dashun Xu, Haiyun Zhao
'Y > O. This corresponds to a more aggressive control program which encourages early identification and quarantine of exposed individuals.
To incorporate early quarantine we modify the integral equations model in Feng et al. (2006) by allowing a fraction 'Y > 0 of infected individuals to be quarantined at their early exposure. The remaining fraction 1 = 1 - 'Y is moved into the latent class E and these individuals will either progress to enter the infectious class J or get quarantined at some later stage age s > O. The quarantined (infected) individuals will progress to the disease stage and will remain isolated. The effect of early quarantine will in general reduce the prevalence of the disease since the quarantined individuals will have a reduced transmission rate due to limited contacts with the general population. We adopt the same assumption as in Feng et al. (2006) to assume that b = 1. Then our model reads:
S(t)
=
lot p,Ne-/-L(t-s)ds -lo\(s)s(s)e-/-L(t-S)dS + Soe-/-L t ,
E(t)
=
1lot >..(s)S(S)PE(t - s)k(t - s)e-/-L(t-s)ds + E(t),
Q(t) = 11t
17
>..(s)S(s) [-PE(T - S)k(T - S)]PE(t - TIT -
xe-/-L(t-S)dsdT + 'Y
It
s)
>..(s)S(S)PE(t - s)e-/-L(t-S)ds + Q(t),
J(t) = 1lot lo1" >..(s)S(s) [-FE(T - S)k(T - s)]Pr(t - T)l(t - T) xe-/-L(t-S)dsdT + J(t), H(t)
=
1lot loU lo7 >..(s)S(s) [-FE(T - S)k(T - s)] X
[-Pr(u - T)i(u - T)]Pr (t - ulu - T)e-/-L(t-S)dsdTdu
t +11 lo7 >"(s)S(S)[-FE(T - S)k(T - s)]Pr(t - T)e-/-L(t-S)dsdT
+'Y R(t) =
r r>"(s)S(S)[-FE(T - s)]Pr(t - T)e-/-L(t-S)ds + H(t) '
io io
It17
>"(s)S(S)[-FE(T - s)][l - Pr(t - T)] xe-/-L(t-S)dsdT + R(t), (2.5)
The Uses of Epidemiological Models in the Study of . . . where >..(t)
I(t)
+ (1 -
155
p)H(t)
N . In this model, PE,P] : [0,00) -+ [0,1] describe the durations of the exposed (or latent) and infectious stages, respectively. More precisely, Pi(S) (i = E, I) gives the probability that the disease stage i lasts longer than S time units (or the probability of being still in the same stage at stage age s). Therefore, the derivative - Pi (8) (i = E, 1) gives the rate of removal from the stage i at stage age 8 by the natural progression of the disease. These duration functions have the following properties =
(3
It is assumed that latent individuals of stage age
8 are quarantined according to a given distribution described by k(8) : [0,00) -+ [0,1] with k(O) = 1 and k(oo) = o. That is, k(8) denotes the probability that exposed individuals have not been quarantined at stage age 8. Similarly, the function l(8) : [0,00) -+ [0,1] with l(O) = 1 and l(oo) = 0 describes the probability that infectious individuals have not been isolated at stage age 8. f.1 is the natural death rate and the disease-induced mortality is ignored. All variables and parameters are listed in Table 1. X(t) = Xo(t)e-/l t + Xo(t) (X = Q, I, H, R) which depends only on initial data (see Feng et al. 2006 for more detailed description of these functions). Obviously X(t) -+ 0 as t - 00. It can be shown that under standard assumptions on initial data and parameter functions the system (2.5) has a unique nonnegative solution defined for all positive time.
3
Reproductive numbers of the general model
We can calculate the effective (or control) reproductive number R c , and the basic reproductive number Ro in the absence of control. To see the biological meaning of the expression of Rc we introduce the following quantities:
(3.1)
156
Zhilan Feng, Dashun Xu, Haiyun Zhao Table 1: Definitions of frequently used symbols
Symbol S(t)
Definition Number of susceptible individuals at time t
SQ(t)
Number of susceptible individuals quarantined at time t
E(t)
Number of exposed (not yet infectious) individuals at time t
Q(t)
Number of quarantined (exposed) individuals at time t
I(t) H(t) R(t)
Number of susceptible individuals at time t
N
Total population size (constant)
Number of isolated (infectious) individuals at time t Number of recovered individuals at time t
C(t)
Number of cumulative new infections at time t
>.(t)
Force of infection at time t
f3
Transmission coefficient
ai, a
Rate at exposed individuals become infectious
8i , 8
Rate at which infectious individuals recover
f-t
Natural death rate
X,
Rate of quarantine, isolation
p
Isolation efficiency (0
b
Fraction of contacts infected (b = 1 in this paper)
~
p
~
1) ~
~
'Y
Fraction of infected with early quarantine (0
Pi(S), Pi(S) k(s), l(s)
Probability that stage i lasts longer than s time units, i = E, I
TE, TI
'Y
1)
Probability of not being quarantined, isolated at stage age s Mean of PE(S) = e- as , PI(S) = e- lis (TE = l/a, TI = 1/8)
In
Probability an exposed becomes infectious: Similar to T E , adjusted by quarantine:
roo
Jo
Probability an infectious person recovers: Similar to
~, adjusted by isolation:
Mean time in exposed stage:
In
In
oo
oo
[-?E(s)]e-I,tsdt
[~?E(s)k(s)]e-/LSdt
In
00
[-?I(s)]e-/LSdt
[-~I(s)l(s)]e-/LSdt
oo
PE(:)e-/LSdt
1
00
Same as 'VE, adjusted by qUar:ntine:
In
PE(s)k(s)e-/LSdt
PI(s~e-/LSdt Same as 'VI, adjusted by isolatio:: 10 PI(s)l(s)e-/LSdt Mean time in infectious stage:
oo
00
The basic reproductive number The reproductive number under control measures EDA or GDA
Exponential or gamma distribution assumption
EDM or GDM
Exponential or gamma distribution model
The Uses of Epidemiological Models in the Study of . . .
157
and
(3.2)
TE and TEk represent respectively the probability and the "quarantineadjusted" probability that exposed individuals survive and become infectious. 7J and 7Jl represent respectively the probability and the "isolationadjusted" probability that infectious individuals survive and become recovered. V E and V Ek represent respectively the mean sojourn time (death-adjusted) and the "quarantine-adjusted" mean sojourn time (death-adjusted as well) in the exposed stage. Vr and Vrl represent respectively the mean sojourn time (death-adjusted) and the "isolationadjusted" mean sojourn time (death-adjusted as well) in the infectious stage. Using (3.1) and (3.2) we can write Rc as
(3.3) where
(3.4)
The three components, R r , R rH , RQH in Rc represent contributions from the I class and from the H class through isolation and quarantine, respectively. From 0 ~ k, [ ~ 1 we know that
(3.5) Hence, RrH and RQH are both positive. Clearly, each Ri (i = I,IH, QH) is a product of the transmission rate 1(3 (or (1- p)1(3 or (1- p)--y(3) , the probability of surviving the exposed stage and entering the infectious stage, and the average sojourn time being infectious in the corresponding class (adjusted by natural death). In the absent of control, i.e., k(s) = [(s) == 1 and "( = 0, Rc gives the basic reproductive number: Ro
=
(31
1
00
00
[-FE(s)]e-J.LSds
Pr(s)e-J.LSds
= (3TE V r .
(3.6)
We can express Rc in terms of Ro as follows. Notice from (3.4) that Rc can be simplified to (1- p)(3TEVr + (1- "()P(3TEkVrl. Hence, from (3.6)
Zhilan Feng, Dashun Xu, Haiyun Zhao
158 we get
(3.7) From (3.5) it is easy to see that Rc < Ro. The impact of various (single or combined) control measures represented by" p, X, or 1> on the reduction of Ro can be evaluated using (3.7). System (2.5) always has the disease-free equilibrium (DFE)
U1
=
(Sl, E 1 , Q1, h, HI, RI)
= (N, 0, 0, 0, 0, 0).
If Rc > 1, then there is a unique endemic equilibrium
U * - (S* , E* , Q* , 1* , H* , R*)
(3.8)
with
= ~,
S*
E*
= YDEk >"*S* ,
Q* = (DE-DEk +,DEk)>"*S*, H*
I*=;YTEkDIz>"*S*,
(3.9)
= (TEDI - TEkDIz +,TEkDIz)>"*S*, R* = J:...TE'h>"*S*,
where >..*
p,
= p,(Rc -
1). Obviously, U* exists only if Rc > 1. Similar
techniques used in Feng et al. (2006) can be applied to here to show that the reproductive number Rc determines the existence and stability properties of the equilibrium points of system (2.5).
4
Comparison of models with different sojourn distributions
It is easy to verify that, in the special case when PE(S) and PI(S) are exponential functions (i.e., e- as and e- 8s respectively), the system (2.5) reduces to the the following ODE model which we will refer to as the ED M (exponential distri bu tion model):
S'
= p,N - >..(t)S - p,S + rSQ,
+ a + p,)E, Q' = ,>"(t)S + XE - (a + p,)Q, I' = aE - (1) + 8 + p,)I, H' = aQ + ¢I - (8 + p,)H, R' = 8I + 8H - p,R.
E' = ;Y>"(t)S - (X
(4.1)
The Uses of Epidemiological Models in the Study of . . .
159
where A(t) is the same as given in (2.3). Note that this system has the same dynamic behavior as system (2.2) with b = I, a1 = a2 = a and 01 = 02 = o. If we assume that in the model (2.5) PE(S) and PI(S) are gamma distributed with parameters a and 0 respectively, i.e., m-1
k
(mas) PE (s ) =e -mQs" ~ l' k.
k=O
- -nos PI (s ) -e
n-1
L
k=O
k
(nos) k! '
then the model (2.5) simplifies also to an ODE model which we refer to as the GDM (gamma distribution model):
8' = f-LN - A(t)8 - f-L8, Ei
= ;YA(t)8 -
(X
+ ma + f-L)E1'
Ej=maEj-1-(x+ma+f-L)Ej, Q~ = 1'XE1 - (ma
j=2,···,m,
+ f-L)Q1,
= XEj + maQj-1 - (ma + f-L)Qj, j = 2,··· ,m, Ii = maEm - (¢ + nO + f-L)h, Ij = nOIj _ 1 - (¢ + nO + f-L)Ij, j = 2,··· ,n, Hi = maQm + ¢h - (no + f-L)H1, Hj = noHj _ 1 + ¢Ij - (no + f-L)Hj , j = 2,··· ,n, R' = nOin + noHn - f-LR, Qj
with I
=
2::7=1 I j ,
H
=
(4.2)
2::7=1 H j ,
where A(t) is the same as given in (2.3). We remark that the EDM is a special case of the GDM when m = n = 1. It was demonstrated in Feng et al. (2006) that the EDM and the GDM may provide contradictory predictions. Here we consider similar comparisons for the case l' > o. We first compare the model predictions using the reproductive number Re. From formulas (3.3), (3.4), and the exponential functions for PE and PI we get Rc for the EDM: (4.3) Similarly, using the gamma distribution function for PE and PI we get
Zhilan Feng, Dashun Xu, Haiyun Zhao
160 'Re for the GDM:
(ma)m (3 ~ (n8)j 'Re = (p, + ma)m p, + n8 (p, + n8)j
f;:o
X
[1- P ( 1
.
_ (1- / )(p, + ma)m p, + n8 (II. + ma + X)m p, + n8 + cP t'"
"n-l (no)' 6j=0 (f.I+noH)j ) ] .
~
"n-l 6)=0 (f.I+ n8 )J
(4.4) The plot of'Re for the EDM and GDM is shown in Figure 4.l. It is clear from Figure 4.1 that, similar to the case of 1 = 0 (see Figure 4.1(a) and (b)), the two models again produce inconsistent conclu1.5,,------------
1.5
~-----------
'Rc(~).X=O.l
-
-
'Rc(x). ~=0.1
'\:'. "''.
- - - - 'Rc(x.¢).x=¢
0.5
-
'Rc(¢). X= 0.1 'Rc<X). ~=0.1
\
- - - - 'Rc(X.~).X=~
0.5 (a)EDM
(b)GDM
0.1
0.2
OJ
X or
\ \
\
0.4
0.5
0.1
0.2
~
OJ
0.4
0.5
X or ¢
-
- 'Rc(~). X=r=0.2
-
'Rc(X). ~=r=0.2
-
-
'R c (¢)·x=r=O.2 'R c (X).¢=Y=0.2
-·-·'Rc(X.~).X=~,r=0.2
_._. 'Rc(X. ¢). X=~. Y= 0.2
...... 'Rc(n.X=~=0.2
...... 'Rc(r). X=~=0.2
\ \
~\
0.5
0.5
(c)EDM 0.1
(d)GDM 0.2 0.3 X. ~ or r
0.4
0.5
0.1
0.2 0.3 X. ~ or r
0.4
0.5
Figure 4.1: Comparison of the EDM (see (a) and (c)) and the GDM (see (b) and (d)) on the effect of various control measures in the reduction of the reproductive number 'Re. (a) and (b) are for the case of 1 = 0, and (c) and (d) are for the case of I> O. Both of the cases show that the two models are inconsistent regarding the effectiveness of control measures (represented by X, cP and ,) in the reduction of'Re.
The Uses of Epidemiological Models in the Study of ...
161
sions for 'Y > 0 (see Figure 4.1(c) and (d)). For example, in Figure 4.1(a) and (b), Rc for both models is plotted either as a function of ¢ for a fixed value of X = 0.1, or as a function of X for a fixed value ¢ = 0.1, or as a function of both X and ¢ with X = ¢. For any vertical line except the one at 0.1, the three curves intersect the vertical line at three points that represent three control strategies. The order of these points (from top to bottom) determines the order of effectiveness (from low to high) of the corresponding control strategies since a larger Rc value will most likely lead to a higher disease prevalence. The order of these three points (labeled by a circle, a triangle and a square) predicted by the EDM and the GDM is clearly different for the selected parameter sets, suggesting conflict assessments of interventions between the two models. Similar conclusions can be drawn for the case of 'Y > 0 by observing the curves of Rc as functions of X, ¢, or 'Y shown in Figure 4.1(c) and (d). Other parameter values used in Figure 4.1 are f3 = 0.2, P = 0.8, a = 1/7, and 8 = 1/10, corresponding to a disease with a latency period of l/a = 7 days and an infectious period of 1/8 = 10 days (e.g., SARS). We can also compare the two models by looking at the final epidemic size (or the cumulative number of infections during an epidemic), 0, which is defined by the equation
O'(t) = )..(t)S(t). Let 0(0) = 0 so that O(t) is the cumulative number of new infections at the end of an epidemic. As in Figure 4.1, the final size 0 (i.e., its value at the end of an epidemic) for both models is plotted as a function of one or two control parameters which are shown in Figure 4.2. The case 'Y = 0 is shown in Figure 4.2 (a) and (b)), and the case of'Y > 0 is shown in Figure 4.2 (c)-(f). Again, the EDM and GDM predict inconsistent outcomes for both cases. We also observe that the final size as a function of X and/or ¢ is not very sensitive to changes in 'Y (e.g., 'Y = 0.1 in Figure 4.2(c)(d) and 'Y = 0.5 in Figure 4.2(e)(f)). This is probably because of the value of p being small (or equivalently 1 - P being large), in which case the role of quarantine/isolation is limited. In fact, from the formulas (4.3) and (4.4) we can see that as a function of 'Y
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Zhilan Feng, Dashun Xu, Haiyun Zhao
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for the GDM. These inequalities also suggest that if 1 - P is not small enough, control via early quarantine b) may be impossible.
The Uses of Epidemiological Models in the Study of ...
163
To examine in more detail the difference between the two models, we can also look at Figure 4.3, where the cumulative number G(t) is plotted versus time t for various disease control measures. The EDM predicts that increasing ¢ from 0.1 to 0.2 leads to a reduction of 414 cases out of 662 of the final size, while increasing X from 0.1 to 0.2 leads to a reduction of 388 cases. This suggests that isolation might be more effective than quarantine (under the assumption that control effort is represented by the parameter values). On the other hand, the GDM predicts that the final size is reduced by 542 cases out of a total of 817 cases if ¢ is increased, and it is reduced by 592 if X is increased by the same value. This suggests that quarantine might be more effective than isolation. All parameters except those highlighted in the figures have the same values as in Figure 4.1 . 900
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Zhilan Feng, Dashun XU, Haiyun Zhao
164
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5
Conclusion
In this paper, we extended the results of Feng et al. (2006) by considering early quarantine of exposed individuals h > 0). The qualitative results seem unchanged regarding the difference between the simple and more complex models in predicting the effectiveness of disease control strategies. As we did in Feng et al. (2006), we formulated a general integral equations model (2.5) which assumes arbitrarily distributed dis-
The Uses of Epidemiological Models in the Study of· . .
165
ease stages. This general model reduces to the EDM-a simple ODE model of the SEI R type with quarantine and isolation-when the stage distributions are exponential. By demonstrating that the EDM may generate conflicting outcomes when compared with the GDM, another reduction of the general model when the stage distributions are gamma, we show that simple models (such as the EDM in this paper) may not be appropriate to use for assessing disease control strategies. A more detailed discussion on the drawbacks of the assumption of exponentially distributed disease stages is given in Feng et aI. (2006). Although the gamma distribution can provide in many cases a more reasonable description of the epidemiological processes, it may not solve the problem completely. Our general model allows us to consider other disease stage distributions within the same modeling structure, and hence makes it possible to compare outcomes from simple and more complex models.
References [1) Chowell, G., P. Fenimorea, M. Castillo-Garsowc and C. CastilloChavez. 2003. SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism. J. Theor. BioI. 224: 1-8. [2) Feng, Z., W. Huang and C. Castillo-Chavez. 2001. On the role of variable latent periods in mathematical models for tuberculosis. Journal of Dynamics and Differential Equations. 13: 435-452. [3) Feng, Z. and H.R. Thieme. 2000a. Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: General Theory. SIAM J. Appl. Math. 61 (3): 803-833. [4J Feng, Z. and H.R. Thieme. 2000b. Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages II: Fast disease dynamics and permanent recovery. SIAM J. Appl. Math. 61 (3): 983-1012. [5) Feng, Z., D. Xu, and H. Zhao. 2006. Epidemiological models with non-exponentially distributed disease stages and applications to disease control. Bulletin of Mathematical Biology. To appear. [6] Hethcote, H. and D. Thdor. 1980. Integral equation models for endemic infectious diseases. J. Math. BioI. 9: 37-47. [7J Hethcote, H., H.W. Stech and P. van den Driessche. 1981. Nonlinear Oscillations in Epidemic Models. SIAM J. Appl. Math. 40 (1): 1-9.
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[8] Lipsitch, M., T. Cohen, B. Copper et aI. 2003. Transmission dynamics and control of severe acute respiratory syndrome. Science. 300: 1966-70. [9] Lloyd, A. 200la. Realistic distributions of infectious periods in epidemic models. Theor. Pop. BioI. 60: 59-7l. [10] Lloyd, A. 2001b. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R Soc. Lond. B. 268: 985-993. [l1J MacDonald, N. 1978. Time lags in biological models. SpringerVerlag. New York. [12] McLean, A.R, M.M. Robert, J. Pattison and RA. Weiss. 2005. A case study in emerging infections. Springer-Verlag. Oxford University Press. New York. [13] Plant, RE. and L.T. Wilson. 1986. Models for age-structured populations with distributed maturation rates. J. Math. BioI. 23: 247262. [14J Riley, S., C. Fraser, C.A. Donnelly, A.C. Ghani et aI. 2003. Transmission dynamics of the etiological agent of SARS in Hong Kong impact of public health interventions. Science. 300: 1961-1966. [15] Taylor, H.M. and S. Karlin. 1998. An Introduction to Stochastic Modeling. Third ed. Academic Press. San Diego. [16J Thieme, H. 2003. Mathematics in Population Biology. Princeton University Press. Princeton.
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Assessing the Burden of Congenital Rubella Syndrome and Ensuring Optimal Mitigation via Mathematical Modeling John W. Glasser Division of Viral Diseases National Center for Immunization and Respiratory Diseases Centers for Disease Control and Prevention Atlanta, Georgia 30333, USA E-mail: [email protected]
Maureen Birmingham Department of Communicable Diseases Surveillance and Response, South East Asia Regional Office World Health Organization c/o Ministry of Health N onthaburi 11000, Thailand E-mail: [email protected]
Abstract Congenital rubella syndrome (CRS) comprises a panoply of lesions, some fatal and others diagnosed too long after birth to be readily associated with first trimester maternal infections. Consequently, this syndrome is under-ascertained even by reviewing medical records for compatible conditions and confirming them serologically. Mathematical modeling provides an alternative to such studies, while also permitting health authorities to estimate the burden of this disease relatively easily and to evaluate means by which it might be mitigated. Our estimates of CRS from disease or serological surveillance and demographic information compare favorably to those from retrospective chart reviews in children's hospitals and clinics specializing on afflictions that may result from infection in utero. The CRS burden could be mitigated either by reducing susceptibility to infection on exposure among women of childbearing age (WCBA) or their risk of exposure. The first approach might involve vaccinating girls attending school or mothers delivering in hospital, neither of which however is universal. The second necessarily involves childhood vaccina-
168
John W. Glasser, Maureen Birmingham tion, via routine age-appropriate doses, mass campaigns, or both. But if sufficient coverage is not sustained, the immunization of some children only temporarily protects others, whose susceptibility may persist into adulthood. Unless vaccinated, their own children may be infected while playing with neighbors or attending school, and infect them. The females among these susceptible parents may be pregnant. The potential of childhood vaccination to exacerbate the future burden of CRS has been known since the early 1980s, but casually dismissed for lack of evidence that could not be apparent for some time. However, examples have recently begun appearing in the literature. Romania exemplifies a newly-recognized hazard: rubella occurs annually, but major outbreaks typically occur every 5 years. The period of these multiannual cycles doubled as falling birth rates decreased the supply of susceptible children, increasing the mean age of infection, and concomitantly CRS, a phenomenon that may be anticipated in China due to efforts to curb population growth (Gao and Hethcote 2006). We describe models with which possible strategies for mitigating the burden of CRS have been evaluated in these and other developing countries. Given estimated costs of feasible tactics, policymakers could consider whether to devote their scarce resources to this or other health problems. We encourage monitoring of disease or serology to ensure policy objectives are attained, even if modeling aided in vaccination program design.
1
The diseases
Rubella is a mild rash illness that generally afflicts children. But congenital rubella syndrome (CRS) - which includes spontaneous abortions, stillbirths, brain (microcephaly, mental retardation), cardiac, hearing, liver or spleen (hepatosplenomeglia), and visual (cataracts, glaucoma) deficits - may afflict the progeny of women infected during gestation, particularly their first trimesters. The nature of the lesion or combination presumably is determined by fetal ontogeny when mothers-to-be are infected (Cooper et al. 1969, Ueda et al. 1979, White et al. 1969).
2
Risk of infection
Cutts and Vynnycky (1999) compared a model in which the risk among susceptible people, termed force of infection, was constant (i.e., k in dp(a)/da = k[l- p(a)], where p denotes proportion immune, a age, and dp(a)/da is the rate of change with age), with a model in which this risk was allowed to vary between age intervals. They argued that models with constant risks were best. Because activity and disposition to interact
Assessing the Burden of Congenital Rubella Syndrome and···
169
with others roughly the same age peak during childhood, we estimate the coefficients of catalytic models (Muench 1959) from which age-specific forces of infection can be derived. Insofar as we choose models whose terms have significant likelihood ratios, our approach not only makes more sense biologically, but makes best use of available observations. Because some infections are asymptomatic and immunity is lifelong, serological surveys are preferred to disease surveillance for information about infection, not just of mothers as required for burden assessment, but children, as required to evaluate alternative means of mitigating the burden. New methods for assaying oral fluids (Ramsayet al. 1998) may increase parental willingness to involve children, and data from crosssectional surveys generally can be fitted by quadratic or cubic polynomials. Meanwhile, Farrington's (1990) model compensates for the most common limitations of observations from serological surveys.
3
Assessing the burden
The burden of CRS is under-ascertained via conventional surveillance partly because of spontaneous abortions, but also because infections may be asymptomatic. Even when symptomatic, the interval between maternal infections and stillbirths or diagnoses of birth defects limits appreciation of causality. Several investigators have attempted to ascertain more accurately the burden among living children by reviewing medical records for characteristic lesions and combinations in hospitals and outpatient facilities where care would be sought (Cutts et al. 1997), and determining if children with compatible conditions have antibodies to rubella. Because only living children can seek care, and not all do, these time- and other resource-consuming studies cannot estimate the true burden of CRS. As many as 5% of maternal infections result in spontaneous abortions and stillbirths (Table 1), while an unknown fraction result in mildly affected children who do not require medical care. The parents of still other children lack the resources required to care for their afflictions irrespective of severity. None of these children has a medical record. Because modeling estimates derive from infections among women of childbearing age (WCBA), they exceed those from chart reviews in hospitals and outpatient facilities caring for children with characteristic deficits. Modeling relies on (a) readily available demographic information, (b) estimates of gestational age-specific risks obtained from registries of infected pregnant women (e.g., Miller et al. 1982, Sever et al. 1969), and (c) surveillance for either incidence of rubella disease or prevalence of antibodies reflecting previous exposure to rubella virus. We will describe our methods for assessing the burden of CRS and
John W. Glasser, Maureen Birmingham
170
Table 1: Abortions, still and live births following infection during pregnancy (percent) 'lbtal
5 t14)
::;pontaneous abortions or stillbirths 3 (8)
105 (89)
8 (7)
5 (4)
18
United Kingdom Poland outbreak
258 (31)
523 (63)
45 (5)
826
28 (74)
7 (18)
3 (8)
38
Total ('10)
420 (41)
543 (53)
56 (5)
1019
lJountry and setting US outbreak Urban US
Live births
'l'herapeutlc abortions
29 (79)
37
lteterence Monif et al. 1966 Sever et al. 1969 Miller et al. 1982 Zg6niak -Nowosielska et al. 1996
provide two examples: Only WCBA were surveyed in Morocco, but in urban and rural settings. In Romania, a cross-section of the population by age was surveyed, and our estimate can be compared with passive surveillance.
3.1
Assessment methods
The number of women pregnant and infected in any given year can be approximated as the sum of products of age-specific numbers of women, birth rates, and risks of infection. Demographic information of this sort generally is available from government publications or, increasingly, web sites. Births underestimate pregnancies, but multiplying by 1.05 would correct for spontaneous abortions and stillbirths (Table 1), yielding a reasonable estimate of affected pregnancies. We estimate the risks of maternal infection (a) directly from case notifications or (b) indirectly via catalytic modeling of results from serological surveys. Surveys typically report numbers of sera tested and above thresholds deemed protective in each age group, from which the coefficients of suitable models may be estimated via maximum likelihood (Grenfell and Anderson 1985). We prefer the highest order polynomial observations warrant, but use Farrington's (1990) model when necessary. While catalytic modeling tacitly assumes equilibrium, annual outbreaks upon which multi-annual cycles may be superimposed - the quintessential feature of infectious diseases - affect these assessments minimally (Whittaker and Farrington 2004). Given a suitable catalytic model, one determines the risk of infection in each age interval by subtracting cumulative proportions at their beginnings, p(a), from those at their ends, p(a+n). Dividing by interval widths n, one obtains average annual risks, [p(a + n) - p(a)]jn. MUltiplying by the numbers in each interval and by age-specific live birth rates yields women experiencing term pregnancies and infections the same year by
Assessing the Burden of Congenital Rubella Syndrome and· . .
171
age. As gestation and years last roughly 40 and 52 weeks, respectively, 40/52 of these women experience these events simultaneously. During each week of gestation, we assume 1/40 of their developing fetuses risk infection. But if infected, the risk of being born with CRS varies with gestational age (Glasser 2007, Figure 3). We estimated those risks via logistic regression from published results of a large pregnancy registry (Miller et al. 1982).
3.2
Assessment examples
Our calculations for rural and urban Morocco during 2001 (Tables 2 and 3) are based on age-specific numbers of women, live birth rates, and risks of infection derived from catalytic modeling of observations from serological surveys (Figures 3.1), essentially cumulative proportions infected, differenced as described above. Assuming multiple births share the same fate, the product of women by age and age-specific birth rates are term pregnancies by age of mother. Their products with age-specific risks of infection are women who were pregnant and infected during a given year. Of these, 40/52 were infected while pregnant. Table 2: Estimated infants with CRS born to women in urban Morocco by age , 2001 Age group
J:<'emales X 10- 3
Live births
Fertility
i~=g
~~~
18,i~~
o;ommn
20-24 25-29 30-34 35-39 40-44 45-49 50+ Totals
839 817 795 715 667 517 395
55,702 67,453 62,250 39,356 11,339 1,431 286 :l56,490
0.023548 0.066391 0.082562 0.078302 0.055043 0.017 0.002768 0.000724
Prob. of infection
Intected while pregnant
0.21155!S 0.144676 0.090199 0.054676 0.03324 0.02053 0.012899 0.008181 0.005147
41~
773 567 318 124 23 2 0 :l,:l:l5
Intants
w/CRS 1 93 175 129 72 28 5 0 0 503
Table 3: Estimated infants with CRS born to women in rural Morocco by age , 2001 Age group
Females xlO- 3
Live births
~ertiIity
i~=i~
~~g
29,~~g
0.0003!S7 0.036957 0.093082 0.104365 0.103795 0.093468 0.03682 0.010789 0.003324
20-24 25-29 30-34 35-39 40-44 45-49 50+ Totals
753 638 516 374 355 289 256
70,091 66,585 53,558 34,957 13,071 3,118 851 :lI:l,UtJS
Prob. of infection
Infected while pregnant
O.11l!{j~
25~
0.056628 0.036372 0.027486 0.022575 0.019197 0.016538 0.014314 0.012408
392 282 186 103 33 7 2 1,267
Intants
w/CRS 1 58 89 64 42 23 8 2 0 :2E7
John W. Glasser, Maureen Birmingham
172
0.8
~'"
0.6
~
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t
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50
40
50
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Figure 3.1: Catalytic models (short and long dashes, alternating dots and dashes) fitted to observations (symbols) from serological surveys among WCBA in urban (a) and rural (b) Morocco during 2002. As the FOr among children are essentially indeterminate, evidently limiting serological surveys to WCBA also limits dynamic modeling to mitigate the burden of CRS. Tabulated risks of infection were calculated from Farrington's (1990) model, which was designed to compensate for such limitations, but those from the catalytic model whose FOr is a quadratic function of age are similar. Subjecting the progeny of these women to gestational age-specific risks of CRS yields annual estimates of 503 and 287 afflicted children in urban and rural Morocco, respectively. These are 196.1 and 105.5 per 105 live births, exceeding by an order of magnitude the 8.1 to 12.7 per 105 live births Bloom et al. (2005) estimated via chart review. While
Assessing the Burden of Congenital Rubella Syndrome and· . .
173
rural women have more births, urban ones have greater risks of infection. In Romania, where risks were estimated from catalytic modeling of results from a cross-sectional serological survey (Glasser 2007, figure 6a), a similar calculation (Table 4) yields 453 afflicted children, 190.8 5 per 10 live births. This compares with 124 passive surveillance reports of suspected CRS the following year, of which only 5 were IgM positive. While modeling better estimates the burden among living children than chart review or passive surveillance, it still underestimates the burden by about 5% due to stillbirths and spontaneous abortions resulting from rubella infection. Table 4: Estimated infants with CRS born to women by age in Romania, 2003
4
Age group
Females xlO- 3
Live births
Fertility
Prob. of infection
10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50+ Totals
834 872 959 911 754 695 842 796 593
463 35,178 92,931 70,697 27,388 8,116 2,369 155 0 237,297
0.000565 0.040335 0.096879 0.077581 0.036301 0.011676 0.002813 0.000195 0
0.17816 0.110626 0.0637 0.035071 0.018874 0.010018 0.005127 0.002185 0
Infected while pregnant 13 599 911 381 80 13 2 0 0 1,998
Infants wieRS 3 136 206 87 18 3
0 0 0 453
Mitigation strategies
Strategic options for mitigating the burden of CRS are: vaccinating WCBA, protecting them - with probability termed vaccine efficacy from infection upon exposure or reducing their risk of exposure. Tactical options include vaccinating children on attaining particular ages, conducting mass campaigns within age ranges, or targeting girls, WCBA, or mothers post-partum (Robertson et al. 1997, Table 2). Protecting adolescent girls and WCBA from infection if exposed has smaller indirect effects than does protecting children (below), but achieving similar coverage among adolescents and adults is more difficult because they are relatively inaccessible. Childhood vaccination can exacerbate the burden (Anderson and May 1983; Hethcote 1983). Because young children are most likely to contact others (Figure 4.1), vaccinating them averts the most secondary infections. Those temporarily protected via herd immunity, as this indirect effect is called, remain susceptible. Because their risks of infection
174
John W. Glasser, Maureen Birmingham 14
12 10
~
i 0
8
6
u
4
2
0
10
20 Age (years)
30
40
4.1: Ratios of contacts by and of people aged x in Sao Paulo. Contacts were estimated as quotients of reported infections of individuals aged i by those aged j, from a case-control study (Camargo et al. 2000), and products of probabilities that individuals aged i were susceptible and j infectious. The first of these is from a statistical synthesis of five serological surveys (Glasser 2007, Figure 2) and second are quotients of laboratory-confirmed case reports and mid-1997 populations (de Moraes et al. unpublished).
wane with age, children who were neither immunized nor infected may still be susceptible on attaining reproductive age, when their risks of infection wax again. Women may be pregnant when their own earlier child or a neighbor's most likely infected in school infects them. Because vaccination of some children allows others to remain susceptible, childhood vaccination can increase susceptibility among future WCBA. Consequently, programs must be designed and implemented carefully or combine complementary strategies (e.g., mothers post-partum together with children) to attain the primary policy objective, reduction in CRS.
5
Evaluating the options
Demographically-realistic dynamic modeling provides the best means of assessing alternative vaccination programs (Anderson and Grenfe111986; Edmunds et al. 2000; van der Heijden et al. 1998). While surveying WCBA suffices for burden assessment, cross-sectional surveys are needed to ascertain forces of infection, from which may be estimated (a)
Assessing the Burden of Congenital Rubella Syndrome and· . .
175
infection rates, given assumptions about mixing (Grenfell and Anderson 1985; Hethcote 1996), and (b) initial conditions (i.e., proportions immune or susceptible by age). Statistical principles dictate sampling in proportion to dp(a)/da, change with age in proportion positive, implying that surveys should be cross-sectional (Figure 3.1 b ). We compare contemplated vaccination programs by simulating their impact on rubella in dynamic, age-structured populations via classic susceptible, exposed, infectious, and recovered (SEIR) models with passive protection and seasonal forcing. We assume risks of infection are not gender-specific, but age-specific proportions female enable us to project the population and calculate CRS from predicted infections via the same information used in our burden assessments. Because contemporary demographic and cross-sectional serological information enabled model predictions to converge quickly on available rubella surveillance, Romania provides a credible example of dynamic modeling aiding in the design of vaccination programs to mitigate the burden of CRS. Should policy objectives be attainable via alternative means, one can determine which strategy is optimal by calculating net costs per case averted (i.e., (program costs less savings by virtue of cases averted]/cases averted). Using a national registry, AI-Awaidy et al. (2006) estimated the costs of caring for children with CRS in Oman. The averages among their children - 84% of whom had ocular, 84% auditory/speech defects, 70% neurological manifestations, and 42% cardiac defects - were $18,644 for medical care and $98,734 including loss of productivity. In addition to cases averted, calculations of this sort require program costs, including administration as well as vaccine. Costs of administering routine age-appropriate vaccinations may be limited to child health centers and personnel. Campaigns have opportunity costs insofar as health care workers are diverted from their regular duties, and require publicity and transportation.
5.1
Dynamic modeling
Our model populations are not only age-structured and dynamic, but open (i.e., include immigration and emigration). The system includes persons protected by maternal antibodies, V(a, t); who are susceptible, W(a, t); harboring latent infections (i.e., infected, but not yet infectious), X(a, t); infectious, Y(a, t); and immune following disease or vaccination, Z(a, t). These equations describe the rates at which persons of age a transit these five epidemiological states at time t:
-av + -av
aa
at
=
V(O, t) - [0" + {.t(a)
+ f(a) -
((a) {I - cp(a)}] V(a, t),
176
aw aa
John W. Glasser, Maureen Birmingham
+ aw = aV(a, t) + W(O, t) at
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[Q:v(a, t)
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ax aa
+ ax = A (a, t) W (a, t)
ay aa
+ ay
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+ az = Q:v(a,t)W(a,t)+pY(a,t)
at
at
= "I X
+ A(a, t) + J-L(a) + E(a)
- [r + J-L (a)
(a, t) - [p + 8 (a)
+ da) -
+ J-L (a) + E (a)
da) ¢ (a)] X (a, t), -
~ (a) ¢ (a)] Y (a, t),
at
- [J-L (a)
+ da) -
da){l - ¢ (an] Z (a, t).
The transition processes and their respective per capita rates are vaccination, v (a), whose efficacy is Q:; infection, A (a, t); becoming infectious, "I; and recovering, p. Age at vaccination is gamma distributed with parameters approximated from hypothetical recommendations (Figures 3 and 6) or estimated - via the method of moments - from recent surveys. Newborns with immune mothers are protected via passively-acquired antibodies they lose at rate a; others are susceptible. Individuals immigrate and emigrate at per capita rates ~ (a) and E (a), respectively, with probabilities ¢ (a) of being susceptible and 1 - ¢ (a) immune, and die at per capita rates J-L (a), which disease increases by 8 (a). As demographic and disease processes invariably interact, we also model population dynamics via actual age and gender distributions and vital statistics insofar as possible. Immune and susceptible children are born at rates
V(O,t) = and
W(O,t)
J =J
8(a)p(a,f)Z(a,t)da
8(a)p(a,f)8(a,t)da,
respectively, with per capita birth rate e(a), proportions female p (a, f), and susceptible and total populations S (a,t) = W (a, t) + X (a, t) + Y (a, t) and N (a, t) = V (a, t) + S (a, t) + Z (a, t), respectively. The age-specific risks of infection among susceptible persons aged A (a, t) = J.B (a, a') [Y (a', t) IN (a', t)] da', where the infection rates .B (a, a'), are products of transmission probabilities, given contact between susceptible and infectious persons aged a and a' Pr [TIC (a, a')], and rates of contact C (a, a'), and term in square brackets is the probability of contacting an infectious person aged a' at time t.
5.2
Modeling example
We fit various catalytic models to numbers of sera analyzed and positive by age, and chose the best fit that made sense biologically, which
Assessing the Burden of Congenital Rubella Syndrome and . . .
177
involves considering implied age-specific risks of infection (Glasser 2007, Figure 6b), oX (a, t) in the equation below. Then we assume effective contacts were a convex combination of preferential and proportionate mixing (Hethcote 1996): assuming preference to associate with others roughly the same age (e.g., 0 < E ~ 1), we estimated activities that would yield the observed forces of infection with f3 (a, a') = 8 (a, a') x E x b (a) + (1 - E) Jb (a) b (a'), where b (a) or b (a') are age-specific activities and 8 (a, a') is the Kronecker delta (i.e., 1 when a = a' and 0 otherwise). Having performed these calculations (Glasser 2007, Figure 7), we estimated ~o as the dominant eigenvalue of the next generation matrix (Diekmann et al. 1990). Observing that ~, the average number of secondary infections, must be less than one lest the sizes of successive generations infected increase, we set ~ = (1 - Pc) ~o = 1 and solved for Pc, the fraction that must be immunized or, if infected, isolated effectively to control outbreaks. With forces of infection from Farrington's model, our estimates of ~o range from 3.32 to 17.2 depending on E. With E = 0.6, ~o is 4.3, whereupon Pc = 1- (1/~o) = 0.77, a calculation our simulations affirm (i.e., coverage of 0.8 with a vaccine that is 95% effective eliminates rubella, but coverage of 0.7 does not). For sustained seasonal cycles, models ofthis sort must be forced (i.e., external phenomena such as school calendars affect internal ones such as infection rates). Our infection rates are forced via harmonic functions, f3t = f30 + asin (wtt + 8) + Et, where Et is a sequence of uncorrelated (0, a 2 ) variates, the amplitude of a is small relative to the variance of Et, and Wt is frequency in radians (i.e., 21f /365.25). While adjusting f3o, elsewhere written as f3 (a, a'), we also estimated age-specific harmonic coefficients, f31 and f32: a sin (wtt + 8) = f31 sin (wtt) + f32 cos (Wtt) , where f3f + f3~ = a 2 and tan (8) = f32/ f31' After estimating these vectors, our simulated infections resembled those reported (Figure 5.1) and we were ready to answer policy questions experimentally. Having attained the criteria recommended by the World Health Organization for initiating rubella vaccination (WHO 2000), Romanian health authorities considered switching from single antigen measles to measles, mumps, and rubella (MMR) vaccine, but wondered if catch-up was necessary. First, we simulated I-dose childhood vaccination programs attaining 70% and 80% coverage among children aged 1 year, demonstrating the intermediate herd-immunity threshold mentioned above (results not shown). Then we simulated 80% coverage with and without catch-up from 2-14 years beginning a year later (Figure 5.2). Elimination is attained more quickly with a campaign, but occurs without one (Figure 5.3). Authorities vaccinated adolescent girls in Bucharest during 2002, but were not contemplating further targeted vaccination. Nonetheless, the enormous age range employed in Latin American campaigns motivated
John W. Glasser, Maureen Birmingham
178
(a) 2500 2000
.,'"
8 1500 11000 ~
,<,:~-".--
500
. . . . : . ,\
/: ~ ---,--:'S~~~ 200
400 600 Time (days since 1 Jan 02)
800
1000
800
1000
(b) 2500 2000
'"
u~ 1500
1
1000 500
200
400 600 Time (days since 1 Jan 02)
Figure 5.1: Comparison of a) observed and b) predicted rubella among children 5-9 years old, and adolescents 10-14 and 15-19 years old in Romania (short, long and alternating dashes and dots, respectively), 200204. Weekly "observations" were made from quarterly reports via SAS' procedure expand. The last quarter arrived after harmonic coefficients had been estimated, providing an unusual opportunity to validate the model: predictions fit the last quarter as well as earlier ones. Overall R2 ::::: 0.7; in our experience, this is quite good for a mechanistic model.
us to simulate this strategy, increasing the age distribution's mean and variance (Figure 5.4). As indirect effects are small (Figure 4.1), the benefit of reducing susceptibility of WCBA to infection on exposure should be proportional to reproductive value (Fisher 1930). But our Romanian model demonstrates unequivocally that benefits do not increase with age, especially where women cease bearing children at relatively young
Assessing the Burden of Congenital Rubella Syndrome and . . .
179
(a) 0.14 0.12
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,,
,, ,, ,,
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Figure 5.2: Hypothetical age distributions (a) and cumulative age distributions (b) at which a single dose of rubella-containing vaccine might be administered (short dashes, age in months) and a catch-up campaign conducted (long dashes, age in years) in Romania. ages (Table 5). Vaccinating women reduces CRS, but its impact on rubella is minimal. Only childhood vaccination can eliminate rubella, the cause of CRS. Modelers have demonstrated two coverage thresholds (Anderson and May 1983; Hethcote 1983): Below the first, relying on natural immunity is best. Thus, in poor countries, rubella vaccination should not compete with other health problems for scarce resources. Between thresholds, vaccinating adolescent girls and young women, and above the second, vaccinating children are the optimal strategies, respectively.
John W. Glasser, Maureen Birmingham
180
(a) 10000 8000
.,'"
u~ 6000
~
'B
e
4000
p..
2000
1000
2000
4000 3000 Time (days)
5000
6000
5000
6000
(b) 1000 800
1000
2000
4000 3000 Time (days)
Figure 5.3: Simulated consequences of a one-dose vaccination program attaining 80% coverage among 1 year-old children (a) without and (b) with a month-long catch-up campaign beginning one year later that attained the same coverage among children 2-14 years old. Curves are for ages 5-9, 10-14, and 15-19 years. Table 5: Average annual costs (not including outreach, testing, administration) and benefits of vaccinating progressively older susceptible WCBA (Figure 5.4). Mean±l<7 24±4 years old 30±4 years old 36±4 years old
Costs Doses 7,818 7,270 6,685
Benefits (% reduction) Rubella CRS 3% 42% 2% 43% 1% 43%
Assessing the Burden of Congenital Rubella Syndrome and···
181
(a) 0.1
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- - - - - - - - -:-- =-= -=-= - - -.::. ./
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Age (years)
Figure 5.4: Age-distributions of hypothetical vaccination programs targeting adolescent girls and adult women of increasing age. The gamma parameters were estimated from means of 24, 30 and 36 years ± one standard deviation. The abscissa is age and ordinates are proportions and cumulative proportions.
Consequently, countries that begin by vaccinating adolescent girls and young women typically switch to young children as coverage increases. As the per capita cost of vaccinating depends on age, a surrogate for accessibility, cost-benefit analysis is required to locate these thresholds.
182
6
John W. Glasser, Maureen Birmingham
Monitoring the impact
Because vaccination can exacerbate the CRS burden, authorities should monitor the susceptibility profile of WCBA or age- and gender-specific incidence of rubella after implementing any childhood vaccination program, even ones designed via modeling (Massad et al. 1994, 1995), to ensure (a) susceptibility to or incidence of rubella is not increasing among WCBA or (b) CRS is indeed declining, and any secondary policy objectives also are being attained. If not, one should consider alternative tactics, in which exercises modeling also can assist. The possible consequences of attending only to disease among children range from wasting resources (Morice et al. 2003) to exacerbating the burden (Panagiotopoulous 1999). While almost 30 years of vaccination attaining coverage of 0.8 by the administrative method doses administered/target population certainly altered the epidemiology of rubella in Costa Rica (Glasser 2007, 4), we cannot tell if it affected CRS (Figure 6.1). Because children vaccinated in the public sector may be counted more than once or omitted from the target population by virtue of age or migration, or be may vaccinated in the private sector, the WHO recommends checking, and possibly correcting administrative coverage estimates via periodic surveys (http://www.who.int/immunization_monitoring/routine/immunizatiOILcoverage/en/index.html/). Because of the potential to exacerbate
1980
1985
1990
1995
2000
Year
Figure 6.1: Cumulative number of cases of CRS since vaccination began in Costa Rica and fitted linear, exponential and logistic functions. If vaccination reduced the rate at which cases accumulated, one could conclude that it had the desired impact. But which function fits best is unclear, suggesting the impact of vaccination is equivocal.
Assessing the Burden of Congenital Rubella Syndrome and . . .
183
the burden of CRS, countries must demonstrate their ability to vaccinate more than 80% reliably before including rubella in their childhood vaccination programs. The WHO also recommends monitoring susceptibility profiles after vaccination begins (WHO 2000).
7
Summary
Congenital rubella syndrome is under-ascertained even by reviewing medical records for compatible conditions and confirming them serologically. Mathematical modeling not only obviates the need for such studies, permitting health authorities to estimate the burden of this disease relatively easily, but also to evaluate means by which it might be mitigated safely. We describe methods for assessing and evaluating possible strategies for mitigating the burden of CRS. Given estimated costs of feasible tactics, policymakers could consider whether to devote their scarce resources to this or other health problems. We advocate monitoring after beginning to vaccinate against rubella, especially children, to ensure that policy objectives are attained.
Acknowledgments We have learned about rubella from Sharon Bloom, Carlos CastilloSolorzano, Carolina Danovaro, Brad Hersh, Charley LeBaron, Jose Cassio de Moraes, Mona Marin, Ana Morice, Adriana Pistol, Alexandru Rafila, Susan Reef, Susan Robertson, and Peter Strebel. We are grateful to Zhien Ma, Jianhong Wu, and Yicang Zhou for the opportunity to participate in the China-Canada joint program on Modeling Infectious Diseases held at Xi'an Jiaotong University during spring of 2006. Jim Alexander, Charley LeBaron, and Susan Goldstein read earlier drafts of this manuscript and suggested improvements. Nonetheless, the opinions expressed are our own, not these colleagues' or our employers', the Centers for Disease Control and Prevention and World Health Organization, respectively, and we take full responsibility for any errors.
References [1] AI-Awaidy, S, Griffiths, UK, Nwar, HM, Bawikar, S, AI-Aisiri, MS, Khandekar, R, Mohammad, AJ, Robertson, SE 2006. Vaccine 24: 6437-45. [2] Anderson, RM, May, RM 1983. Vaccination against rubella and measles: quantitative investigations of different policies. J Hyg (Lond) 90: 259-325.
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[3] Anderson, RM, Grenfell, BT 1986. Quantitative investigations of different rubella vaccination policies for the control of congenital rubella syndrome (CRS) in the United Kingdom. J Hyg (Lond) 94: 305-33. [4] Bloom, S, Rguig, A, Zniber, A, Bouazzaoui, N, Zaghloul, K, Reef, S, Zidouh, A, Papannia, M, Seward, J 2005. Congenital rubella syndrome burden in Morocco: a rapid retrospective assessment. Lancet 365: 135-41. [5] Camargo, MCC, de Moraes, JC, Souza, VAUF, Matos, MR, Pannuti, CS 2000. Predictors related to the occurrence of a measles epidemic in the city of Sao Paulo in 1997. Rev Panam Salud Publica 7: 359-65. [6] Cooper, LZ, Ziring, PR, Ockerse, AB, Fedun, BA, Kiely, B, Krugman, S 1969. Rubella: clinical manifestations and management. Am J Dis Child 118: 18-29. [7] Cutts, FT, Robertson, SE, Diaz-Ortega, J-L, Samuel, R 1997. Control of rubella and congenital rubella syndrome (CRS) in developing countries, part 1: burden of disease from CRS. Bull World Health Organ 75: 55-68. [8] Cutts, FT, Vynnycky, E 1999. Modeling the incidence of congenital rubella syndrome in developing countries. Int J Epidemiol 28: 117684. [9] de Moraes, JC, Glasser, JW, Pannuti, CS, Souza, VAUF, Hersh, BS, de Quadros, CA manuscript. 1997. Measles outbreak in metropolitan SaoPaulo, Brazil: Evaluation of vaccination strategies via mathematical modeling. Unpublished. [10] Diekmann, 0, Heesterbeek, JAP, Metz, JAJ 1990. On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations. J Math BioI 28: 503-22. [11] Edmunds, WJ, van de Heijden, OG, Eerola, M, Gay, NJ 2000. Modeling rubella in Europe. Epidemiol Infect 125: 617-34. [12] Farrington, P 1990. Modeling forces of infection for measles, mumps and rubella. Stat Med 9: 953-67. [13] Fisher, RA 1930. The Genetical Theory of Natural Selection. Oxford University Press, Oxford. [14] Gao, L, Hethcote, HW 2006. Simulations of rubella vaccination strategies in China. Mathematical Biosciences 202: 371-85. [15] Glasser, JW 2007. Clinical and public health applications of mathematical models. Mathematical Modeling of Infectious Diseases: Dy-
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namics and Control, Grenfell, BT, Ma, S, Xi, Y, eds. Institute for Mathematical Sciences, NationalUniversity of Singapore. [16] Grenfell, BT, Anderson, RM 1985. The estimation of age related rates of infection from case notifications and serological data. J Hyg (Lond) 95: 419-36. [17] Hethcote, HW 1983. Measles and rubella in the United States. Am J Epidemiol 117: 2-13. [18] Hethcote, HW. 1996. Modeling heterogeneous mixing in infectious disease dynamics. 215-228 in: Isham, V, Medley, G, eds. Models for Infectious Human Diseases: Their Structure and Relation to Data. Cambridge University Press. [19] Massad, E, Burattini, MN, Azevedo Neto, RS, Yang, HM, Coutinho, FAB, Zanetta, DMT 1994. A model-based design of a vaccination strategy against rubella in a non-immunized community of Sao Paulo state, Brazil. Epidemiol Infect 112: 579-94. [20] Massad, E, Azevedo-Neto, RS, Burattini, MN, Zanetta, DMT, Coutinho, FAB, Yang, HM, Morales, JC, Pannuti, CS, Souza, VAUF, Silveira, ASB, Struchiner, CJ, Oselka, GW, Camargo, MCC, Omoto, TM, Passos, SD 1995. Assessing the efficacy of a mixed vaccination strategy against rubella in Sao Paulo, Brazil. Int J Epidemiol 24: 842-50. [21] Miller, E, Cradock-Watson, JE, Pollak, TM 1982. Consequences of confirmed maternal rubella at successive stages of pregnancy. Lancet 2: 781-84. [22] Monif, GRG, Hardy, JB, Sever, JL 1966. Studies in congenital rubella, Baltimore 1964-65. 1. Epidemiologic and virologic. Bull Johns Hopkins Hosp 118: 85-96. [23] Morice, A, Carvajal, X, Leon, M, Machado, V, Badilla, X, Reef, S, Lievano, F, Depetris, A, Castillo-Solozano, C 2003. Accelerated rubella control and congenital rubella syndrome prevention strengthen measles eradication: the Costa Rican experience. J Infect Dis 187: SI58-63. [24] Muench, H 1959. Catalytic Infection Models in Epidemiology. Harvard University Press. [25] Panagiotopoulous, T, Antoniadou, I, Valassi-Adam, E 1999. Increase in congenital rubella occurrence after immunization in Greece: retrospective survey and systematic review. Br Med J 319: 1462-66. [26] Ramsay, ME, Brugha, R, Brown, DWG, Cohen, BJ, Miller, E 1998. Salivary diagnosis of rubella: a study of notified cases in the United Kingdom, 1991-94. Epidemiol Infect 120: 315-19.
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[27] Robertson, SE, Cutts, FT, Samuel, R, Diaz-Ortega, J-L 1997. Control of rubella and congenital rubella syndrome (CRS) in developing countries, part 2: vaccination against rubella. Bull World Health Organ 75: 69-80. [28] Sever, JL, Hardy, JB, Nelson, KB, Gilkeson, MR 1969. Rubella in the Collaborative Perinatal Research Study. Amer J Dis Child 118: 123-32. [29] Veda, K, Nishida, Y, Oshima, K, Shepard, TH 1979. Congenital rubella syndrome: correlation of gestational age at time of maternal rubella with type of defect. J Pediatr 94: 763-65. [30] van der Heijden, OG, Conyn-van Spaendonck, MAE, Plantinga, AD, Kretzschmar, MEE 1998. A model-based evaluation of the national immunization programme against rubella infection and congenital rubella syndrome in The Netherlands. Epidemiol Infect 121: 653-71. [31] Whitaker, HJ, Farrington, CP 2004. Estimation of infectious disease parameters from serological survey data: the impact of regular epidemics. Stat Med 23: 2429-43. [32] White, LR, Sever, JL, Alepa, FP 1969. Maternal and congenital rubella before 1964: frequency, clinical features, and search for isoimmune phenomena. J Pediatr 74: 198-207. [33] World Health Organization 2000. Rubella vaccines: position paper. Wkly Epidemiol Rec 75: 161-72. [34] Zgoniak-Nowosielska, I, Zawilinska, B, Szostek, S 1996. Rubella infection during pregnancy in the 1985-86 epidemic: follow-up after seven years. Eur J Epidemiol 12: 303-8.
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Persistence of Vertically Transmitted Parasite Strains which Protect against More Virulent Horizontally Transmitted Strains* Thanate Dhirasakdanon, Horst R. Thieme* Department of Mathematics and Statistics, Arizona State University Tempe, AZ 85287-1804, U.S.A. E-mail: [email protected]@math. asu. edu
Abstract The question whether a vertically and a horizontally transmitted parasite strain can coexist under complete cross-protection is investigated in a host-parasite model with susceptibles and infectives only. It is shown that coexistence is possible even if the vertically transmitted strain would go extinct on its own provided that it is considerably less virulent than the horizontally transmitted strain. While the vertical transmitted strain is without benefit to the host as such, it protects the host against the more harmful horizontally transmitted strain. The coexistence is shown in the form of uniform strong persistence of the host and both parasite strains.
1
Introduction
There is a wide range of pathogens which are both horizontally and vertically transmitted (see [2, 6, 7, 8, 9] and the references mentioned there). Common sense suggests and mathematical models prove that a parasite which is only vertically transmitted cannot persist (unless it is beneficial to its host under certain circumstances, i.e. it is not always parasitic). In this paper we will demonstrate that a parasitic strain which is only vertically transmitted can persist in the presence of a more virulent horizontally (and also perhaps vertically) transmitted strain if the two strains provide complete cross-protection against each other, i.e. a host which is infected by one strain cannot be infected by the other ·partially supported by NSF grant DMS 0314529.
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Thanate Dhirasakdanon, Horst R. Thieme
strain. Cross-protection that is at least partially effective has been found in the woodland grass Brachypodium sylvaticum where vertical infection by the fungus Epichloe sylvatica makes the plants less susceptible to infection by horizontally transmitted strains [8]. Coexistence of the two strains (and the host) has numerically been observed for mass action incidence [7]. Coexistence at equilibrium has analytically been established first for mass action incidence and a logistic birthrate [7] and later for general incidence (including standard incidence) and general birth and death rates [3]. Here we will prove dynamic coexistence (uniform strong persistence of the host and both parasite strains). We also show global stability of the coexistence equilibrium for standard incidence and a host birth rate which linearly depends on host density. If the host birth rate is nonlinear, the coexistence equilibrium is unstable in certain parameter regions. (See Subsection 5.1 for more details, though proof and discussion of the local stability results will be presented in another publication.) The coexistence of the vertically strain and the horizontally transmitted strain is of interest for parasite evolution. At carrying host capacity (in absence of the parasite), the vertically transmitted strain has a replacement ratio which is strictly smaller than 1 while the horizontally transmitted strain has a replacement ratio which exceeds 1. The coexistence of the two strains is a counterexample to the principle of Ro-maximization that strains with higher basic replacement ratio drive strains with lower basic replacement ratio into extinction (see [12] for a survey). In view of the competitive exclusion principle, there are two consumers (the two parasite strains) and one resource (the host); still the two consumers coexist because horizontal and vertical transmission offer two different routes of resource utilization. We mention an apparent paradox [3]: at endemic equilibrium, the ratio of infections by the horizontally transmitted strain to infections by the vertically transmitted strain is a decreasing function of the coefficient of horizontal transmission. An analogous paradox has been observed in one-strain models where, at equilibrium, the ratio of horizontal infections to vertical infections is a decreasing function of the coefficient of horizontal transmission [6, 3].
2
A model with horizontal and vertical transmission
To set the stage we first formulate a one-strain model. Without the disease, the popUlation with density N(t) at time t develops as
N' = ((3(N) - J-l(N))N,
(2.1)
Persistence of Vertically Transmitted Parasite Strains which· . .
189
where fJ(N) and JL(N) are the per capita reproduction and mortality rates (of healthy individuals). Assumption 2.1. fJ(N) is a decreasing positive function of N ~ 0, JL(N) is an increasing positive function of N ~ O. Both are continuously differentiable, fJ'(N) - JL'(N) < 0 for all N > O. fJ(N) - JL(N) is positive for N = 0 and negative for large N > O. It follows from these assumptions that there exists a unique number
K > 0 such that fJ(K) - JL(K)
=
O.
(2.2)
K is called the carrying capacity of the host population in absence of the disease, because N(t) -> K as t -> 00 provided N(O) > O. The disease divides the population into a susceptible part, with density 8(t), and an infective part, with density I(t),
N=8+I, 8' =fJ(N)(8 + q(l - p)I) - JL(N)8 I'
(JC(~)8I + qpfJ(N)I -
(JC~)8I,
(2.3)
JL(N)I - aI.
The infection is vertically transmitted at the probability p, p E [0,1]. Infected individuals reproduce at the reduced rate qfJ(N), q E [0,1]. a is the additional per capita rate of dying from the disease. The parameter (J is a compound parameter whose exact interpretation depends on the specific transmission mode of the parasite. In fungal plant diseases, (J factors in the average spore production of a typical infected plant and the conditional probability that an infection occurs once a spore has landed on a susceptible plant. In sexually transmitted diseases, (J combines the average sexual activity of a typical sexually active person and the conditional probability that a given sexual contact between a susceptible and an infective individual actually leads to an infection. The parameter (J will be of central importance in our analysis, and we call it the horizontal transmission coefficient. The contact function C(N) describes how the per capita amount or rate of contacts depend on the host population density N. These may be direct contacts as in sexually transmitted diseases or indirect contacts as through spores in fungal plant diseases. Again the precise interpretation depends on the type of disease. 1/N is the conditional probability that a given contact made by a susceptible individual actually occurs with an infective individual. In fungal plant diseases, C(N) is proportional to the probability at which a given spore lands on host plants rather than on the soil (or
Thanate Dhirasakdanon, Horst R. Thieme
190
somewhere else where it is wasted) provided that the host plant density is N. At low host plant densities, this probability should be roughly proportional to the plant density which suggests that C(O) = O. In sexually transmitted diseases, C(N) is proportional to the number of sexual contacts a typical sexually active person makes in a population with density N. Some models assume that C(N) is basically independent of N unless the population density is so low that a deterministic model like ours is not valid anyway. This assumption results in what is sometimes called standard (or frequency-dependent) incidence [5, 2.1] and is a special case of assuming C(O) > O. The studies in [6, 7] assume mass action (or density-dependent) incidence where C(N) is proportional to N such that C(N)jN does not depend on host density. Our analysis includes both standard and mass action incidence and all reasonable interpolations between these two extremes. A collection of contact functions that have been used in the literature can be found in [11, Sec.19.1]; another example, C(N) = (In(a + LIN), has been suggested for insect diseases [1, App.B]. We replace the equation for S by an equation for N,
N' =({3(N) - p,(N))N I'
o-G(N)t - 1)1
((1 - q){3(N) + ex) I,
+ qp{3(N)I -
p,(N)I - exI.
(2.4)
We introduce the fraction of infective individuals, I
(2.5)
f= N· ,
I'
N
By the quotient rule, f = N - f N ' following form in terms of Nand f,
=
N' f (TI' - N)· The model takes the
N' =N({3(N) - p,(N) - ((1- q){3(N)
f' = f ([aC(N)
+ ex) f) , (2.6)
- ex - (1 - q){3(N)] (1 - f) - q(1 - p){3(N)).
Assumption 2.2. All parameters are non-negative, q > 0 (the disease does not sterilize), p < 1 (vertical transmission typically is imperfect). C(N) is an increasing function of N ~ 0, C(N) > 0 for N > o. C(N) is continuously differentiable at N > O. .
3
The persistence equilibrium
The origin is an equilibrium where both the host and the parasite are extinct. Potentially there are equilibria of three other types: the parasite
Persistence of Vertically Transmitted Parasite Strains which· . .
191
extinction equilibrium (K,O) with the carrying host capacity K > 0, the host extinction equilibrium (0, f#) with f# > 0, where the host is extinct and the parasite persists (not in absolute density but in proportion), and the persistence equilibrium (N*, f*) where both host and parasite persist.
3.1
Uniqueness and existence
There is at most one persistence equilibrium [3]. We restrict our consideration to imperfect vertical transmission, p < 1. This excludes that all hosts are infective at equilibrium. The following equation can be derived for the persistence equilibrium, a* := aG(N*) = (
q(l - p)(3* + a - q(3*
p,*
+ 1) ((1 -
q
)(3*
+ a)
(3.1)
,
where (3* = (3(N*) is a decreasing and p,* = p,(N*) an increasing function of N* and (3(N*) - p,(N*) is a strictly decreasing function of N* (Assumption 2.1). For details see [3]. We define R(N)
=
aG(N) + qp(3(N) . p,(N) + a
(3.2)
R(N) is the basic replacement ratio of the parasite at host population density N, i.e., the average number of new infective hosts produced by one infective host in a completely susceptible population of density N. Notice that iL(rJHo: is the mean length of the infective period. aG(N) is the average rate at which a typical infective individual produces new infections by horizontal transmission if it is introduced into a completely susceptible population of density N. qp(3(N) is the average rate at which a typical infective individual produces new infections by vertical transmission if the population density is N. Recall the carrying host capacity Kin (2.2).
Theorem 3.1 ([3]). There exists at most one equilibrium (N*, f*) at which both host and parasite persist, N*, f* > O. If the persistence equilibrium exists, N* depends in a strictly decreasing wayan a while the fraction of infectives f* depends in a strictly increasing wayan a. The persistence equilibrium exists, with 0 < N* < K, if and only if the following two conditions are satisfied:
(a) R(K) > 1, (b) either
+ a - q(3(O) ~ 0 p,(O) + a - q(3(O) > 0
p,(0)
or { aG(O)
< (
and
q(l - p )(3(0) p,(0) + a - q(3(O)
+ 1) ((1 -
q)(3(O)
+ a).
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Thanate Dhirasakdanon, Horst R. Thieme
Notice that condition (b) is satisfied if C(O) = 0 which includes the case of mass action incidence. Condition (a) guarantees that 1* > 0 while (b) guarantees that N* > O. If (a) holds but not (b), then the disease drives the host into extinction [14, 3].
3.2
Global stability of the persistence equilibrium
We state that the persistence equilibrium, when it exists, represents the long term behavior of the host-parasite dynamics.
Theorem 3.2. Let the assumptions (a) and (b) of Theorem 3.1 be satisfied. Then the persistence equilibrium (N* , 1*) is locally stable and all solutions N, f of (2.6) with N(O) > 0 and f(O) E (0,1] satisfy N(t) ----+ N* and f(t) ----+ 1* for t ----+ 00. Proof· The local stability of (N*, 1*) follows by linearization and a straightforward application of the Routh-Hurwitz criterion. The convergence of f and N is proved in [14, Sec.3].
o The proofs of the following global results can be found in [14, Sec.3]. The local results follow from standard linearized stability arguments in two dimensions.
Theorem 3.3. Assume that R(K) < 1. Then the equilibrium (K,O) is locally asymptotically stable and f(t) ----+ 0 and N(t) ----+ K for every solution of (2.6) with N(O) > 0, f(O) E [0,1]. Corollary 3.4. A completely vertically transmitted parasite (O' = 0) dies out, unless vertical transmission is perfect, p = 1, and the parasite is completely harmless, q = 1 and 0: = O. Proof. Let 0' = O. Recall that (3(K) = fL(K). So the sufficient condition for parasite extinction in Theorem 3.3 is satisfied if 0 < (l-pq)fL(K) +0:, i.e., if pq < 1 or 0: > O. 0
Theorem 3.5. Assume that R(K) > 1. Let 0'0 = O'C(O) , (3o and fLo = fL(O) and also assume fLo + 0: - q{3o > 0 and 0'0> (
Then N(t)
----+
q(l-p){3o +l)((l- q ){3o+o:). fLo + 0: - q{3o
0 as t
f(t)
--+
----+ 00
and
f# :=
0'0 0'0 -
(1 - qp){3o (1 - q){3o -
0: 0:
for every solution with N(O) ~ 0 and f(O) E (0,1].
>0
= (3(0)
Persistence of Vertically Transmitted Parasite Strains which· . .
4
193
The multiple strain model
We extend our model to allow for multiple strains of the parasite. We assume that there is cross-protection between the strains, i.e., a host that has been infected by one strain cannot be infected by another strain. We also assume that a host that has been infected one way (horizontally or vertically) cannot be again infected by the same strain the other way. In this section, we will consider arbitrarily many strains, let us say n, for some basic investigations. But soon we will restrict the consideration to two strains with the second strain only vertically transmitting. We will see that, if the horizontal transmission coefficient of the first strain is sufficiently high and the second strain is less virulent, the two strains can coexist. As before, N denotes the total density of hosts, S the density of susceptible, uninfected, hosts, I the total density of infected hosts, while I j denotes the density of hosts infected with strain j, n
n
S' = (S +
"£ qk(l - Pk)h ) (3(N) -
p,(N)S (4.1)
k=l
-
C(N)S ~ N
I
~ak k,
k=l
Ij
= I j (C(~)S aj + qjpj(3(N) - p,(N) - aj).
The parameters and parameter functions have the same meaning as before, but the epidemiologic parameters now carry an index which denotes the parasite strain. In the same way as for the one-strain model, we rewrite the system in terms of the total host density, n
k=l
Ij =Ij (a j C<;) (N -
~ Ik) + qjpj(3(N) -
(4.2)
p,(N) - a j ).
We introduce the fraction of strain j infective individuals, I·J f j -- N'
(4.3)
and the fraction of infective individuals n
(4.4)
194
Thanate Dhirasakdanon, Horst R. Thieme
. . . terms B y th e quotIent ru1e, f'j = f j (II TJ - NN ' ) . We reWrIte the system III of the total host density and the fractions of strain j parasites, n
N ' =N(f3(N) - M(N) -
2:= h((l- qk)f3(N) + a k )), k=1
fi =fj(ajC(N)(l-
1) -
(1- qjpj)f3(N) -
aj
(4.5)
n
+ 2:= h((l -
qk)f3(N)
+ ak)).
k=1 We derive a differential equation for the fraction of infected hosts, f, n
l' =(2:= ajfJ )C(N)(l -
n
f) - (1- f)
j=1
2:= fJ(1- qjpj)f3(N) j=1
n
- ff3(N)
2:= qj(l- pj)fj -
n
(1 - f)
j=1
2:= ajh
(4.6)
j=l
iI, ... ,
/j
Theorem 4.1. Let N and /n be non-negative numbers, 2::7=1 ~ 1. Then there exists a unique non-negative solution of (4.5) on lR+ such that N(O) N, fj(O) }j for j 1, ... , n. Moreover 2::7=1 fj(t) ~ 1 for all t ;;:: 0 and N(t) ~ max{N, K} where K is the
=
=
=
carrying capacity for the parasite-free host population, f3(K) Finally lim SUPt---> 00 N(t) ~ K.
= /-l(K).
Proof. We notice that
with locally Lipschitz continuous functions G j : lR+. ----; R So we a have an ODE system with a locally Lipschitz continous vector field and a unique solution on a maximal interval [0, b). By the form of the equations, fJ(t) = fJ(O) exp(J~ 1>j(s)ds) with 1>j = Gj(N, II,···, fn) as long as the solution exist. So fj ;;:: 0 as long as the solution exists and the same holds for N. So b > 0 can be chosen such that limSUPt--->b(N(t) + 2::7=1 fj(t)) = 00 if b < 00. We employ (4.6) and use Pj ~ 1, fj ;;:: 0, to obtain the differential inequality n
l' ~ (2:= ajfJ) C(N)(l j=l
f) - (1 - f)f3(N)
2:=(1 - qjpj)fJ j=l
n
- (1 - f)
n
2:= ajh j=l
(4.7)
Persistence of Vertically Transmitted Parasite Strains which· . .
195
This implies that, as long as the solution of (4.5) exists, d
-
-
dt (1 - f) ~ (1 - f)¢(t) with a continuous function ¢ : [0, b) inequality, 1 - 1(t)
~ (1 -
--+
R
We solve the differential
1(0)) exp(fot ¢(s)ds).
Since 1 - 1(0) ~ 0, also 1 - 1(t) ~ 0 for all t E [O,b). N satisfies the differential inequality,
N' :::;; N(f3(N) - f.L(N)). Let t E (0, b) and N = max[O,tj N(t). By [11, Lemma A.6J, N = N(O) or there exists some s E (0, tJ such that N'(s) ~ 0 and N(s) = N. SO o : :; N'(s) = N(f3(N) - f.L(N)). Since f3(N) - f.L(N) < 0 for N > K, N :::;; K. So N :::;; max{N(O), K}. Since this estimate does not depend on t E [O,b), N is bounded on [O,b) and N(t) :::;; max{N(O),K} for all t E [0, b). This implies that b = 00 and the estimate holds for all t ~ O. By the fluctuation method [4J [11, Prop.A.22J, there exists a sequence tj --+ 00, N(tj) --+ N OO , N'(tj) --+ 0 as j --+ 00. This implies 0:::;; N OO (f3(NOO) - f.L(NOO)). Since the right hand side of this inequality would be negative for N OO > K, N OO :::;; K. 0 In the remainder of this paper we assume that no strain sterilizes the host and that every strain has imperfect vertical transmission, i.e.,
• qj > 0 and Pj < 1 for all j = 1, ... , n. Theorem 4.2. There exists some c > 0 such that n
lim sup LJj(t) :::;; 1 - c t---+oo
j=1
for all solutions of (4.5) with N(O) ~7=1 iJ(O) :::;; 1.
~
0, iJ(O)
~
0, j
= 1, ... , n, and
A formula for c > 0 is found in the subsequent proof, see (4.8).
Proof. Set ~ = minj=1 qj(1 - Pj) and a = maxj'=1 CTj. Then ~ the following inequality is obtained from (4.6),
l' :::;;aC(N)(1 -
f) -
> 0 and
~f3(N)f.
Let 100 limsuPHOO 1(t). By the fluctuation met~od ([4J ~r [11, Prop.A.22J), there exists a sequence tk --+ 00 such that f(tk) --+ foo and
Thanate Dhirasakdanon, Horst R. Thieme
196
/,(tk) ----> 0 as k ----> and (3 decreasing,
Since limsuPt_+<X) N(t) ~ K and C is increasing
00.
o ~ o-C(K)(1- 1
00
We solve this inequality for -00
f
1
00
)
-
~(3(K)l°o·
,
o-C(K)
1
~ o-C(K) + ~(3(K) < .
(4.8) D
Lemma 4.3. Let OjC(O) ~ (1- qjPj){3(O) + aj for j = 1, ... , n. Then, for all solutions with N(O) = 0, fJ (t) ----> 0 as t ----> 00, j = 1, ... ,n.
Proof. Let N(O) written as
=
O. Then N(t)
=
0 for all t ~ 0 and (4.6) can be
n
l' = L fJ (O"jC(O) (1 - 1) -
(1 - f){3(0) (1 - qjpj)
j=l
(4.9)
- lqj(1 - Pj )(3(0) - (1 - f)aj). By assumption,
n
l' ~ - L
fJqj (1 - Pj )(3(0)f.
j=l Set ~ = minj=l, ... ,n qj(1 - Pj). Then ~ > 0 and /' ~ -~(3(O)p. This implies that f(t) ----> 0 as t ----> 00. D Theorem 4.4. Let O"jC(O) ~ (1- qjPj )(3(0) + aj for j = 1, ... ,n. Then the host population is uniformly persistent: there exists some E: > 0 such that lim inft---> 00 N(t) ~ E: for all solutions with N(O) > O.
Proof. We use the language and the results in Section Appendix: Our state space is X = {(N,it, ... ,fn) E lR~+1,L:?=lfJ ~ 1}. We split up X as X = Xl l±I X 2 (disjoint union) with Xl = {N > O} n X and X 2 = {N = O} n X. Then X is closed and X 2 is compact. Both Xl and X 2 are forward invariant. By Lemma 4.3, (lJ .... , 0) is globally asymptotically stable for X 2 . By Theorem 4.1, every solution in X tends to the compact set X n {N ~ K}. System (4.5) has the form of Lemma A.7 with x = (N, it, ... , fn). By assumption, 91 (0, ... ,0) = (3(0) - jL(O) > O. By Lemma A.7, (0, ... ,0) is a uniform weak repeller for X n {Xl> O} = X n {N > O} = Xl. By Proposition A.6, the singleton set containing (0, ... ,0) is an isolated invariant set for X. By Theorem A.4, X 2 is a uniform strong repeller for Xl which is equivalent to the statement of the theorem. D
Persistence of Vertically Transmitted Parasite Strains which· . .
5
197
The two strain model with one strain only vertically transmitted
We restrict our consideration to two parasite strains. The second strain is only vertically transmitted while the first strain is transmitted horizontally and possibly vertically too. Somewhat imprecisely, we will speak about the first strain as the horizontally transmitted strain (HT strain) and about the second strain as the vertically transmitted strain (VT strain). System (4.2) specializes to 2
N' =N((3(N) - {l(N)) - L1k((1- qk)(3(N) k=l 1~
=h ( (JG(N)
N
-h - 12 N
+ O!k),
+ qlPl(3(N) - {l(N) -
)
O!l ,
(5.1)
1~ =h (q2P2(3(N) - {l(N) - 0!2)' and the system (4.5) specializes to 2
N' =N((3(N) - {l(N) -
L Jk((l- qk)(3(N) + O!k)) , k=l
J{ =h((JG(N)(l- h - h) - (l-QlPl)(3(N) - O!l 2
+ L h((l - qk)(3(N) + O!k)) ,
(5.2)
k=l
J~ =12(-(1- Q2P2)(3(N) - 0!2 2
+ :E Jk ((1 - qk)(3(N) + O!k)). k=l
(J is again called the coefficient of horizontal transmission. We assume that both strains do some harm to the host, O!j > 0 or qj < 1 for j = 1,2. However, neither strain sterilizes the host, i.e. qj > 0 for j = 1,2. Further vertical transmission is imperfect for both strains, pj < 1 for j = 1,2.
5.1
Coexistence equilibrium
By the last equation of (5.1), an equilibrium where both parasite strains and the host coexist satisfies
o =q2P2(3(N*) -
{l(N*) - 0!2·
(5.3)
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Thanate Dhirasakdanon, Horst R. Thieme
Since the right hand side of this equation is strictly decreasing, we learn that N* is uniquely determined and does not depend on (1. We write (1* = (1C(N*), {3* = {3(N*), p,* = p,(N*). So {3* and p,* do not depend on (1 either, while (1* is proportional to (1. We define
(5.4) R2 (N) is the basic replacement number of the VT strain at population density N and is the special case of (3.2) for the VT strain ((1 = 0). The following is shown in [3J. Theorem 5.1. A (uniquely determined) coexistence equilibrium exists if and only if the following assumptions are satisfied:
(a) R2(0) > l. (b) The vertically transmitted second strain is less harmful than the horizontally transmitted first strain in the following way,
where N* is the unique solution ofR2(N*)
= l.
(c) The horizontal transmission coefficient is large enough,
Remarks 5.2. The equation R2(N*) = 1 in Theorem 5.1 (b) is equivalent to (5.3) by which the condition in Theorem 5.1 (b) is equivalent to
(5.5) The condition in Theorem 5.1 (c) is equivalent to
Notice that the right hand side of this inequality is strictly decreasing. Since N* ~ K, (5.6) also holds if N* is replaced by K. Since {3(K) = p,(K), the condition in Theorem 5.1 (c) implies that
(5.7) which is equivalent to condition (a) in Theorem 3.1 for q = ql, P = Pl. Let us assume that condition (b) in Theorem 3.1 also holds. Then we
Persistence of Vertically Transmitted Parasite Strains which· . . have a boundary equilibrium (N~, present. By (3.1),
ff, 0)
199
where only the HT strain is
Since the right hand side of this equation is decreasing, N~ < N*. We have the following stability results which will be proved and discussed elsewhere. Theorem 5.3. If the coexistence equilibrium exists, i.e. conditions (a), (b), (c) in Theorem 5.1 hold, it is locally asymptotically stable if
(i) the horizontal transmission coefficient (J" is sufficiently large, or (ii) the per capita birth rate f3 does not depend on the population density N. Under standard incidence, i.e. if the contact function C is constant, the coexistence equilibrium can be unstable for the following scenario: • P2 and q2 are close enough to 1, i.e.
the VT strain is almost perfectly vertically transmitted and causes almost no fertility reduction,
and • ql (I-pI) is close enough to 0, i.e. the HT strain is almost perfectly
vertically transmitted or sterilizes the host almost completely. Differently from the case of standard incidence, the coexistence equilibrium is locally asymptotically stable under mass action incidence (i.e. C(N)jN does not depend on N) if P2 is sufficiently close to 1. Whether or not the coexistence equilibrium is locally asymptotically stable whenever it exists is still an open question for mass action incidence.
5.2
Dynamic coexistence
The criteria for coexistence of both parasite strains and the host at equilibrium that were proved in Theorem 5.1 also guarantee dynamic coexistence. Theorem 5.4. The following are equivalent:
(i) There exists a coexistence equilibrium.
Thanate Dhirasakdanon, Horst R. Thieme
200
(ii) The horizontally and vertically transmitted strains coexist in the sense that there exists some c > 0 such that lim inf I j ( t) ;? c,
j
t~oo
= 1,2,
for all solutions of (5.1) with h(O) > 0,12(0) > 0, N(O) ;? h(O) + 12(0) . Existence of a coexistence equilibrium is necessary for the dynamic coexistence in (b) due to a general result [13, Thm.1.3.7]. The sufficiency is shown in Section 7. A global stability result can be shown if C and (3 do not depend on the population density N. Theorem 5.5. Assume that C and (3 are positive constants and that the coexistence equilibrium x* = (N*,Ii,In with N*,Ii,Ii, > 0 exists. Then all solutions of (5.1) with I 1 (0),h(0) > 0, N(O) ;? h(O) + h(O) converge towards the coexistence equilibrium.
The proof will be given in the next section.
6
Global stability for constant contact function and per capita birth rate
We consider the special case that C(N) and (3(N) do not depend on N. Notice that Assumption 2.1 implies that p'(N) > 0 for all N > O. In the following we show that whenever an endemic equilibrium exists where the VT strain is present (i.e. U > 0), then this equilibrium attracts all solutions with h (0) > 0 and 12(0) > O. Depending on a further threshold condition the host population goes extinct or converges to a positive limit. The equations for the strain fractions (5.2) become independent of the host equation,
((1 - h - h)aC + h,l + 12,2 - 1'1), f~ = 12 (f1l1 + 12,2 - 1'2), f{
=
h
(6.1)
where ,j =(1 - qj)(3 + aj,
i'j Notice that
=,j
i'j > 'j.
+ qj(l -
Pj)(3
= (1 -
qjPj)(3 + aj.
(6.2)
Persistence of Vertically Transmitted Parasite Strains which· . .
201
Lemma 6.1. The equilibrium ut, 0) with 0 < ft :::;; 1 exists if and only if !JC - 1'1 > 0, in which case
ft
=
!JC - 1'1 < 1. !JC - 1'1
(6.3)
Proof. ut,O) exists with 0 < ft if and only if (1- ft)!JC+ ftl'1 -1'1 = ~g::t > 0 and numerator and denominator have
o if and only if ft =
the same sign. Since 1'1 > 1'1, it :::;; 1 if and only if !JC - 1'1 > 0, in which case ft < 1. D
Lemma 6.2. An equilibrium 1 exists if and only if
Uf, U) with fi > 0, U > 0 and fi + U : :;
1. the equilibrium ut, 0) in Lemma 6.1 exists, and
2. ftl'1 - 1'2 > 0, where ft is given by (6.3). The equilibrium is unique. In case Uf, U) exists, we have 1'1 > 1'1 > 1'2 > 1'2, fhl + UI'2 = 1'2, and fi + U < 1. Proof. Suppose the equilibrium ut, 0) exists and ft :::;; 1, we have 1'1 > 1'1 > 1'2 > 1'2· Since
It 1'1 -
1'2 > O. Since
o<
o<
#
A
f 1 1'1 - 1'2 =
!JC - 1'1 (!JC - 1'1)"(1 - (!JC - 1'1)1'2 C 1'1 - 1'2 = C !J - 1'1 !J - 1'1 !JCbl - 1'2) - 1'1 (1'1 - 1'2) !JC - 1'1 A
we have !JC bl - 1'2) - 1'1 (1'1 - 1'2) > O. Define
f~ = !JCbl - 1'2) - 1'1(1'1 - 1'2) !JCbl - 1'2)
> o.
(6.4)
Since
!JCbl - 1'2) - 1'1 (1'1 - 1'2) < !JCbl - 1'2) - 1'1 (1'1 - 1'2) < !JCbl - 1'2), we also have i.e.,
U <
1. Define
fi
to be such that fhl
+ UI'2
- 1'2 = 0,
(6.5)
Thanate Dhirasakdanon, Horst R. Thieme
202
Then we have flo > because
"12 -
f~ "12
"11
= (1 - f 2 ) n. > o. (II' f2) is an equilibrium "11
(1 - ff - fn(J"C + if'Yl + f~'Y2 - 1'1 = (1 - if - fn(J"C - (1'1 - 1'2) (by (6.5))
=
f~ )(J"C -
(1 - 1'2 -'Y:2'Y2 -
(1'1 - 1'2)
(by (6.5))
(ry1 - 1'2 - UbI - 'Y2) )(J"C - 'Yl (1'1 - 1'2) 'Yl = bl - 1'2)(J"C - (1'1 - 1'2) - 'Ylbl - 'Y2)(J"CU = 0 'Yl
(by (6.4)).
Since (1 - fi - f2)(J"C = 1'1 - 1'2 > 0, we have fi + 12 < 1. Conversely, suppose the equilibrium (If, 12) exists with fi > 0, 0, and fi + 12 ~ 1. Then (1 -
ff -
fn(J"C
if'Yl
+ if'Yl + f~'Y2 - 1'1 = 0,
+ f~'Y2 -
1'2
=
12 > (6.6) (6.7)
O.
We rewrite (6.7) as (1 - f2)bl - 1'2) = (1 - fi - f2hl + 12(1'2 -1'2) and see that 1'1 > 'Yl > 1'2 > 'Y2. We combine (6.6) and (6.7),
(1 - if - fn(J"C - (1'1 - 1'2)
= O.
(6.8)
We rearrange (6.7) as
(1 - if - f~h1 = - f~bl - 'Y2) + 'Yl - 1'2.
(6.9)
We combine (6.8) and (6.9),
(- f~bl - 'Y2)
+ 'Yl -
1'2)(J"C - 'Yl (1'1 - 1'2)
=
0,
(6.10)
which we rearrange as
Since 'Y1 > 1'2, we have
(J"C> 'Yl(1'l - 1'2) _ 1'lb1 - 1'2) + 1'2(1'1 - 'Yd
,
1'1 - 1'2 and so the equilibrium
-
,
'Y1 - 'Y2
'
> 'Yl,
(It, 0) in Lemma 6.1 exists. By (6.11),
Persistence of Vertically Transmitted Parasite Strains which· . .
203
Lemma 6.3. Every solution h(t), 12(t) with
> 0,
liminf h(t) t -H:X> lim sup J(t)
liminf 12(t) t----o,. ex;
lim sup (h(t)
=
t---+oo
> 0,
and
+ 12(t)) < 1,
t--+CX)
converges to the interior equilibrium (tl' U). Proof. We have
If =
F 1, 1~
= F 2, where
F 1(h,12) =h((1- h - 12)a-G + h/1
+ 12/2 -11 -
q1(1- pd,B)
((1 - h - h) (aC - Id - 12 b1 - 12) F2(h, h) =12 (in1 + 12/2 -/2 - q2(1 - P2),B) =h
=12( -(1- h -
12h2 + hb1 -/2) -
q1 (1 - pd ,B) ,
q2(1- P2),B).
From the assumption, the w-limit of the solution is contained in D
{(h h)
=
E]R2 :
Define p: D ~]R by p(h,12) aC -11 12
=
0 < h, 0 < 12, h
+ 12 < 1}.
hh(l~h-h)' Then
F p 1
=
11 -/2
pF2
= _ 12 +
11 -/2
h
1 - h - 12
1 - h - 12
q1(1- P1)f3 12(1 - h - h) ,
q2(1 - P2),B h (1 - h - h) ,
and so OpF1 oh OpF2 012
11 -/2 (1- h - 12)2 11 -/2 (1- h - 12)2
q1 (1 - P1),B 12(1- h - 12)2' q2(1 - P2),B
11(1 - 11 - 12)2'
Hence OpF1 oh
+
OpF2 q1(1- P1),B _ q2(1 - P2),B <0 012 =-12(1-h-12)2 h(1-h-12)2 .
Since this is a planar dissipative system, the solution converges towards an equilibrium in D, actually to (tl' U) because there is only one interior equilibrium. D Lemma 6.4. For every solution, we have limsuPt-+oo f(t) < 1. If the interior equilibrium (tl' U) exists, we also have liminft-+oo h(t) > 0 and liminft-+oo 12(t) > 0 whenever h(O), 12(0) > O.
Thanate Dhirasakdanon, Horst R. Thieme
204
Proof. The first assertion follows from Theorem 4.2. Now assume that the interior equilibrium (fl' N) exists. Let
= {(h,h) E]R2 : 0 ~ II, 0 ~ 12, h + h Xl = {(fl, h) EX: h > 0, h > O}, X 2 = {(fl, h) EX: h = 0 or h = O}. X
~ I},
There are two equilibria in X 2, (0,0) and (ft, 0). Every solution in X with h(O) = 0 converges (0,0) and every solution in X with h(O) = 0, h (0) > 0 converges to (ft, 0). In particular, the two equilibria form an acyclic set in X2. We rewrite system (6.1) in the form of Lemma A.7. From Lemma 6.1 and Lemma 6.2, we have gl(O,O)
= aD - 1'1 > 0
and
g2(ft,0) = ftn -
1'2> O.
By Lemma A.7, (0,0) is a uniform weak repeller for X n {II > O} and (ft,O) a uniform weak repeller for X n {h > O}. On X n {h = O} we have f~ = h((h -1){2 - q2f3(1- P2)), which is negative whenever h E (0,1]. Hence (0,0) is locally asymptotically stable for X n {II = O}. By Proposition A.6, (0,0) is an isolated invariant set for X. It is easy to see that (ft, 0) is locally asymptotically stable for X n {h = O}. Since (ft, 0) is a uniform weak repeller for X n {fz > O}, (ft,O) is an isolated invariant set for X by Proposition A.6. By Theorem A.4, X 2 is a uniform strong repeller for Xl, in particular lim inft-> = h(t) > 0 and liminft ---+= fz(t) > 0 whenever II(O) > 0 and h(O) > O. 0
Theorem 6.5. If the unique equilibrium (fl,Jn with f1 > 0, N > 0, f1 + N ~ 1 exists {i.e. if the conditions in Lemma 6.2 are satisfied}, then it is locally asymptotically stable and every solution of (6.1) with h(O) > 0 and fz(O) > 0 converges to this equilibrium. Proof. The local stability follows from a standard linearized stability analysis in two dimensions. The convergence results follow from Lemma 6.3 and Lemma 6.4. 0
We return to the full system (5.2) which, by (6.2), can be rewritten in this special case as N' = N(f3 - f.t(N) - f1"l1 - hY2),
f{ = h ((1- h - h)aG + II'YI + fz'Y2 - 1'1),
f~ = fz (f1"l1 + f2'Y2 - 1'2)'
(6.12)
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Corollary 6.6. The unique coexistence equilibrium (N*, J;, f2) of (5.2) with fi > 0, f2 > 0, fi + f2 ~ 1, and N* > 0 exists if and only if q2P2(3 > p,(0) + a2 and the unique equilibrium (ii, U) of (6.1) exists with fJ > 0 and Ii + U ~ 1. Actually fi = Ii and f2 = U. If this equilibrium exists, every solution of (5.2) with h (0) > 0, 12(0) > 0 and N(O) > 0 converges to this equilibrium as t -+ 00.
Proof· By (6.2) and Lemma 6.2, q2P2(3 > p,(0) (3 - p,(0) - i2 > 0 and to
+ a2
is equivalent to
(6.13) which is a necessary and sufficient condition for the existence of a solution N* > 0 to the equation (6.14) Cf. the first equation in (6.12). By Theorem 6.5, h (t) -+ Ii and 12(t) -+ U· By (6.13) there exist T ~ 0 and c > 0 such that (3 - p,(0) h (t)'"n - 12(th2 ~ c for t ~ T. Since p, is continuous, there is 0 > 0 such that (3 - p,(x) - h(t)'"n- 12(th2 ~ c/2 > 0 for x E [0,0] and t ~ T. Since N' = N((3 - p,(N) - hll - 1212) and since N(t) > 0 for all t, we have Noo ~ 0 > O. By the fluctuation method [4] [11, Prop.A.22] and (6.7) and (6.2), we have
o=N°O ((3 -
p,(N°O) - fill - f~'2)
=Noo ((3 - p,(Noo ) - fill -
f~'2)'
Since N°O ~ Noo > 0, (6.14) holds for both N°O and Noo in place of N*. Since N* is the unique solution of (6.14), we have N°O = Noo = N* which implies that N(t) -+ N* as t -+ 00. 0 Corollary 6.7. If q2P2(3 ~ p,(0) + a2 and the equilibrium (ii, U) of (6.1) with Ii > 0, U > 0 and Ii + U ~ 1 exists, then every solution of (5.2) with h(O) > 0, 12(0) > 0 and N(O) ~ 0 satisfies N(t) -+ 0, h(t) -+ Ii, 12(t) -+ f2 as t -+ 00.
Proof. Let h(t), 12 (t), N(t) be the solution with h(O) > 0, 12(0) > 0 and N(O) ~ O. From Theorem 6.5 we have (h, h) -+ (ii, U). By Lemma 6.2, we have fiTI + UI2 = i2' By the fluctuation method [4] [11, Prop.A.22], we have from Lemma 6.2 and (6.2) 0= N°O
((3 - p,(N°O) -
= N°O (_p,(N°O)
fill -
+ q2P2(3 -
f~'2) = N°O ((3 - p,(N°O) - i2) a 2)'
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Since f.1 is strictly increasing, we have q2P2(3 - f.1(x) - a < 0 for x > O. Hence NCO = 0 and so N(t) ----; O. D
7
Uniform strong coexistence
This section is devoted to the proof that the HT and VT strain and the host uniformly coexist, whenever the coexistence equilibrium exists. By assumption we always have three equilibria: the origin (0,0,0), (K, 0, 0) and the coexistence equilibrium (N*, fi, f2) where all components are positive. Another possible equilibrium is (N", ff, 0), at which the host and the HT strain persist, i.e., the first two components are positive. This equilibrium corresponds to the equilibrium (N*, j*) for the one strain model considered in Section 2 and Section 3. The dynamics of the twostrain model on the invariant set {!z = O} are the same as the dynamics of the one-strain model. If C(O) > 0, there are two more possible boundary equilibria: (0, It, 0) and (0, ii, 12). We will use the results of Section 6 for the invariant set {N = O}. The parameters 'Yj and ij, (3 and C in Section 6 must then be understood as being evaluated at N = 0, and the equilibrium coordinates ft and ii, 12 in Section 6 are the same as in the boundary equilibria just mentioned. The dynamics of the host-parasite model with two strains restricted to the invariant set {N = O} are the same as the dynamics of two dimensional model considered in Section 6 (with the exception of Corollary 6.6 and Corollary 6.7). Let us summarize some previous results (Corollary 3.4, Theorem 3.2 Theorem 3.5). Consult Definition A.5. Lemma 7.1. (a) Solutions with N(O) > 0, fICO) = 0, !z(0) ~ 0 converge to (K, 0, 0). (K, 0, 0) is locally asymptotically stable for {fI =
O}. (b) There exists at most one equilibrium (N", f", 0) with N", f" > O. If it exists, it attracts all solutions with N(O) > 0, fICO) > 0, !z(0) = 0 and is locally asymptotically stable for {!z = O}.
(c) There exists at most one equilibrium (0, ii, 12) with f'j > O. If it exists, it attracts all solutions with N(O) = 0, fICO) > 0, !z(0) > 0 and is locally asymptotically stable for {N = O}. (d)
There exists at most one equilibrium (0, ft, 0) with ft > O. If it exists, it attracts all solutions with N(O) = 0 = !z(0) and h(O) >
o.
If it exists and (N", ff, 0) does not exist, it attracts all solutions with !z (0) = 0 and fI (0) > 0 and is locally asymptotically stable for {!z = O}.
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If it exists and (0, ii, U) does not exist, then it attracts all solutions with N (0) = and h (0) > and is locally asymptotically stable for {N = o}.
°
°
The last statement can be derived from the Poincare-Bendixson theory and a standard linearized stability analysis in the plane. For the next results consult Definition A.I.
Lemma 7.2. For the purpose of this lemma, let X = {(N,Jl, h) E ~t; h + 12 ~ 1}. If (N*, fi, fn exists, then: (a) (K, 0, 0) is a uniform weak repeller for X n {h > O}. (0,0,0) is a uniform weak repeller for X n {N > O}.
(b) Let (N~, ff, 0) exists. Then it is a uniform weak repeller for X n {12 > O}. (c) Let (0, fr, 0) exist. Then (0,0,0) is a uniform weak repeller for X n {h > O}. If (O,ff,U) does not exist, (O,fr,O) is also a uniform weak repeller for X n {N > O}.
(d) If (0, ii, U) exists, it is a uniform weak repeller for X n {N > O} and (0, fr, 0) is a uniform weak repeller for X n {12 > O}. Proof. The system (5.2) has the form in Lemma A.7 with x = (N, h, h). (a) As mentioned in Remark 5.2, the existence of the coexistence equilibrium implies that
By Lemma A.7, (K,O,O) is a uniform weak repeller for X n {X2 > O}
=
xn {h > O}. Let us turn to (0,0,0). By assumption, 91 (0,0,0) = 13(0) - JL(O) > O. By Lemma A.7, (0,0,0) is a uniform weak repeller for X n {Xl> O} =
Xn{N>O}. (b) As mentioned in Remark 5.2, N~ < N*. = 0, this implies
Since N* satisfies
q2P2!3(N*) -JL(N*) - a2
q2P2!3~ - JL~ - a2 > 0,
where !3~ = !3(N~) and JL~
93(N~, ff, 0)
= -
=
JL(N~). Then
(1 - q2P2)!3~ - a2
> -!3~ + JL~ + ff((1 -
+ ff((1 - qd!3~ + ad ql)!3~ + al) = 0,
with the last equality following from the first equation in (5.2) which is evaluated for N = N~ and h = ff, 12 = O. By Lemma A.7, (N~,ff,O) is a uniform weak repeller for X n {X3 > O} = X n {12 > O}.
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(c) Since (0, ff, 0) exists, o-C(O) - 1'1 second equation in (5.2),
> 0 by Lemma 6.1. By the
92(0,0,0) = o-C(O) - 1'1 > O. By Lemma A.7, (0,0,0) is a uniform weak repeller for X n {X2 > O} = X n {II> O}. If (0, ii, U) does not exist, then ff'Y1 :::;; 1'2 by Lemma 6.2. Since the coexistence equilibrium exists,R2(0) > 1 and q2P2f3(0) - p,(0) - 0:2 > 0 by Theorem 5.1. Then
gl (0, ff, 0) =13(0) + p,(0) - ff'Y1 ;;::: 13(0) - p,(0) - 1'2 =Q2P2f3(0) - p,(0) - 0:2 > O. By Lemma A.7,
(o,ff, 0) is a uniform weak repeller for
X n {Xl> O}
=
Xn{N>O}. (d) In this case, o-C(O) > 1'1 and ff'Y1 > 1'2. Then
g3(O, ff, 0) = -(1 - Q2P2)f3(O) - 0:2 + ff'Yl
=
-1'2 + ffil > O.
By Lemma A. 7, (0, ff, 0) is a uniform weak repeller for X X n {h > O}. Further
n {X3 > O} =
2
gl (0, if, fn = 13(0) - p,(0) -
L f~«l - Qk)f3(O) + O:k). k=l
Since N = 0, fj = ff solve the third equation in (5.2),
gl (0, if, f~)
=
Q2P2f3(0) - p,(0) - 0:2 > 0
by Theorem 5.1 (a). By Lemma A.7, (0, ff, U) is a uniform weak repeller for X n {Xl> O} = X n {N > O}. 0
Proposition 7.3. If (0, ff, 0) does not exist, there is some c
> 0 such
that (i) liminft-+oo N(t) ;;::: c for all solutions with N(O) > O.
> 0, h(O) > O. (iii) liminft-+oo h(t) ;;::: c for all solutions with N(O) > O,h(O) > 0, h(O) > O. (ii) liminft-+oo h(t) ;;::: c for all solutions with N(O)
Proof. Assume that (0, ff, 0) does not exist. By Lemma 6.1, o-C(O) :::;; (1 - QlPr)f3(O) + 0:1 and (i) follows from Theorem 4.4 with 0-2 = O. For (ii), we choose the state space X = {(N, h, h) E lR~; h + h :::;; 1; N > O}. By (i) and Theorem 4.1, all solutions in X are absorbed in a
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compact set which is contained in X. We split X = XIl±! X 2 with Xl = X n {h > O} and X 2 = X n {h = O}. The only equilibrium contained in X 2 is (K, 0, 0). (K, 0, 0) is globally stable for X 2 by Lemma 7.1 (a) and a uniform weak repeller for Xl by Lemma 7.2 (a). By Proposition A.6, it forms an isolated invariant set for X which is trivially acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (ii) . For (iii), we choose the state space X = {(N, h, h) E ~~; h + 12 ~ 1; N > 0, h > O}. By (i) and (ii) and Theorem 4.1, all solutions in X tend to a compact set which is contained in X. We split X = Xl l±J X 2 with Xl = X n {12 > O} and X 2 = X n {12 = O}. It follows from (i) and (ii) and [13, Thm.1.3.7] that the equilibrium (NH,ff,O) exists. It is the only equilibrium contained in X 2 and is globally asymptotically stable for X 2 by Lemma 7.1 (b). By Lemma 7.2 (b), it is a uniform weak repeller for X I and so forms an isolated invariant set by Proposition A.6 which is trivially acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl which implies (iii).
o
rtF,
Proposition 7.4. If (0, 0) exists, there is some c > 0 such that liminft-.oo h(t) ~ c for all solutions with h(O) > O.
Proof. We split up the state space X = {(N, h, h) E ~~, h + 12 ~ O} as X = XIl±! X 2 with Xl = {(N,h,12) E X;h > O} and X 2 = {(N, h, h) E X; h = O}. By Theorem 4.1, all solutions in X tend to the compact set X n {N ~ K}. X 2 only contains the equilibria (0,0,0) and (K, 0, 0). By Lemma 7.1 (a), all solutions in X 2 with N(O) > 0 converge towards (K, 0, 0) and (K, 0, 0) is locally asymptotically stable for X 2 . By Lemma 7.2 (a), (K, 0, 0) is a uniform weak repeller for Xl and thus an isolated invariant set for X by Proposition A.6. One readily checks that all solutions in X 2 with N(O) = 0 converge to (0,0,0), and (0,0,0) is locally asymptotically stable for X 2 n {N = O}. By Lemma 7.2 (a), (0,0,0) is a uniform weak repeller for {N > O}. Under the assumptions of this proposition, it is also a uniform weak repeller for {h > O} by Lemma 7.2 (c). So {(O,O,O)} is an isolated invariant set for X by Proposition A.6. Obviously M = {(O, 0, 0), (K, 0, O)} is acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl which implies the statement.
o
Proposition 7.5. If (0, some c > 0 such that
rtF, 0)
exists, but not (0, Ii, 12), then there is
(i) lim inft-+oo h (t) ~ c for all solutions with h (0)
> 0,
(ii) liminft-.oo N(t) ~ c for all solutions with h(O) > 0, N(O) > 0,
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Thanate Dhirasakdanon, Horst R. Thieme
(iii) lim inf t---+ 00 h(t) ;? c for all solutions with h(O) > 0, N(O) > 0, h(O) > O. Proof. (i) follows from Proposition 7.4. For (ii), we take the state space X = {(N, h, h) E lR~; h + h ~ 1, h > O}. By Proposition 7.4 and Theorem 4.1, X contains a compact set to which all solutions in X tend. We split up X = Xl I±I X 2 with Xl = {(N, h, h) E X; N > O} and X 2 = {(N, h, h) E X; N = O}. All solutions in X 2 converge to (0, it, 0) and (0, it, 0) is locally asymptotically stable for {N = O}. Since (0, fl, 12) does not exist, (0, ft, 0) is a uniform weak repeller for {N > O} and in particular for Xl by Lemma 7.2 (c). By Proposition A.6, {(O, it, is an isolated invariant set and obviously acyclic. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies assertion (ii). For (iii), we take the state space X = {(N, h, h) E lR~; h + h ~ 1,N> O,h > O}. We split up X = X I I±IX2 with Xl = {(N,h,h) E X; h > O} and X 2 = {(N, h, h) E X; h = O}. By (i) and (ii) and Theorem 4.1, all solutions in X tend to a compact set contained in X. X 2 contains the equilibrium (N~,if,O) which exists by (ii) and [13, Thm.1.3.7] and is globally asymptotically stable for X 2 . By Lemma 7.2 (b), (N~,jf, 0) is a uniform weak repeller for X I and so an isolated invariant set for X. By Theorem A.4, X 2 is a uniform strong repeller for X I. This implies the assertion. 0
on
Proposition 7.6. Assume that the equilibria (0, it, 0) and (0, fl, i~) exist, but not (N~, if, 0). Then there exists some c > 0 such that (i) liminft---+oo h(t) ;? c for all solutions with h(O) > 0, (ii) liminft---+oo h(t) ;? c for all solutions with h(O) > 0, 12(0) > O. (iii) liminft---+oo N(t) ;? c for all solutions with h(O) > 0, h(O) > 0, N(O) > 0, Proof. (i) follows from Proposition 7.4. For (ii), we take the state space X = {(N,h,h) E lR~;h + h ~ 1,h > O}. By Proposition 7.4 and Theorem 4.1, all solutions in X converge to a compact set which is contained in X. We split X = Xl I±I X 2 with Xl = X n {h > O} and X 2 = Xn{h = O}. Since (N~, itO) does not exist, (0, i#,O) is globally asymptotically stable for X 2 by Lemma 7.1 (d). By Lemma 7.2 (d), it is a uniform weak repeller for X 2 and so forms an isolated invariant set for X. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (ii). For (iii), we choose the state space X = {(N, h, h) E lR~; h + 12 ~ 1, h > 0,12 > O}. By (ii) and Theorem 4.1, all solutions in X tend to a compact set which is contained in X. We split X = Xl I±I X 2 with
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Xl = X n {N > O} and X 2 = X n {N = O}. By Lemma 7.1 (c), (0, ii, U) is globally asymptotically stable for X 2 . By Lemma 7.2 (d), it is a uniform weak repeller for Xl and so forms an isolated invariant set by Proposition A.5. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (iii).
o Proposition 7.7. Assume that (0, ff, 0), (0, ii, U) and (N~, ff, 0) exist. Then there exists some
(i) lim inft->oo h (t) ~
E
E
> 0 such that
for all solutions with
h (0) >
(ii) lim inft-+oo N (t) ~ E and lim inf t-+oo 12 (t) ~ h(O) > 0, 12(0) > 0 and N(O) > 0,
E
0,
for all solutions with
Proof. (i) follows from Proposition 7.4. For (ii), we take the state space X = {(N, h h) E lR~; h + 12 ::::; 1, h > O}. By Proposition 7.4, all solutions in X converge to a compact set which is contained in X. We split X = X l I±JX2 with Xl = X n {N > 0,12 > O} and X 2 = X n {N = o or 12 = O}. By Lemma 7.1, (0, ii, U) is globally asymptotically stable for X n {N = 0, 12 > O}, (N~, ff, 0) is globally asymptotically stable for X n {N > 0, 12 = O} and (0, ff, 0) is globally asymptotically stable for X n {N = 0,12 = O}. This implies that (O,ff,O) is a uniform weak repeller for {N > O} and {h > O} and thus an isolated invariant set for X. By Lemma 7.2 (d), (0, ii, U) is a uniform weak repeller for {N > O} and forms an isolated invariant set for X. (N~, ff, 0) is locally asymptotically stable for X n {h = O} by Lemma 7.1 (b) and a uniform weak repeller for Xn{h > O} by Lemma 7.2 (b). By Proposition A.5, it forms an isolated invariant set for X. The set M consisting of these three equilibria is acyclic in X 2 as one can see from the dynamics described before. By Theorem A.4, X 2 is a uniform strong repeller for Xl. This implies (ii). 0
Appendix: Elements of persistence theory Let F : lR+. ~ lR n be locally Lipschitz and consider the ODE system x' = F(x). A set X c lR+. is called forward invariant, if all solutions with x(O) E X are defined for all t ~ 0 and x(t) E X for all t ~ O. X is called invariant, if all solutions with x(O) E X are defined for all t E lR and x(t) E X for all t E R The distance from a point x to a set Y is given by d(x, Y) = inf{llxYII;Y E Y}.
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Definition A.I. We assume that X is a forward invariant subset of ~+, X = Xl U X 2 , Xl n X2 = 0, with X 2 being a relatively closed subset of X and Xl forward invariant. Let Y2 ~ X2· Y2 is called a uniform weak repeller for X I if there exists some is > 0 such that limsupd(x(t), Y2 ) ~ <5 t-+oo
for all solutions x(t) with x(O) E Xl· Y 2 is called a uniform strong repeller for Xl if there exists some <5 such that lim inf d(x(t), Y 2 ) ~ <5
>0
t-+oo
for all solutions x(t) with x(O) E Xl.
Definition A.2. We assume that X is a forward invariant subset of ~+, and Z ~ X an invariant set. Then Z is called an isolated invariant set in X if there exists some open subset U of ~n such that Un X contains no invariant subset other than Z. Definition A.3. Let Y
~ ~+.
An equilibrium x* E Y is called chained
in Y to an equilibrium y* in Y, x* ~ y*, if there exists a solution x' = F(x) which is defined for all t E ~ and takes all its values in Y such that x(t) -----> x* as t -----> -00 and x(t) -----> y* as t -----> 00 and there exists some t E ~ such that x(t) i= x* and x(t) i= y*. A set M of equilibria in Y is called cyclic in Y if there exists some x* E M with x* ~ x* or if there exist xi, ... ,x'k in M such that xi ~ X
2 2: ... 2: x'k 2: xi,
and acyclic in Y if it is not cyclic in Y.
The following result is a special case of [10, Thm. 4.6].
Theorem A.4. Let F : ~+ -----> ~n be locally Lipschitz continu01.ts, X ~ Assume that X is forward invariant for x' = F(x). Let X = Xl UX2, Xl nX2 = 0, with X 2 being a relatively closed subset of X and Xl forward invariant. Assume that there exists a compact set C in ~n, C ~ X, to which every solution of x' = F(x) in X tends: d(x(t), C) -----> 0 as t -----> 00. Let M be a finite set of equilibria in X 2 • Assume that every solution that starts in X 2 and stays in X 2 for all forward times converges to one of the equilibria in M _ Assume that every equilibrium in M forms an isolated invariant set in X and is a weak repeller for Xl and that M is acyclic in X 2 Then X 2 is a uniform strong repeller for Xl-
~+.
We present a condition for a set to be an isolated invariant set.
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Definition A.5. Let x* E Y ~ lR:;:' be an equilibrium, F(x*) = O. Then x* is called locally stable for Y if the following holds: (i) For every c > 0 there exists some 8 > 0 with the following property: If x: lR+ --+ Y is a solution of x' = F(x), defined for all t ;? 0 with values in Y, and Ilx(O)-x*11 < 8, then Ilx(t)-x*11 < c for all t;? o. x* is called locally asymptotically stable for Y if is is locally stable and
(ii) There exists some 80 > 0 with the following property: If x : lR+ --+ Y is a solution of x' = F(x), defined for all t ;? 0 with values in Y, and Ilx(O) - x* II < 80 , then x(t) --+ x* for t --+ 00. x* is called globally asymptotically stable for Y if is is locally stable and
(iii) x(t) --+ x* for t --+ 00 for all solutions x : lR+ defined for all t ;? 0 with values in Y.
--+
Y of x'
=
F(x),
--+ IRn be locally Lipschitz and X ~ lR:;:' be forward invariant for the solutions of x' = F(x). Assume that X = Y1 I±J Y2 where Y1 is also forward invariant and X and Y2 are closed. Let x* E Y2 be an equilibrium, F(x*) = o. Assume that {x*} is a uniform weak repeller for Y1 and x* locally asymptotically stable for Y2. Then {x*} is an isolated invariant set for X.
Proposition A.6. Let F : IR:;:'
Proof. Let 81 > 0 be such that limsupt-+oollx(t) - x*11 > 81 for all solutions x: lR+ --+ Y 1 . Let 82 > 0 be such that limt-+oo x(t) = x* for all solutions x: lR+ --+ Y2 with Ilx(O) - x* II < 82 . Let 8 = ~ min(8 1 , 82 ). Let Me X n B8(X*) be an invariant set. We have M n Y1 = 0 because if Xo E M n Y1 and x : lR+ --+ Y1 is the solution with x(O) = Xo, then there is t E lR+ such that Ilx(t) - x* II > 81 > 8, and so x(t) (j. M, contradicting the invariance of M. Hence M C Y2 n B8(X*). Now suppose (to get a contradiction) that we can pick Zo E M ~ Y2 , Zo -I- x*. Let z : IR --+ M be the solution (defined on all lR) with z(O) = zoo We have z(t) E Y2 for all t E lR because if there is t' E lR such that z(t') E Y 1 , then there is t" > t' with Ilz(t") - x*11 > 81 > 8, again contradicting the invariance of M. Let c = ~llz(O) - x*11 and choose 8' > 0 such that Ilx(t) - x* II < c for every solution x : lR+ --+ Y2 with Ilx(O) - x*11 < 8'. Since M c B8(X*) and B8(X*) is compact, the a-limit set of z, a(z), is not empty and we have a(z) c M n Y2 c X n B8(X*). Since Y2 is closed, a(z) C Y2 n B8(X*).
214
Thanate Dhirasakdanon, Horst R. Thieme
Now pick Xo E a(z) C Y2 n Bt5(x*) and let x : 1R+ ---; a(z) be the solution with x(O) = Xo. Then limt_oo x(t) = x*, and since a(z) is closed, we have x* E a(z). Therefore there exists f < 0 such that Ilz(f) - x*11 < 6'. But then we have Ilz(O) - x*11 < c = ~llz(O) - x*ll, so we get a contradiction. Therefore M = {x*}. 0 The following easy lemma will be used again and again to show that a point is uniform weak repeller. Lemma A.7. Let gj : 1R+ ---; 1R be continuous, j = 1, ... , m. Consider the system of differential equations xj = Xjgj(x), j = 1, ... , m, x(t) = (Xl(t), ... xm(t)). Let X ~ 1R+ be forward invariant. Let x EX, j E {1, ... ,m}, Xj = 0, and gj(x) > O. Then x is a uniform weak repeller for X n {x j > O}.
Proof. Set c = gj(x)/2. By continuity, choose 6 > 0 such that Igj(x) gj(x)1 < c whenever Ix - xl < 6. So gj(x) ;;:: c whenever Ix - xl < 6. Suppose that x is not a uniform weak repeller for X n {Xj > O}. Then there exist a solution with Xj(t) > 0 for all t ;;:: 0 and limsuPt-oo Ix(t)xl < 6. Then x' (t) liminf ----L-() ;;:: liminfgj(x(t)) ;;:: t-oo Xj t t-oo
This implies Xj(t) ---;
00
as j ---;
00,
f
> O.
a contradiction.
o
Acknowledgement The authors thanks Stanley H. Faeth and Karl-Peter Hadeler for helpful discussions.
References [1] BRIGGS, C.J., H.C.J. GODFRAY, The dynamics of insect-pathogen interactions in stage-structured populations, The American Naturalist 145 (1995), 855-887 [2] BUSENBERG, S.N., K.L. COOKE, Vertically Transmitted Diseases: Models and Dynamics, Springer, Berlin Heidelberg 1993 [3] FAETH, S.H., K.P. HADELER, H.R. THIEME, An apparent paradox of horizontal and vertical disease transmission, J. Bio!. Dyn. 1 (2007), 45-62 [4] HIRSCH, M.W., H. HANISCH, J.-P. GABRIEL, Differential equation models for some parasitic infections: methods for the study of
Persistence of Vertically Transmitted Parasite Strains which· . .
215
asymptotic behavior, Commun. Pure Appl. Math. 38 (1985), 733753 [5] HETHCOTE, H.W., The mathematics of infectious diseases, SIAM Review 42 (2000), 599-653 [6] LIPSITCH, M., M.A. NOWAK, D. EBERT, RM. MAY, The population dynamics of vertically and horizontally transmitted diseases, Proc. R. Soc. Lond. B 260 (1995),321-327 [7] LIPSITCH, M., S. SILLER, M.A. NOWAK, The evolution of virulence in pathogens with vertical and horizontal transmission, Evolution50 (1996), 1729-1741 [8] MEIJER, G., A. LEUCHTMANN, The effects of genetic and environmental factors on disease expression (stroma formation) and plant growth in Brachypodium sylvaticum infected by Epichloe sylvatica, GIKGS 91 (2000), 446-458 [9] SAIKKONEN, K., P. WALl, M. HELANDER, S.H. FAETH, Evolution of latency in foliar fungi, Trends in Plant Science 9 (2004),275-280 [10] THIEME, H.R., Persistence under relaxed point-dissipativity (with application to an epidemic model), SIAM J. Math. Anal. 24 (1993), 407-435 [11] THIEME, H.R, Mathematics in Population Biology, Princeton University Press, Princeton 2003 [12] THIEME, H.R, Pathogen competition and coexistence and the evolution of virulence, Mathematics for Life Sciences and Medicine (Y. Takeuchi, Y. Iwasa, K. Sato, eds.), Springer, Berlin Heidelberg 2007 [13] ZHAO, X.-Q., Dynamical Systems in Population Biology, Springer, New York 2003 [14] ZHOU, J., H.W. HETHCOTE, Population size dependent incidence in models for diseases without immunity, J. Math. Biol. 32 (1994), 809-834
216
Richards Model: A Simple Procedure for Real-time Prediction of Outbreak Severity* Ying-Hen Hsieh Department of Applied Mathematics, Chung Hsing University Taichung 401, Taiwan, China E-mail: [email protected]
Abstract We propose to use Richards model, a logistic-type ordinary differential equation, to fit the daily cumulative case data from the 2003 severe acute respiratory syndrome outbreaks in Taiwan, Beijing, Hong Kong, Toronto, and Singapore. This model enabled us to estimate turning points and case numbers during each phases of an outbreak. The 3 estimated turning points are March 25, April 27, and May 24. Our modeling procedure provides insights into ongoing outbreaks that may facilitate real-time public health responses when faced with infectious disease outbreak in the future.
1
Introduction
Prediction of the future is a risky but tantalizing endeavor in any discipline in the scientific studies of natural phenomena, be it that of climate change, seismic movement, or occurrence of deadly diseases, not to mention the ascertaining of social phenomena such as economic trends and market volatility. In recent decades, the utilization of mathematical models in the studies of infectious diseases (e.g., [2]) for the purpose of public health prevention and control has placed the predictive abilities of the models in high demanding, especially for newly emerging disease outbreaks where public health policy makers must decide on the best 'This research is supported by NSC of Taiwan under grant (95-125-M005-003). The Singapore part of the work was carried out while the author visited Institute of Mathematical Sciences, National Singapore University. The article was written while the author visited the School of Mathematics at University of New South Wales, Sydney, Australia, funded by Taiwan CDC grant (DOH95-DC-1407)
Richards Model: A Simple Procedure for Real-time Prediction··· 217 course of intervention measures as crucial scientific knowledge regarding the disease outbreak is being gathered and observations or theories can be tested as understanding of the phenomenon develops (e.g., [1, 30, 6, 23]). For novel infectious diseases such as the severe acute respiratory syndrome (SARS) outbreak of 2003, the importance of proper prediction of the disease severity at the early stages of the outbreak became even more evident [31, 4, 26, 10]. In a 1972 paper on predictions of future human populations, Keyfitz [16] made the distinction between two types of prediction. One is a "projection" which is a consequence of a set of assumptions; the other is a forecast, an unconditional statement of what will happen, albeit perhaps with a measure of the uncertainty. The two are related in the sense that, often, the methods for projection provide means with which forecasts are possible. In the aftermath of the SARS outbreak, for example, Massad et al. [23] attempted to analyze the distinction between forecasting and projection models as assessing tools for the estimation of the impact of intervention strategies, by providing a projection of what would have happened with the course of SARS epidemic if the universal procedures to reduce contact were not implemented in the affected areas. In an endeavor to assess the effectiveness of intervention measures during the SARS pandemic, Zhou and Yan [38] used Richards model, a logistic-type model [32], to fit the cumulative number of SARS cases reported daily in Singapore, Hong Kong, and Beijing. In that article, they obtained estimates for the cumulative case number and basic reproduction number for each affected area. However, only partial case data during the outbreak was used which influenced the accuracy of the result. More seriously, the inflection point of the logistic curve, which could provide vital information pertaining to the changing trends of the epidemic and possibly indicating changes in intervention and control, was not discussed. Hsieh et al. [12] proposed to use Richards model, along with the complete Taiwan SARS case data from the beginning of the outbreak to its end, to obtain an estimate of the accumulative case number. Moreover, the inflection point of the S-shaped epidemic curve was obtained which indicates the turning point of the outbreak in Taiwan when the daily number of infections starts to decrease. More recently, Hsieh and Cheng [14] use the SARS case data of Greater Toronto area (GTA) to demonstrate that even for a multi-staged epidemic, Richards model still can be used for real-time prediction of outbreak severity as well as real-time detection of turning points. In this work, we will give a complete overview of Richards model as a useful tool for public health purposes of instantaneous ascertaining of a short and ongoing disease outbreak. We will introduce some basics of Richards model in the next section. In Section 3, we will demonstrate
218
Ying-Hen Hsieh
the use of model in outbreaks where the cumulative case curve exhibits an S-shaped curve by using the SARS data of Taiwan, Beijing, and Hong. In Section 4 we will make use of the SARS data of GTA and Singapore to demonstrate that the same procedure can be used for realtime prediction of an outbreak with multiple waves. Finally, we give some remarks in Section 5.
2
Logistic and Richards models
The logistic model was first proposed by Verhulst [34] in 1838 to model population growth after reading Thomas Malthus' An Essay on the Principle of Population [22]. The model equation, also known as Verhulst equation, is as follows:
l'(t)
=
rl[l-
~],
(2.1)
where let) is the population size in question at time t, r is the intrinsic growth rate, and K is "scarrying capacity". In his 1995 book How Many People Can the Earth Support, Cohen [5] explained that Verhulst attempted to fit a logistic curve based on the logistic function to 3 separate censuses of the population of the United States of America in order to predict future growth. Interestingly, all 3 sets of predictions failed. This equation is also sometimes called the Verhulst-Pearl equation following its rediscovery by Pearl in 1920's (see, e.g., [28]). Pearl, together with Reed, used Verhulst's model to predict an upper limit of 2 billion for the world population. This was passed in 1930 [29]. A later attempt by Pearl and an associate Sophia Gould in 1936 then estimated an upper limit of 2.6 billion. This was passed in 1955. Alfred J. Lotka also derived the equation again in 1925, calling it the law of population growth [20]. In 1959, Richards [32] proposed the following modification of the logistic model to model growth of biological populations:
l'(t) = rl[l- (i)a]. K
(2.2)
The additional of the parameter a provide a measure of flexibility in the curvature of the S shape exhibited by the resulting solution curve. As a model for the growth of an epidemic outbreak, let) is the cumulative number of infected cases at time t in days, K is the carrying capacity or total case number of the outbreak, r is the per capita growth rate of the infected population, and a is the exponent of deviation from the standard logistic curve. Unlike models with several compartments commonly used to predict the spread of disease, the Richards model considers only the cumulative infective population size with saturation in growth as
Richards Model: A Simple Procedure for Real-time Prediction··· 219 the outbreak progresses, caused by decreases in recruitment because of attempts to avoid contacts (e.g., wearing facemask) and implementation of control measures. The basic premise of the Richards model is that the daily incidence curve consists of a single peak of high incidence, resulting in an S-shaped epidemic curve and a single turning point of the outbreak. These turning points, defined as times at which the rate of accumulation changes from increasing to decreasing or vice versa, can be easily located by finding the inflection point of the epidemic curve, the moment at which the trajectory begins to decline. This quantity has obvious epidemiologic importance, indicating either the beginning (i.e., moment of acceleration after deceleration) or end (i.e., moment of deceleration after acceleration) of a phase. The analytic solution of (2.2) is (2.3) It is trivial to show that ti is the only inflection point (or turning point denoting deceleration after acceleration) of the S-shaped epidemic curve obtained from this model. Moreover, tm = ti + (lna)jr in (2.3) is equal to the inflection point ti when a = 1, and approximates ti when a is close to 1.
3
Single wave out break
The Richards model fits the single-phase SARS outbreak in Taiwan [12] well. We give below the parameter estimation results and the theoretical epidemic curve for Taiwan SARS outbreak of February 23-June 12, 2003, using Richards model from [12] in Table 1 and Figure 3.1, respectively. The result indicated that the infection occurred on May 3, and the estimate for the maximum case number of K = 343.3 [95%CI: (340,347)] is merely 0.8% off the actual total case number of 346. Moreover, the case number data used was sorted by onset date. Given a mean SARS incubation of approximately 5 days [37], the inflection point for SARS in Taiwan could be traced back to 5 days before May 3, namely April 28. On April 26, the first SARS patient in Taiwan died. Starting April 28, the government implemented a series of strict intervention measures, including household quarantine of all travellers from affected areas [17]. In retrospect, April 28 was indeed the turning point of the SARS outbreak in Taiwan. It is also interesting to note that, using this method, relatively accurate estimates for the turning point of the epidemic and the final epidemic size can be obtained fairly early [14]. In this instance, estimate of turning point on May 3 can be obtained using case data of up to May 10, while CI interval for total case number of (298, 370) is obtained using
Ying-Hen Hsieh
220
Table 1: Estimates of parameters in Richards model using cumulative confirmed SARS case data in Taiwan (N=346) of selected time periods. t m =66.6 implies the turning point of epidemic is May 3. (Source: [12]) a 10.0632
875.8
95% C.L (0*,147247)
204.9
(185.2,224.6)
0.2737
4.7745 2.4169
253.1
(232.1,274.2)
67.508
0.1483
1.2326
334.2
(298.2,370.2)
67.432
0.1419
1.1694
342.1
(321.5,362.6)
66.6187
0.1359
1.0731
343.4
(339.7,347.1)
Time Period 2/25 - 4/28
tm
r
78.2193
1.1421
2/25 - 5/05 2/25 - 5/10
65.5108 66.9819
0.5343
2/25 - 5/15 2/25 - 5/20 2/25 - 6/15
K
*max(O, lower bound)
350F==::::::::::::::::================::;;iiiiii ,----------- . : • Real Data 300 --2/25-6/15.---------§-:..-----\ i 2/25-5/5 ~ 250 i • • • • 2/25-5/10--------Ir....,...-~~~~-"i ~:: . :--. _.--2/25-5/15 ---_ . ----.
.~ 200
1 u
150~----------_#:..-------~
100~---------~--------~ 50~--------~---------~
oL-__c=~~__________~ 2/25 3/7
3/17 3/27 4/6
4/16 4/26 5/6 Date
5/16 5/26 6/5
6/15
Figure 3.1: The theoretical epidemic curve for Taiwan SARS outbreak during February 23-June 12, 2003, using Richards model. Turning point is May 3. (Source: [12])
data up to May 15. This indicates that, if no deviation from the actual events had occurred, the authority could detect the turning point (for the better) of the outbreak about one week after its occurrence. Furthermore, a range for the final epidemic size could be estimated a month before the end of the outbreak. The real-time predictive potential of this procedure will be discussed in more details in the Conclusions section. We also note that although we did use the additional laboratory confirmed case data in Taiwan as detailed in [13], estimation studies have shown that the accuracy of the procedure will not be compromised even
Richards Model: A Simple Procedure for Real-time Prediction··· 221 if we did use the additional case data. For the purpose of illustration and comparison, we perform the same the 2003 procedure to the SARS data from other affected areas. epidemic, the largest outbreak of SARS occurred in Beijing in the of 2003. Multiple importations of SARS to Beijing initiated transmission in several healthcare facilities. The outbreak in Beijing March 5, and by late April daily hospital admissions for SARS exceeded 100 for several days. According to [18], total 2,521 cases of probable SARS occurred. We reconstruct the daily incidence data from epidemic curve given in Figure 1 of [18] and obtain the incidence data of 2380 cases with hospitalization dates between March 5 and 29 in 3.2. We also note that the official cumulative case number in is 2631 as published by World Health Organization (WHO) website (see or [36]). 180 150 120
'" "S"
i
90 60 30 0 3/5
3/15
3/25
4/4
4/14
4/24
5/4
5114
5/24
Date
Figure 3.2: The daily SARS incidence curve by hospitalization date for SARS outbreak during March 3-May 29, 2003. (Source: The data was used to estimate the parameters in Richards model. The parameter estimates and the resulting theoretical epidemic curve are in Table 2 and Figure 3.3, respectively. The estimate for total case number of J{ = 2352 [95%CI: 2369)] is somewhat less accurate than that of Taiwan SARS, due to the fact that not all probable cases (totaling 1521) were accounted for in the data used. Moreover, the epidemic curve data of was by hospitalization date, which was affected by variance in
Ying-Hen Hsieh
222
Table 2: Estimates of parameters in Richards model using cumulative confirmed SARS case data of 2380 cases in Beijing during March 5-May 29, 2003. ti=51.92 implies the turning point of epidemic is April 26. Period 3/5 - 4/25
a 7.62
tm
t;
K
0.846
78.14
75.74
25385.5
3/5 - 4/30
0.751
6.77
54.30
51.75
1798.1
(1707.7 - 1888.4)
3.53
55.41
52.24
2097.3
(2039.6 - 2155.0)
r
(0'
95% C.L 7012482.0)
3/5 - 5/05
0.398
3/5 - 5/10
0.321
2.80
55.47
52.27
2198.5
(2164.3 - 2232.7)
3/5 - 5/15
0.274
2.33
55.27
52.19
2264.7
(2238.2 - 2291.2)
3/5 - 5/20
0.242
1.98
54.90
52.07
2315.3
(2291.9 - 2338.6)
3/5 - 5/29
0.219
1.74
54.45
51.92
2351.8
(2334.8 - 2368.8)
*max(O, lower bound)
2500
j. 2000
- -
Real Data
I
/.
I
J J
t
§<=I 1500 §
u
.--Ul·U!·!
"",.p"
/"
3/5-5/29
..0
i
...
,r
J
,
1000
500
o 3/5
I",'
I
I
...,,:,,,.''''''"
.d/.!!-........
3112 3/19 3/26 4/2
4/9 4116 4/23 4/30
517 5/14 5/21 5/28
Date
Figure 3.3: The theoretical epidemic curve for Beijing SARS outbreak during March 3-May 29, 2003, using Richards model. Turning point is April 26.
the time it took for each case to be hospitalized, also could result in inaccuracy in the estimates. Assuming that it takes at least 24 hours for a symptomatic SARS case to be hospitalized (see e.g., [13]), the turning point of April 26 by hospitalization date leads us to conclude that the turning point for SARS infections in Beijing had occurred on
Richards Model: A Simple Procedure for Real-time Prediction· .. 223 or before April 20. It is interesting to note that, on April 17, 123 fever clinics were set up in all secondary and tertiary hospitals in Beijing to monitor suspected SARS cases with onset of symptoms [27]. Perhaps more importantly, on April 20, the outbreak was announced publicly by the Chinese government for the first time, thus alerting the domestic population as well as the international community to the presence of this possibly fatal infectious disease epidemic, most likely leading to improved infection control and decrease in contact rate. We again note that, similarly accurate estimation results can be obtained using only data from March 5 to May 5, merely 10 days after the actual turning point around April 26. Next we turn our attention to Hong Kong, where the second largest clusters of SARS infections occurred. Using the official epidermic curve data by onset date from the Hong Kong Department of Health website, we fit the cumulative case data of 1755 total cases from February 15 to May 31 to Richards model and obtained the resulting parameter estimates in Table 3 with the corresponding theoretical epidemic curve in Figure 3.4. The estimated final epidemic size of K = 1742 [95%CI: (1730, 1753)] shows a slight underestimate, perhaps due to the presence of multiple superspreading events (SSEs) in Hong Kong [33]. The turning point of March 26 implies the turning point for infections had occurred by March 21. Riley et al. [33] estimated that the Amoy Gardens SSE, at the peak of the outbreak in Hong Kong, had occurred on March 19 (95%CI, March 18 to March 20) and had infected 331 [95% CI: (295,331)] people. Moreover, they also estimated that the estimated reproduction number at time t in the absence of SSEs, defined to be the average number of infections caused by one typically infectious individual at time t
Table 3: Estimates of parameters in Richards model using cumulative confirmed SARS case data in Hong Kong during February 15-May 31, 2003. ti =38. 76 implies the turning point of epidemic is March 26. Period 2/15 - 4/01
0.593
2/15 - 4/11
K
95%C.I. (969.1 1179.7)
a 4.83
tm 41.92
0.180
1.19
40.61
39.65
1452.3
(1372.5 - 1532.1)
2/15 - 4/21
0.123
0.61
35.66
39.62
1625.7
(1575.2 - 1676.1)
2/15 - 5/01
0.108
0.43
31.56
39.40
1685.7
(1655.8 - 1715.6)
2/15 - 5/11
0.100
0.33
28.18
39.18
1713.3
(1639.5 - 1733.1)
2/15 - 5/21
0.095
0.26
24.85
38.96
1731.0
(1716.4 - 1745.6)
2/15 - 5/31
0.092
0.21
21.96
38.76
1741.8
(1730.4 - 1753.3)
r
ti 39.27
1074.4
Ying-Hen Hsieh
224 1800 1600
H.
Real Date
I
/
r
- - 2/15-5/31
/<
I; 1200
7
~
~ 1000
:l
:/
~
u~
..... uM. . .·u.r·W·
./
1400
]
..
-
,........
.'1
800
,I
/..'
400 200
o
1
. . i
600
.:;;Y .#.
2/5 2/22 3/1 3/8 3/153/223/294/54/124/194/265/3 5/10 5/17 5/24 5/31 Date
Figure 3.4: The theoretical epidemic curve for Hong Kong SARS outbreak of February 15-May 31, 2003, using Richards model. Turning point is March 26. excluding SSEs and denoted by Rf ss , dropped sharply to 1.0 [95% CI: (0.7, 1.2)] by 21 March, which can be attributed to increased awareness of the infection by the general population, leading to voluntary drops in contact rates and to improved control measures in hospitals. However, control measures such as school closures and recommendations against unnecessary travel could also have played an important role. Here the turning point can be detected using data of February 15 to April 21, almost one month after its occurrence.
4
Outbreaks with multiple waves
All of the previous examples exhibit a single S-shaped epidemic curve, indicating one wave of infection. However, the SARS outbreak in the greater Toronto area (GTA) in Canada during February 23 to June 6 was know to have two phases. Recently, Hsieh et al. [14] proposed a multi-staged Richards model, a variation of the S-shaped Richards model, which makes a distinction between two types of turning points. Other than the previous inflection point of the S curve, there is a second type of turning point in a multi-wave epidemic curve where the growth rate of the number of cumulative cases begin to increase, which signals
Richards Model: A Simple Procedure for Real-time Prediction· .. 225 the beginning of the next wave. For a multistage Richards model, one stage for each of the S-shaped segments results from the multiple waves of infection during this outbreak. Stages are distinguished by turning points (or inflection points), denoting acceleration after deceleration at the end of each S-shaped segment, the local minima of the corresponding incidence curves. For an n-phase epidemic outbreak, n -1 local minima separate the n phases. For illustration, the incidence curve for GTA contains two peaks (local maximum or turning point of the first type) and one valley (local minimum or turning point of second type). The multistage Richards model procedure requires the following five steps: 1. Fit the Richards model to cumulative cases on successive days by using a standard least-square routine. For single-phase outbreaks, parameter estimates (a, T, ti, K) will converge as the trajectory approaches the carrying capacity K, as demonstrated in the Taiwan, Beijing, and Hong Kong SARS outbreaks.
2. If estimated parameters remain convergent until no more new cases are detected, the outbreak has only one phase. However, if the estimates begin to diverge from heretofore fixed values, one knows that a turning point denoting the start of a second phase has occurred. 3. Locate the turning point, tmin, separating two S-shaped phases of the epidemic as the local minimum of the incidence curve. This is the curve for I"(t) given in the equation (2.2). 4. Fit the Richards model to the cumulative case curve again, but starting from tmin + 1, the day after the start of second phase. The estimated parameters (a, T, ti, K) will again converge as the curve approaches the carrying capacity K for the second phase. 5. Repeat steps 2-4 in the event more phases occur until the outbreak ends. By considering successive S-shaped segments of the epidemic curve separately, one can estimate the maximum case number, K, and locate the turning points, thus providing an estimate for the cumulative number of cases during each phase. Using this procedure and the GTA daily SARS case number by onset date obtained from the Public Health Agency of Canada (PHAC) website (http://www.phacaspc.gc.ca/sarssras/pdf-ec/ec_ 20030808.pdf), the parameter estimates of the two waves of the GTA outbreak were obtained in [14] and given below in Tables 4 and 5, with the corresponding theoretical epidemic curve give in Figure 4.1. The number of cases during the first phase ending on April 27 (or April 26) is 143 (see Table 4), well approximated by our estimate for
Ying-Hen Hsieh
226
Table 4: Parameter estimates for Phase 1 of GTA outbreak (2/23-4/27, total number of cases is 143). ti=30.35 implies the turning point of the first phase is March 25. (Source: [14]) Time Period 2/23 - 3/25
r 0.859
a 4.835
tm 26.93
ti 25.09
K 60.10
95% C.l. 54.71-65.49
2/23 - 4/04
0.146
0.689
27.51
30.06
140.53
115.88-165.17
2/23 - 4/14
0.152
0.773
28.81
30.50
142.78
137.34-148.22
2/23 - 4/24
0.147
0.718
28.19
30.45
143.99
141.76-146.21
2/23 - 4/26
0.146
0.710
28.08
30.43
144.14
142.19-146.09
2/23 - 4/27
0.146
0.710
28.08
30.43
144.14
142.19-146.09
2/23 - 4/28
0.146
0.709
28.08
30.43
144.14
142.42-145.86
2/23 - 4/30
0.144
0.693
27.86
30.40
144.41
142.85-145.96
2/23 - 5/02
0.142
0.664
27.47
30.35
144.84
143.40-146.29
Table 5: Parameter estimates for Phase 2 of GTA outbreak (4/28-6/6, cumulative number of cases is 249). ti=26.37 corresponds to the turning point of the second phase on May 24. (Source: [14]) Time Period
r
a
tm
ti
K
95% C.l.
4/28 - 5/25
0.557
3.866
27.02
24.59
223.37
199.67-247.07
4/28 - 5/27
0.350
2.393
28.33
25.84
244.36
220.53-268.18
4/28 - 5/29
0.236
1.554
29.22
27.36
271.28
240.94-301.62
4/28 - 5/31
0.321
2.202
28.88
26.43
252.53
244.32-260.74
4/28 - 6/02
0.352
2.448
28.90
26.36
249.51
245.70-253.33
4/28 - 6/04
0.359
2.508
28.92
26.36
248.96
246.67-251.25
4/28 - 6/06
0.367
2.576
28.95
26.37
248.52
246.98-250.07
the carrying capacity, K = 144.14 [95%CI: (142.19,146.09)]. Note that the number of 142 has been added to the estimates for K and its 95% confidence interval (see [14]). The turning point of March 25 indicates the turning point of the first wave of infection occurred around March 20. Satisfactory estimates for case number and turning point can be obtained using epidemic data of up to April 4, 10 days after the turning point had occurred. The divergence of parameter estimates soon after April 28 indicates that the second turning point had occurred around April 27, or the start of second wave of infections five days earlier on April 22. Our results corroborate the assessment of Health Canada, which pinpointed
Richards Model: A Simple Procedure for Real-time Prediction· .. 227 300 Real data
250 ~
- - 2/23-4/26 -4/27-6/12
200
1
/
"'_._u_ .;: ..........
Q)
1
150
r
J'.1
u 100 I
."
(
I'
50
.....
;
'7
,A
o 2/23
• ..u •..,........
3/4
3113 3122 3/31 4/9
4118 4127
5/6
5115 5/24
6/2
6/11
Date
Figure 4.1: The theoretical epidemic curve for GTA SARS outbreak during February 23-June 6, 2003, using Richards model. Turning points are March 25, April 27, and May 24. (Source: [14])
April 21 as the start of the second phase of the outbreak [35]. The parameter estimation of the second phase yields the estimated case number of 249 [95%CI: (247,250)]' exactly the actual case number in the GTA outbreak. The estimated turning point ti = 26.36 pinpoints to May 24, or a turning point for SARS infections 5 days earlier on May 19. This finding further corroborates Health Canada's assertion that, among the 79 cases that resulted from exposure at the hospital where the index patient of the second phase stayed, 78 had exposures that occurred before May 23 [35]. Note also that good estimates can obtained by using data that end just 3 days after the turning point, on May 27, giving an accurate prediction (K = 244.36 [95%CI: 240.53-268.18]) of the actual cumulative case number. Zhou and Yan [38] had shown that Richards model fits the singlephase SARS outbreaks in Hong Kong and Beijing well, but not as satisfactorily for the Singapore outbreak. Here we fit the Singapore daily SARS case data by onset date, obtained from Singapore Ministry of Health (MOH) website, to the multi-staged Richards model. The results are given in Tables 6 and 7, and Figure 4.2. The first wave corresponds to the hospital cluster at Tan Tock Seng Hospital (TTSH), a 1400-bed acute care hospital [15], where 105 total
Ying-Hen Hsieh
228
Table 6: Parameter estimates for Phase 1 of Singapore outbreak (2/253/28, total number of cases is 103). ti=19.28 implies the turning point of the first phase is March 16. a 8.771
tm 35.763
ti 27.74
47.06
95% C.l. 0* -32212.20
0.999
5.157
21.410
19.77
88.16
84.45-91.86
2/25 - 3/25
0.646
3.217
21.374
19.57
93.20
89.11-97.29
2/25 - 3/26
0.524
2.526
21.223
19.45
96.14
91.70-100.60
2/25 - 3/27
0.432
1.999
20.950
19.35
99.27
94.41-104.10
2/25 - 3/28
0.382
1.701
20.669
19.28
101.50
96.68-106.40
2/25 - 3/29
0.329
1.385
20.18
19.19
104.50
99.29-109.80
Time Period 2/25 - 3/21
0.271
2/25 - 3/23
r
K
*max(O, lower bound)
Table 7: Parameter estimates for Phase 2 of Singapore outbreak (3/285/05, total number of cases is 204). t i =8.56 implies the turning point of the second phase is April 6. Time Period 3/28 - 4/15
0.181
a 0.003
tm -22.76
ti 9.21
K 192.80
3/28 - 4/20
0.170
0.021
-14.18
8.46
200.79
181.00-220.60
3/28 - 4/25
0.162
0.027
-13.49
8.79
203.90
192.86-214.90
3/28 - 4/30
0.162
0.024
-13.92
8.94
204.10
197.27-210.90
3/28 - 5/05
0.16
0.02
-15.83
8.56
203.50
200.14-206.90
r
95% C.l. 0*-324.50
*max(O, lower bound)
secondary cases occurred between March 4 to April 5 [3]. The number of cases during the first phase ending on March 28 of 103 is again well approximated by our estimated carrying capacity, K = 101.50 [95%CI: (96.68, 106.40)], using the case data of up to March 28. The turning point of March 16 indicates the turning point of the first wave of infection occurred around March 11. We note that in TTSH, isolation of infectious cases and admission of any new suspected or probable cases to isolation facilities were implemented from March 13. Moreover, one of the infected healthcare workers (HeWs) at TTSH (index case B in [8]) with onset of symptoms on 7 March and provisionally diagnosed to have dengue fever, later was admitted on 10 March to Ward 8A where she in turn infected 21 persons before she was isolated on March 13. On the same day, the Singapore MOH alerted all hospitals and doctors to look
Richards Model: A Simple Procedure for Real-time Prediction· .. 229 250
J' 200
... '"
-
Real data - 2/25-5/5
I "
"/
......... ,.,.-
".,
._ e/
1'"
150
, / ,/
..../.. ~
.~
]
S
... .... , .,:...
100
<':'~
::i
0/' '
U
./ 1
50
o ~_r
/'
Y
./e •
2/25 3/2 3/7 3/12 3/173/22 3/27 4/1
4/6 4/11 4/164/21 4/26 5/1
Date
Figure 4.2: The theoretical epidemic curve for Singapore SARS outbreak during February 25-May 5, 2003, using Richards model. Thrning points are March 16, March 27, and April 6.
out for cases of pneumonia who had recently travelled to Hong Kong, Hanoi or Guangdong province [8]. The MOH also advised travellers returning from these areas to seek medical attention if they developed flu-like symptoms, all of which helped to alleviate the spread of SARS infections. Satisfactory estimates for case number and turning point can be obtained using epidemic data around 10 days after the turning point had occurred. Here, the divergence of parameter estimates soon after March 28 indicates that the second turning point had occurred around March 28, or the start of second wave of infections occurred five days earlier before March 22. The second wave could probably be attributed to multiple events. One such event is index case D in [8], a 60-year-old ex-patient of TTSH who was admitted on 5 March to Ward 5A (the same ward as index case A) at TTSH, and discharged on 20 March with no clinical manifestations of SARS. He was readmitted to an open ward (Ward 57) at Singapore General Hospital (SGH), on 24 March for steroid-induced gastrointestinal bleeding and a diabetic foot ulcer. It was only on 5 April when chest x-ray showed evidence of pneumonia that he was clinically diagnosed as a probable SARS case, by which time several family members and HeWs in the wards he stayed in had been identified [8]. Another is
230
Ying-Hen Hsieh
a 90-year-old woman (index case F in [8]) who had been warded next to a SARS patient in Ward 7D in TTSH on March 16-17, was discharged to a private nursing home (Orange Valley Nursing Home) and then admitted to Changi General Hospital (CGH) on March 25 when she subsequently fell ill again with breathing difficulty. This index case led to a small cluster of 7 cases linked to the nursing home and CGH. The parameter estimation of the second phase yields the estimated case number of 203.50 [95%CI: (200.14, 206.90)], exactly the actual case number of 204 in the Singapore outbreak. Again, the number of 102 has been added to the estimates for K and its 95% confidence interval. The estimated turning point pinpoints to April 6, or a turning point for SARS infections 5 days earlier on April 1. By this time, multiple intervention and control measures had already been in place. On March 26, decision taken to close all childcare centers, pre-schools, primary and secondary schools, junior colleges, centralized institutes and madrasahs from 27 March 2003 to 6 April 2003. Other measures taken during this time period include the establishment of an Inter-Ministry Working Group on March 28 to look into further measures to contain SARS; the Civil Aviation Authority of Singapore (CAAS) directing all airlines at Changi Airport to ask departing passengers the three WHO-recommended questions on symptoms of SARS and contact history before departure with health alert notice given to inbound passengers from affected areas starting on arch 30; and from March 31 on, for all inbound flights from affected areas, nurses were stationed to check passengers who appeared unwell and those with fever sent to TTSH for assessment (see the Appendix in [8]). Although by that time (march 28) the second wave had already started, which was unknown to the authority, these measures contributed to containing the second phase.
5
Conclusions and remarks
The Richards model fitted all data well, allowing us to study retrospectively the significance of various events occurring at different times in each affected area during the SARS outbreak. Through this procedure, we can pinpoint retrospectively the key turning points for the spread of disease during a single- or multi-phase outbreak. Given incidence by onset or hospitalization date during the outbreak, one can use our procedure to forecast the eventual severity of current phases of the outbreak in real-time by estimating the carrying capacity, K. However, accuracy depends on having the incidence data for some time past the inflection point ti and no new waves of infection in the future. However, when a new wave occurs, the divergence in the estimation will immediately alert us to the occurrence of a turning point of second type, as demon-
Richards Model: A Simple Procedure for Real-time Prediction· .. 231 strated with the GTA and Singapore multi-phase outbreak in Section 4. Furthermore, such estimates are possible shortly after the inflection point (or turning point) had occurred. In Table 8, we give a summary comparison of the date when the case data used will be sufficiently accurate to pinpoint the turning point of the current outbreak, as well as the estimated final case number. Table 8: Comparison of estimated turning points and total case numbers (rounded off to integers) in SARS affected areas in 2003. Affected area
TUrning
Case
Estimation
(duration)
point
no.
date
(95%CI)
Taiwan (2/25-6/15)
5/3
346
5/15
334 (298, 370)
Estimate
Beijing (3/3-5/29)
4/26
2380
5/5
2097 (2040, 2155)
Hong Kong (2/15-5/31)
3/26
1755
4/21
1626 (1575, 1676)
GTA 1 (2/23-4/27)
3/25
143
4/4
141 (116, 165) 244 (221, 268)
GTA 2 (4/28-6/6)
5/24
249
5/27
Singapore 1 (2/25-3/28)
3/16
103
3/27
102 (97, 106)
Singapore 2 (3/28-5/5)
4/6
204
4/25
204 (193, 215)
Several observations can be drawn from the table. First, in most cases, the turning point was detected around ten days after it had occurred. The exceptions are Hong Kong, which took almost one month, and the second phase of GTA, which took only 3 days. The former can be explained by the fact that the spread of disease continue to persist even after the turning point on March 26 and did not end until more than two month later on May 31. The short amount of time it took for turning point to be detected seems to indicate the decisive nature of the intervention measures implemented (see previous section for a discussion) as the GTA outbreak ended within 13 days on June 6 after the turning point (May 24) had occurred. The estimates of the total case numbers are quite accurate, with the exception of Beijing, and perhaps Hong Kong to a less degree. As mentioned earlier, the inaccuracy in Hong Kong estimation is probably caused by the superspreading event at Amoy Gardens, whose feature was not captured by Richards model. The inaccuracy in the Beijing estimation is most likely caused by the use of case data by hospitalization date, which resulted in loss of information on the actual spread of infections. This highlights the importance of swift and accurate data collection, especially for the purpose of real-time prediction. We further note that, although the outbreak in each affected area occurred almost simultaneously within a time period of a little over two weeks, the turning points varied significantly. In that respect, Singapore
Ying-Hen Hsieh
232
seemed to have responded most swiftly and the outbreak there would have ended quickly with minimal loss if not for the second wave of infections in several hospital clusters due to undetected cases. The same can be said, to some degree, of the outbreak in GTA as well, which underscore the importance of swift identification and correct diagnosis when faced with a novel infectious disease. The results of the parameter estimation can also be used to compute the basic reproduction number Ro, or the average number of infections caused by one typically infectious individual in an entirely susceptible population, for each outbreak by using the formula Ro = exp[rT], where T is the duration of infectiousness. Using T = 8.4 from [19], we construct Table 9 to compare the affected areas. The results for the second phases of GTA and Singapore outbreaks are not valid for the purpose of initial stage estimation, due to the nature of multi-phase outbreak which distorts the the epidemiologic parameters (contact rate and transmission probability) of the initial phase. Table 9: Comparison of basic reproduction numbers Ro for SARS in some affected areas in literature computed using Richards model and generation time of SARS infection T = 8.4 (Lipsitch et al. [19)) Affected area
Reference
r
Ro
Taiwan
Hsieh et al. [12J
0.136
3.08
Beijing
Zhou and Yan [38J
0.16
3.8
Beijing
[this articleJ
0.219
6.29
Hong Kong
Zhou and Yan [38J
0.09
2.1
Hong Kong
[this article]
0.092
2.17 2.7
Singapore
Zhou and Yan [38J
0.12
Singapore (phase 1)
[this articleJ
0.382
24.7
GTA (phase 1)
[this article]
0.146
3.41
The estimates of Ro for Hong Kong from both [38] and this article agree almost exactly, and within range of the estimate of 2.7 [95%CI: SS , in (2.2,3.7)] for the basic reproduction number excluding SSEs, [33]. However, the results for Beijing differ significantly. For Beijing, since we have used the complete Beijing case data by hospitalization date in this work, while [38] only used partial Beijing case data from April 21 on, it is reasonable to suggest that the loss of information from the daily incidence data of March 3 to April 20 had an effect on the accuracy. Furthermore, it can be deduced that the loss of information on the initial explosive nature of the outbreak in Beijing led to a significant underestimate of the basic reproduction number. The larger basic reproduction numbers for Taiwan as compared with Hong Kong may be
Rt
Richards Model: A Simple Procedure for Real-time Prediction··· 233 attributable to the relatively higher percentage of nosocomial infections in Taiwan [9J. The estimates of Ro for phase 1 of Singapore outbreak differ significantly, 2.7 in [38J compare to our estimate of 24.7. We note that it has been reported in an epidemiological study of Singapore SARS outbreak [8J that the first four index cases (index cases A, B, and C in [8]) in Singapore can be directly traced to have infected, respectively, 21, 22, and 26 cases by March 20 when index patient C was isolated. Hence our unusually high estimate might be a reasonable reflection of a series of superspreading events which had occurred during the initial stage of the Singapore outbreak (as opposed to the superspreading event of community infection cluster at Amoy Gardens in Hong Kong, which happened when the outbreak was already well underway). On the other hand, the estimate by [38J was obtained using the Singapore onset data after March 17, by which time most of the infections caused by the first three index cases (A, B, and C) had already occurred [8J. This further demonstrates the difficult dilemma of dealing with stochastic superspreading events [7], especially for real-time forecasts. The fact that our result for basic reproduction number of Singapore SARS outbreak seems to capture the impact of SSEs occurring at the early stages of the outbreak, while our result for Hong Kong agrees with the estimate by [33J of the basic reproduction number excluding SSEs in Hong Kong, where SSEs occurred in the later stages of the outbreak there, offers hope that simple models can indeed be useful, when properly utilized. Mathematical models have been used to predict the course of epidemics, albeit with mixed results [24J. The easily implemented procedure, which can be run with any commercially available software which includes a subroutine for nonlinear least-square estimation, described can be extended to analysis of turning points and severity of multiphase epidemics while ongoing. During an outbreak such as SARS, to which available data were limited and uncertain, a simple model that requires only the most basic and perhaps only easily obtainable data under these circumstances offers our best chance to a practical solution to the understanding, prediction, and timely control of the outbreak. However, one must understand that mathematical models do not provide accurate numerical predictions and can be used to forecast only in fairly gross terms [21J. The accuracy of predictions depends heavily also on the assumption that no stochastic events occur in the remaining days that could significantly alter the course of the current phase of an outbreak. Detecting the occurrence of a second turning point or start of a second phase, as outlined in Step 2 of our multi-staged Richards model procedure in Section 4, is especially useful as it allows us to recognize early that an epidemic is worsening, as demonstrated here with GTA and
234
Ying-Hen Hsieh
Singapore SARS. Though predicated on the availability and accuracy of case onset data, this procedure could be a valuable tool to public health policymakers for responding to future disease outbreaks with multiple turning points.
References [1] RM. Anderson, Epidemiological models and predictions. Trop. Geogr. Med. 40 (1988), no.3, 30-39. [2] RM. Anderson, R.M. May, Infectious diseases of humans. (1991), Oxford: Oxford University Press. [3] M.LC. Chen, Y.-S. Leo, B.S.P. Ang, B.-H. Heng, P. Choo, The Outbreak of SARS at Tan Tock Seng Hospital - Relating Epidemiology to Control, Ann. Acad. Med. Singapore 35 (2006),317-325. [4] B.C.K. Choi, A.W.P. Pak, A simple approximate mathematical model to predict the number of SARS cases and deaths, J. Epidemo Com. Health. (2003). [5] J.E. Cohen, How Many People Can the Earth Support? (1995), New York: W. W. Norton. [6] C. Fraser, S. Riley, R.M. Anderson, N.M. Ferguson, Factors that make an infectious disease outbreak controllable, PNAS 101 (2004), no.16, 6146-615l. [7] A.P. Galvani, RM. May, Dimensions of superspreading, Nature 438(2005), 293-295. [8] K.-T. Goh, J. Cutter, B.-H. Heng, S. Ma, B.K.W. Koh, C. Kwok, C.-M. Toh, S.-K. Chew, Epidemiology and Control of SARS in Singapore, Ann. Acad. Med. Singapore 35 (2006), 301-316. [9) M.S. Ho, LJ. Su, Preparing to prevent severe acute respiratory syndrome and other respiratory infections, Lancet Infect. Dis. 4 (2004), 684-689. [10] Y.-H. Hsieh, C.W.S. Chen, Re: Mathematical modeling of SARS: Cautious in all our movements, J Epidem Com Health, (2003), eletter available online at http://jech.bmjjournals.com/ cgi/ eletters /57/6 /DC1 #66. [11] Y.-H. Hsieh, C.W.S. Chen, S.B. Hsu, SARS outbreak, Taiwan. Emerg. Infect. Dis. 10 (2004), no.2, 201-206. [12] Y.-H. Hsieh, J.-Y. Lee, H.L. Chang, SARS epidemiology. Emerg. Infect. Dis. 10 (2004), no.6, 1165-1167.
Richards Model: A Simple Procedure for Real-time Prediction··· 235 [13] Y.-H. Hsieh, C.C. King, M.S. Ho, C.W.S. Chen, J.Y. Lee, F.C. Liu, Y.C. Wu, J.S.J. Wu, Quarantine for SARS, Taiwan. Emerging Infectious Diseases 11 (2005), no.2, 278-282. [14] Y.-H. Hsieh, Y.S. Cheng. Real-time forecast of multi-wave epidemic outbreaks. Emerg. Infect. Dis. 12 (2006), no.l, 122-127. [15] L.Y. Hsu, C.C. Lee, J.A. Green, B. Ang, N.I. Paton, L. Lee, et al., Severe acute respiratory syndrome (SARS) in Singapore: clinical features of index patient and initial contacts. Emerg. Infect. Dis. 9 (2003), 713-717. [16] N. Keyfitz, On future population. J. Am. Stat. Assoc. 67 (1972), 347-363. [17] M.L. Lee, C.J. Chen, LJ. Su, K.T. Chen, C.C. Yeh, C.C. King, et al. Use of quarantine to prevent transmission of severe acute respiratory syndrome-Taiwan, 2003, MMWR Morb Mortal Wkly Rep. 52 (2003), 680-683. [18J W. Liang, Z. Zhu, J. Guo, Z. Liu, X. He, W. Zhou, D.P. Chin, and A. Schuchat (Beijing Joint SARS Expert Group), Severe Acute Respiratory Syndrome, Beijing, 2003, Emerg. Infect. Dis. 10 (2004), no.l, 25-3l. [19J M. Lipsitch, T. Cohen, B. Cooper, J.M. Robins, S. Ma, L. James, et al., Transmission dynamics and control of severe acute respiratory syndrome. Science 300 (2003), 1966-1970. [20J A.J. Lotka, The Size of American Families in the Eighteenth Century: And the Significance of the Empirical Constants in the PearlReed Law of Population Growth, Journal of the American Statistical Association, 22 (1927), No. 158, 154-170, [21J F.E. McKenzie, Smallpox models as policy tools. Emerg. Infect. Dis. 10 (2004), 2044-2047. [22J T. Malthus, An Essay On The Principle Of Population, (1798 1st edition) with A Summary View (1830), and Introduction by Professor Anthony Flew. Penguin Classics. ISBN 0-14-043206 [23] E. Massad, M.N. Burattini, L.F. Lopez, F.A.B. Coutinho, Forecasting versus projection models in epidemiology: The case of the SARS epidemics. Med. Hypothesis 65 (2005), 17-22. [24J R.M. May, Uses and abuses of mathematics in biology. Science 303 (2004), 790-793. [25J Ministry of Health, People's Republic of China. http://www.moh.gov.cn/zhgl/yqfb/index.htm [26] H. Nishiura, Mathematical modeling of SARS: cautious in all our movements. J Epidem Com Health (2003).
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[27J X. Pang, Z. Zhu, F. Xu, J. Guo, X. Gong, D. Liu, Z. Liu, D.P. Chin, D.R. Feikin, Evaluation of control measures implemented in the Severe Acute Respiratory Syndrome outbreak in Beijing, 2003, JAMA 290 (2003), 3215-322l. [28J R. Pearl, L.J. Reed, The Population of an Area Around Chicago and the Logistic Curve (in Notes) Journal of the American Statistical Association, 24 (1929), no.165, 66-67. [29J R. Pearl and L.J. Reed, The Logistic Curve and the Census Count of 1930 (in Special Articles) Science, New Series, 72 (1930), no.1868, 399-40l. [30] T.C. Porco, P.M. Small, S.M. Blower, Amplification dynamics: predicting the effect of HIV on tuberculosis outbreaks, J Acquir Immune Defic Syndr. 28 (2001),437-444. [31J O. Razum, H. Becher, A. Kapaun, T. Junghanss, SARS, lay epidemiology, and fear, Lancet. 361 (2003), 1739-1740. [32J F.J. Richards, A flexible growth function for empirical use, J. of Experi. Botany. 10 (1959), 290-300. [33J S. Riley, C. Fraser, C.A. Donnelly, A.C. Ghani, L.J. Abu-Radda, A.J. Hedley, et al., Transmission dynamics of the etiological agent of SARS in Hong Kong: Impact of public health interventions. Science 300 (2003), 1961-1966. [34J P.F. Verhulst, Notice sur la loi que la population pursuit dans son accroissement, Correspondance mathematique et physique 10 (1838), 113-12l. [35] T. Wallington, L. Berger, B. Henry, R. Shahin, B. Yaffe, B. Mederski, et al. Update: Severe acute respiratory syndromeToronto, 2003, Can Commun Dis Rep. 29 (2003), 113-117. Available at http://www.hcsc.gc.ca/pphb-dgspsp/publicat/ccdrrmtc j03vo129 j dr2913ea.html [36J World Health Organization. Cumulative number of reported probable cases of SARS. Available from: http: j j www. who .int j csr j sars j country j 2003..07 _09 j en j [37J World Health Organization. Consensus document on the epidemiology of severe acute respiratory syndrome (SARS) [monograph on the InternetJ. Available from: http://www.who.int/csr/sars/en/WHOconsensus.pdf. [38J G. Zhou, G. Van, Severe acute respiratory syndrome epidemic in Asia. Emerg Infect Dis. 9 (2003), 1608-1610.
237
The Basic Reproduction Number and the Final Size of an Epidemic James Watmough Department of Mathematics and Statistics University of New Brunswick Fredericton, New Brunswick, Canada E3B 5A3 Email: [email protected]
1
Introduction
In their seminal study of a simple disease transmission model, Kermack and McKendrick [14] derived expressions for the likelihood and severity of an outbreak of infection. These two quantities remain the most useful means of evaluating the e_ectiveness of disease control measures, yet many open questions remain as to how these quantities can be computed for more complicated models. In the context of deterministic models, the likelihood of an outbreak is related to the stability of a trivial solution known as the disease-free equilibrium, and is usually expressed in terms of the basic reproduction number, no. The severity of the outbreak is reflected in the peak and the final size of the epidemic. The latter is the simplest to obtain analytically and is the focus of the early pioneering of Kermack and McKendrick. This chapter discusses no and the final size of an epidemic for a general single-group model similar to the one proposed by Kermack and McKendrick [14]. The model is generalized to allow state dependent contact rates in the manner recently proposed by Brauer [5].
2
A general disease transmission model
With most infections, the probability of transmission depends on the time elapsed since infection. An influenza infection, for example, begins with latent phase of roughly two days, during which the probability of transmission is very low. This is followed by an infectious phase lasting several days, and finally recovery with partial immunity [17]. A disease transmission model, therefore, must classify infected individuals by their time since infection. One method of incorporating infection-age
238
James Watmough
structure into a model is to divide the population into subpopulations of individuals at dLerent stages of the disease. For the influenza example given Ro and the final size of an epidemic 2 above, the population is subdivided into four compartments: susceptible (S), exposed and latently infected (E), infectious (I), and recovered with partial immunity (R), giving the classic, four compartment, SEIR model. A further subdivision into symptomatic and asymptomatic infections is illustrated in Section 6. A more general approach is to use a continuous variable for the infection-age of an individual and to specify the density of individuals as a function of both chronological time and infectionage. The transmission rate is then a function of the infection-age of an individual. Many infection-age models, including that of Kermack and McKendrick, are proposed as systems of evolution equations satisfied by the infection-age distribution. The model introduced in this chapter follows the lead of Diekmann and Heesterbeek [7] and uses the incidence of infection at time t, rather than the infection-age distribution, as the main variable of interest. This seems a more natural approach, as the epidemiological data is more often in the form of an incidence curve than an infectionage distribution of the population. Moreover, the most important step in the formulation of a model is the specification of the incidence of infection as a function of the current numbers of susceptible and infected individuals. Let i(t) denote the incidence of infection at time t, measured as infections per unit time. In practice, what is reported is the number of people developing symptoms, and the incidence of infection must be back-calculated from this; however, the details of this calculation are beyond the scope of this chapter. Let B(T) denote the probability an individual infected at time (t - T) is still alive and infected at time t. The current infection-age distribution of infected individuals in the population can be computed from the historical incidence: the number of individuals of infection-age T at time t is given by B(T) i (t - T). The function B incorporates both recovery from infection and mortality. The rate that infectious individuals make contacts su_cient to transmit infection depends on both their infection-age, T, as mentioned above, and on the population size. Let N(t) denote the total population at time t, and let S(t) denote the number of people still susceptible to infection at time t. For simplicity, many deterministic disease-transmission models assume the transmission rate, or number of new infections caused by a single infected individual per unit time, can be expressed as the product of a contact rate C(N(t)) and a transmission probability per contact (3( T). Assuming that contacts are equally likely between each pair of individuals in the population, only the fraction S(t)/N(t) of all contacts by an infected individual are with susceptible individuals, and the transmission rate at time t for an individual infected at time (t - T)
The Basic Reproduction Number and···
239
is (3(T)C(N(t))S(t)/N(t). Suppose that 10 recently infected individuals are introduced, at time t = 0, into a population free of disease. The incidence at a time t > 0 is the sum of infections due to direct contacts with the original 10 cases together with secondary contacts between susceptible individuals and individuals infected at any time (t-T) with 0 < T < t. Since the number of such infected individuals remaining at time t is B(T) i (t - T), i(t) satisfies the Volterra integral equation
i(t)
=
Set)
(fat "iJ!(N(t), T) i (t -
where
"iJ!(N(t) , T) =
T)dT + "iJ!(N(t), t)Io) ,
(2.1)
C~;~)) (3(T)B(T)
denotes the per capita infection rate at time t for an individual infected at time (t - T). This equation must be coupled with a demographic model for the total population. However, for many diseases, the time scale of the initial outbreak is much shorter than the demographic time scale, and births and non-disease deaths can be ignored. This leads to what is often called an epidemic model, where the change in the number of susceptible individuals is due to infection only, and the change in the total population is due to disease related deaths. Mortality due to disease is modelled by introducing M(T) as the fraction of the individuals infected at time (t - T) who are still alive at time t. The total population N (t) is the sum of the remaining susceptible individuals and the surviving infected individuals. Hence, N(t) and Set) satisfy the pair of equations
N(t) = Set)
+ fat M(T) i (t - T)dT + IoM(t),
:t
Set) = -i (t),
(2.2) (2.3)
together with the initial conditions
S(O) = So.
(2.4)
Note that M(T) ~ B(T), so that the integral in the above expression sums infected individuals as well as those recovered and immune to further infection. For simplicity, assume that M(O) = 1, and that N(O) = No = So + 10 . The model can be extended to include loss of immunity; however, this will not be covered in this chapter. Implicit in this model is a population, R(t), of individuals recovered from infection
James Watmough
240
and immune to further reinfection, which can be computed from the incidence, i (t - T), as follows:
R(t) = N(t) - S(t)
=
lt
-It
B(T) i (t - T)dT
(M(t) - B(t)) i (t - T)dT
+ Io(M(t) - B(t)).
Equations (2.1) through (2.4) constitute the general disease transmission model to be studied in the next sections. Kermack and McKendrick [15] studied the case with the per capita contact rate, C(N)jN, constant, and showed that epidemic-like solutions are possible only if the basic reproduction number defined in the next section is at least one, and that the epidemic will pass leaving a fraction of the population untouched by the infection. This special case, with the contact rate proportional to total population, has become known as mass action incidence.
3
The basic reproduction number
Consider an index case imported into a population composed entirely of susceptible individuals. The initial shape of the incidence curve is approximated by solutions to the linearization of (2.1) about the trivial solution i(t) = 0, S(t) = N(t) = SO. This equilibrium is referred to as the disease-free equilibrium. Note that there is a family of trivial solutions to Equations (2.1) through (2.4) parameterized by So. The linear model is the Volterra integral equation,
1
00
i(t)
=
i(t - T)A(T)dT
+ IoA(t),
(3.1)
where A(T) = SOW(SO,T) = C(So)(3(T)B(T) is the expected rate secondary infections are produced by an infected individual with infectionage T. The initial behaviour of solutions to Equation (2.1) is determined by the roots of its characteristic equation: (3.2) If Equation (3.2) has roots with real part greater than zero, then solutions to (4.1) with i (t) initially small will have i (t) increasing. That is, an outbreak will occur. In contrast, if all roots of (3.2) have negative real parts, then solutions to (3.1) with i (t) initially small will have i (t) remaining small and an outbreak does not occur.
The Basic Reproduction Number and···
241
It can be shown that (3.2) has a single real root, r*, and that all other roots have a real part less than that of r*. Further, this root is positive if and only if ¢(o) > 1 [9, 10, 16]. This root is referred to as the basic reproduction number, R o , and is given by
This number has a clear epidemiological interpretation. Since A(T) is the expected number of secondary infections caused by a person of disease age T, Ro is the expected number of secondary infections due to a single individual over the course of its infection. From the viewpoint of the epidemiologist, the basic reproduction number, R o , is defined as the expected number of secondary infections produced by an index case in a completely susceptible population [1, 7]. If Ro < 1, solutions to Equations (2.1) through (2.4) with 10 sufficiently small monotonically return to the disease-free equilibrium given by i(t) = 0. In contrast, if Ro > 1, then i(t) initially grows exponentially. Ro depends on the disease-free population size, So, through the contact rate C(So). Usually, this function is increasing with So. McNeil [18] discusses several historical epidemics and the effect of increasing population size and contact rates. There are several cautions to this interpretation of Ro. The assumption of homogeneity of contacts is likely invalid during the initial stage of the epidemic and is only valid after several secondary cases arise. The contact rate depends on many factors, such as age, setting (contacts in hospitals dominated the SARS epidemic in Toronto) or geographicallocation [19]. However, Ro remains a useful measure of the likelihood and severity of disease outbreaks.
4
The final size of a simple epidemic
For many diseases there is little disease induced death, and it is reasonable to assume the total population N (t) remains constant at the initial population size No = So + 10 . In this case, the model given by Equations (2.1) through (2.4) simplifies to the pair of equations studied by Kermack and McKendrick [14]:
i(t)
=
~S(t) = dt
Set)
(!at iJ!(No, T)i(t - T)dT +
-i(t),
iJ! (No , t)lo) ,
(4.1)
(4.2)
James Watmough
242
with S(O) = So. This simple model is also obtained if the per capita contact rate, C(N)jN, is constant, a case referred to as mass action incidence. Several results of this model are reviewed in the text of Radcliffe and Rass [21]. In particular, it is shown that Set) and i(t) are positive, and it follows that Set) is decreasing and must have some limit Soo = limt-+oo S (t). Dividing (4.1) through by S (t) and integrating with respect to t from 0 to 00 leads to
100 (I 1roo00 100
t
log ( : : )
=
=
a
=
'li(No, T)i(t - T)dT
'li(No, T)i(t - T)dt dT
T
io
roo
'li(No, T)dT io
(1
+ 'li(No, t)Io) + Rsa J,a
~~
=
~~ (So -
(4.4)
a
i(t)dt +
R I
~o a
00
=
dt (4.3)
(4.5)
i(t)dt + Io)
(4.6)
+ 10)'
(4.7)
Soo
Since the right hand side is finite, the limit SeX) is strictly positive, and the epidemic passes without infecting the entire population. Defining the attack rate, p, as the fraction of the population infected over the course of the epidemic, p
=~ S a
1
00
0
'(t)dt = So - Soo s a '
z
the final size equation can be written
1 ) log ( 1 _ p
=
pRo
Rolo
+ ----s;;'
(4.8)
Figure 4.1 illustrates that (4.8) has a single root between 0 and 1. Since Io « So, the last term involving 10 can be neglected. In this case, (4.8) has the single root p = 0 if Ro < 1, and a single root between 0 and 1 if Ro > 1. Thus, if Ro < 1, then there is no outbreak, and only a few cases arise from contacts with the index case. However, if Ro > 1, then an outbreak occurs, and secondary cases are expected to lead to tertiary cases and so on. The final size relation (4.8) was first derived by Kermack and McKendrick, who noted that Soo was not zero, or equivalently, that the attack rate was strictly less than one, and the epidemic passes without infecting every susceptible individual. The total number of people infected over the course of the epidemic is Kp = K - Soo.
The Basic Reproduction Number and···
243
log (_I ) I-p
p
Figure 4.1: The final size of a simple epidemic.
5
A final size inequality for a general model
The final size relation (4.8) applies if either N(t) is constant or if the per capita contact rate, C(N)/N, is constant. In this section, a final size inequality is derived for a more general model. It is reasonable to assume that C(N) is an increasing function of N with C(N)/N decreasing. The reasoning is that as population size increases a smaller fraction of the population is contacted by each infected individual. Further, assuming that any change in the total population is due to disease death implies that N(t) is decreasing with time t, and it follows that C(N(t))/N(t) is an increasing function of time t, so that
\J!(N(t), T)
~
\J!(No, T).
This bound can be used to estimate the final size of the epidemic in the general case. Let No be the initial population size, so that S(O) = N(O) = No. Repeating the steps preceding (4.7) leads to the following inequality:
1 (I ~ 1 (I 00
log ( : : )
t
\J!(N(t), T)i(t - T)dT + \J!(N(t) , t)Io) dt
=
00
=
t
\J! (No , T)i(t - T)dT + \J!(No, t)Io) dt
R o (So - Soo So
+ 10).
In terms of the attack rate, the final size inequality can be written as follows: log ~ pRo + RsoIo. (5.1)
(_1_) 1-p
a
As seen from the sketch in Figure 4.1, this final size inequality gives a lower bound for the attack rate p. If a lower bound can be placed on
James Watmough
244
N(t), giving an upper bound on 'J!(N(t), T), then this would lead to a lower bound on the attack rate. In practice however, this is not a simple estimate to obtain.
6
Examples
Several examples of simple disease transmission models are given in this section, beginning with the classic SIR and SEIR models and more general compartmental ODE models, and ending with a simple model with a discrete delay. The compartmental ode models are discussed in greater detail by Hethcote [12]. An interesting collection of recent modelling efforts is captured by Gumel [11]. An introduction to ODE models for biology in general can be found in many texts [8, 6,20].
6.1
The constant rate SIR epidemic model
The number of infected individuals at time t is a function of the incidence of infection i(t) and the fraction of infected individuals surviving to infection-age T as follows:
I(t) =
lot i(t - T)B(T)dt + IoB(T).
(6.1)
Three simplifying assumptions lead to a simple SIR epidemic model that is the core of most disease transmission models. First, suppose that B(T) = e-O: T • Substituting this form for B(T) in the expression for I(T) and dLerentiating leads to the di_erential equation
d
dt I(t) = i(t) - "(I(t). Second, suppose j3(T)
= 13 is
a constant. Then (2.1) simplifies to
i(t) = j3C(N(t))S(t)I(t) N(t) . .
. C(N(t)) . N(t) IS a constant, assumed to be unity, then Equations
Fmally, If
(2.1) and (2.3) simplify to the following system of ordinary differential equations.
dS = -j3S1 dt ' dI = j3S1 - aI dt '
-
(6.2) (6.3)
dR
-=aI dt
(6.4)
The Basic Reproduction Number and··· with S(O) = So and 1(0) simple model is
no
= 10 .
245
The basic reproduction number for this
= SO {'XJ {3e-
Jo
C'
dT
= (3So. a
It is clear from (6.3) that the number of infected individuals, let), is increasing if (3S(t) > a and decreasing for (3S(t) < a (see Figure 6.1). Hence, if no > 1, let) initially increases until Set) < a/ (3 and then decreases to zero. If no < 1, then let) decreases to zero. The final size of the epidemic can be determined as follows. First divide (6.2) through by S and integrate to obtain
r
T
(S(T)) J(o S'(t) Set) dt = log S(O) = -(3 J l(t)dt. o Summing (6.2) and (6.3) and integrating leads to the relation
SeT)
+ leT) -
(So
+ 10)=
-a
loT l(t)dt.
Thus,
Set)
+ let) =
So
+ 10 + ~ In (~)
(6.5)
.
Setting limt---+oo let) = 0 gives the final size relation of Equation (4.8). Figure 6.1 shows a solution of (6.5) with So > a/ {3. For this simple example, the peak of the epidemic is
1 max = So
So
+ 10 + no (1 + In no)'
Further discussion can be found in the review of Hethcote [12]. I
a
f3 Figure 6.1: Solution of (6.5) with So > a/{3.
s
246
6.2
James Watmough
Simple compartmental models
The previous results can be applied directly to ordinary di_erential equation models with a single susceptible compartment where all infections arise in a single infective compartment. As a simple example of such a model, consider the SEIR model. dS = -{3S(EE + I), dt dE = {3S(EE + I) - K,E, dt dI dt =K,E-(a+6")I, -
dR = aI. dt
Here, E(t) is the number of latently infected individuals, and R(t) is the number of individuals recovered from infection with full immunity to further infection. Newly infected individuals first enter a latent stage of infection and progress to a fully infectious stage at a rate K,. Infectious individuals succumb to disease death at a rate 6" and recover from infection at a rate a. The parameter . E < < 1 is used to model a partially infectious latent stage. More generally, suppose there are n disease compartments and let x(t) E ~n denote the populations of these compartments at time t. Let b E ~n denote the relative infectiousness of each compartment. The model is given by the following system of ordinary dLerential equations. dS __ SC(N) bT dt N x, dx _ QSC(N)bT dt N x
(6.6) _
Vx.
(6.7)
The n x 1 matrix Q = (1,0"" ,O)T indicates that all new infections arise in the first disease compartment. The equations for R(t) and N(t) are left unspecified as they do not enter into the analysis of the simple case discussed here. For the SEIR model, bT = (E, 1), X=
V=
(6.8)
(:), (:K,
and
a:
6") .
(6.9)
(6.10)
The Basic Reproduction Number and···
247
Denoting the incidence of infection as i(t) as before, (6.7) can be written
!
(eVtx(t))
= eVtQ8(t)~~~(t)) bT x(t) = eVtQi(t).
(6.11)
The exponential of the matrix V is defined by the Taylor series (see, for example, [13]) e V =1 + V
V2
vn
V3
+ -+ -3! + ... + -+ ... 2 n!
Integrating (6.11) gives the population of each compartment, x(t) as follows: x(t) = e-Vtx(O) + i(t - r)e- VT dr.
1t
Since the introduced cases are assumed to be in the latent stage, x(O) Qlo, and
=
(6.12) Let u(r) = e-VTQ. the solution of
~~ =
-Vu with u(O) = Q. This
is interpreted as the distribution, after a time r, of a cohort of infected individuals initially all in the latent stage. It follows that B( r) is the sum of the entries of u(r) and {3(r)B(r) = bTcVTQ. Thus, from (3.3),
1
00
Ro = C(80 )
bT e-VTQdr
= C(80 )bT V- I Q,
Returning to the SEIR example, we find
V-I
= (:;:
Ro = E{3 /'\,
l~a)'
+!!... a
Notice that the (i,j) entry of V-I is the expected time an individual initially in compartment j spends in compartment i over the duration of its infection. Thus, the second term in Ro above is the product of l/a, the expected time in compartment I, and C(80 ), the rate an individual in compartment I produces secondary infections, giving the expected number of secondary infectious produced by an infected individual while in compartment 1.
248
6.3
James Watmough
The SLIAR model for influenza
It is not necessary that the disease progress through the compartments
in sequence as with the simple SE1R model. For example, the results also apply to the SL1AR model proposed by Arino et al. [2] illustrated in Figure 6.2.
Figure 6.2: The SL1AR model for influenza. Here the disease progresses from a latent stage (L) to an infectious stage which is either asymptomatic and mild (A) or symptomatic (I). The model assumes a fraction p of latently infected individuals develop symptoms and progress to I, whereas a fraction (1 - p) have an asymptomatic infectious period. The model is given by the following system of ordinary dLerential equations:
S' L' I' A' R' N'
= -i(t),
(6.13)
= i(t) - r;,L, = pr;,L - 0.1, = (1- p)r;,L - '1]A,
(6.14) (6.15)
= faI +'1]A,
(6.17)
= -(1 - f)al
(6.18)
(6.16)
with the incidence of infection i(t) given by
i(t)
=
~~l~~~;~~~ (tL(t) + (1 -
q)I(t)
+ 8A(t)),
and initial conditions
8(0) A(O)
= 8 0 , L(O) = la, 1(0) = 0, = 0, R(O) = 0, N(O) = No = So + 10 .
This model was proposed for influenza and studied by Arino et al.[2]. There they also extend the model to include treatment and vaccination. The model differs from the simple models of the previous sections by allowing the contact rate to depend on the number of infected individuals,
The Basic Reproduction Number and· ..
249
an issue also discussed by Brauer [4]. The parameter q is a reduction in the number of contacts made by individuals with symptomatic infections. Equations (6.14) through (6.16) can be written in the form
dx
dt
=
SC(N) QbT _ V N _ qI x x,
(6.19)
with (6.20) (6.21)
(6.22)
As before U(T) = e-VTQ, and from (6.12), I(t) = J; i(t - T)U2(T)dT, where U2(t) is the second component of u(t). Integrating (6.18) gives N(t) as follows:
N(T)
loT lot U2(t - T)i(T)dT dt (1- f)a loT iT U2(t - T)i(T)dtdT
=
No - (1 - f)a
=
No -
=
No - (1 - f)a
loT i(T - a) (l~Q U2(t + a - T)dt) da
= No - foT i(T - a)m(a)da
(6.23)
a
where m(a) = (1- f)aJo u2(S). The presence of q in the denominator of (6.19) does not effect R o , and
Ro=f3(~+ (l-q)p + 8(1- P)). K,
a
'f/
(6.24)
The final size of the epidemic can be estimated from (4.8). However, it is dLcult to put bounds on this estimate. A further discussion can be found in Arino et al. [3].
250
6.4
James Watmough
A discrete delay
As a final example, consider the model
i(t) = S(t) C(N(t)) ('JO i(t - T)B(T)dT N(t) iij with B(T) = e-aT and 8 > O. This model is similar to the SEIR model of Section 6.2, with f = 0, except that infected individuals now spend precisely 8 days in a latent stage before becoming infectious. In the SEIR model, individuals progress from E to I at a rate K, resulting in an exponential distribution for the waiting times in E (recall the first component of U(T) was e-I
I(t) =
roo i(t _ T)B(T)dT = Jt
io
i(T)B(t - T)dT.
-00
Differentiation yields
I'(t) = i(t) - o1(t). However, since
we have
I'(t) = S(t) C~~~)) I(t - 8)e-
W )
-
aI(t).
The contact function A(T) is given by T
< 8,
T? 8. Integrating C(SO)A(T) gives the reproduction number as aij
C(So)eno = --''-'----a
(6.25)
As with the SEIR model, C(So) is the rate new infectious are produced by infectious individuals. However, since some individuals recover before becoming infectious, the expected time spent infectious is e- aij la.
7
Further Reading
The text of Diekmann and Heesterbeek [7J is an excellent reference for many results on the final size of an epidemic and the computation of
The Basic Reproduction Number and···
251
the basic reproduction number, no, and the text of Rass and Radcliffe [21] covers these issues for spatial models. Hethcote [12] provides a general review of disease transmission models, including some discussion on both no and the final size of an epidemic. Age-structured population models similar to Equations (2.1) through (2.4) appear in all areas of mathematical biology; many more details of the analysis and application of these types of equations are given in the text of Webb [22].
References [1] Roy M. Anderson and Robert M. May. Infectious Diseases of Humans. Oxford University Press, Oxford, 1991. [2] Julien Arino, Fred Brauer, Pauline van den Driessche, James Watmough, and Jianhong Wu. Simple models for containment of a pandemic. J. Roy. Soc. Interface, 3(8): 453-457,2006. [3] Julien Arino, Fred Brauer, Pauline van den Driessche, James Watmough, and Jianhong Wu. A model for influenza with vaccination and antiviral treatment. J. Theo. Biol., in preparation. [4] Fred Brauer. The Kermack-McKendrick epidemic model revisited. Mathematical Biosciences, 198: 119-131, 2005. [5] Fred Bral.!-er. Some simple epidemic models. Mathematical Biosciences and Engineering, 3(1): 1-15, 2006. [6] Gerda de Vries, Thomas Hillen, Mark Lewis, Johannes Mliller, and Birgitt Schofisch. A course in mathematical biology: quantitative modeling with mathematical an dcomputational methods. Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics, 2006. [7] Odo Diekmann and J.A.P. Heesterbeek. Mathematical epidemiology of infectious diseases. Wiley series in mathematical and computational biology. John Wiley & Sons, West Sussex, England, 2000. [8] Leah Edelstein-Keshet. Mathematical Models in Biology. Random House, New York, 1988. [9] Willy Feller. On the integral equation of renewal theory. The Annals of Mathematical Statistics, 12: 243-267, 1941. [10] James C. Frauenthal. Analysis of age-structure models. In Thomas G. Hallam and Simon A. Levin, editors, Mathematical Ecol- ogy: An Introduction, volume 17 of Biomathematics, 117-148. SpringerVerlag, Berlin, 1986.
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[11] Abba B. Gumel, Carlos Castillo-Chavez, Ronald E. Mickens, and Dominic P. Clemence, editors. Mathematical studies on human disease dynamics: emerging paradigms and challenges. AMS, 2006. [12] Herbert W. Hethcote. The mathematics of infectious diseases. SIAM Rev., 42: 599-653, 2000. [13] Morris W. Hirsch and Stephen Smale. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, Orlando, Florida, 1974. [14] W.O. Kermack and A.G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. R. Soc. London, 115: 700-721, 1927. [15] W.O. Kermack and A.G. McKendrick. Contributions to the mathematical theory of epidemics. III - Further studies of the problem of endemicity. Proc. R. Soc. London, 141: 94-122, 1933. [16] Mark Kot. Elements of Mathematical Ecology. Cambridge University Press, Cambridge, UK, 200l. [17] I.M. Longini, M.E. Halloran, A. Nizam, and Y.Yang. Containing pandemic influenza with antiviral agents. Am. J. Epidem., 159: 623-633, 2004. [18] William H. McNeil. Plagues and Peoples. Anchor Press, 1976. [19] Lauren Ancel Meyers, Babak Pourbohloul, M.E.J. Newman, Danuta M. Skowronski, and Robert C. Brunham. Network theory and SARS: predicting outbreak diversity. Journal of Theoretical Biology, 232: 71-81, 2005. [20] Jim Murray. Mathematical Biology. Springer Verlag, Berlin, 1989. [21] Linda Rass and John Radcliffe. Spatial Deterministic Epidemics, volume 102 of it Mathematical Surveys and Monographs. AMS, Providence, R.I., 2003. [22] G.F. Webb. Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and textbooks in pure and applied mathematics. Marcel Dekker, Inc, New York, 1985.
253
Epidemic Models with Reservoirs K.P. Hadeler* t Department of Mathematics and Statistics Arizona State University Tempe, AZ 85287, U.S.A. E-mail: [email protected]
Abstract A small reservoir is introduced into the classical epidemic models of SIR type. Since some of the models are structurally unstable (in the sense of dynamical systems theory), the dynamics may change completely and maximal prevalence may increase drastically. The effects of the reservoir on the time course of the disease and on endemic states are investigated by analytical methods and discussed in public health terms.
1
Introduction
The great majority of epidemic models, from simple SIR models to sophisticated systems incorporating population structure and control policies, assume the existence of an uninfected stationary state, i.e., of an equilibrium or, in the case of homogeneous systems, an exponential solution where the disease is absent. The notion of a basic reproduction number, Ro, seems to require the existence of such a state, since Ro is, by definition, the average number of new cases caused by one infectious case in a totally susceptible population [8], [4], [14], [2J. In reality there are many situations where, due to a reservoir, there is no such uninfected stationary state. The reservoir may be represented by some infected animal population, with a small but not negligible unidirectional interspecies transmission like in avian flu. In animal husbandry, the reservoir may be represented by a wild animal population which comes into occasional contact with the domestic population, as in swine flu where wild boars may be carriers of the disease. Finally, the * And Department of Mathematics, University of Tiibingen. tSupported by NSF Grant DMS - 0502349
254
K.P. Hadeler
population may be connected by occasional travelers to another population in which the disease is firmly established and which acts like a reservoir for the population considered. The effects of reservoirs on epidemic and ecological models have been systematically investigated in the thesis of Shikha Sengar [13J and in [12J. In [13], [12], it has been assumed that the reservoir may change with time whereby its dynamics depends on the size of the population. This general situation is suited to describe, e.g., sanitary conditions changing with population size. In [9J (see also publications cited therein) classical models are studied where the infectious agent may infect two separate populations which are only coupled by the disease and otherwise totally independent. If the infectious agent can be transmitted in both directions then one can compute a basic reproduction number for the joint population. If, however, transmission occurs only from a reservoir (e.g. animal) population to a host (e.g. human population) then persistence of the disease depends entirely on the reservoir population whatever dramatic effects the disease has on the host population. In this case the dynamics in the host population is not governed by the classical threshold phenomenon. Here we assume that the reservoir is at equilibrium and hence constant. We are interested in the dynamics within the host population. The concept of uninfected state/basic reproduction number is crucial for the standard modeling approach: linearization at the stationary point is the standard mathematical tool and a threshold theorem is the standard mathematical result. If this machinery breaks down, how should we replace it? This question is also interesting in a purely mathematical sense. Some of the classical epidemic models are structurally unstable. Hence introducing an arbitrarily small reservoir may change the global dynamics completely. One can argue that, for a given trajectory (characterized by its initial data), a very small reservoir has only a small effect on the evolution over finite time. Also this argument breaks down since in the classical models the time course of the disease is described by an orbit which does not start at a finite time but starts at the uninfected state at time -00. It seems that also the notions of total size and total cost [6J of the epidemic become meaningless in the presence of a reservoir. We study the effects of small reservoirs systematically, progressing from the simplest model systems to more complicated ones. In Section 2 we consider a reservoir in the simplest SIR model, we compare the systems with and without a reservoir in terms of singular perturbation theory in Section 3. Then we proceed to models with demographic turnover in Section 4, to general homogeneous models for demography and epidemics in Section 5 and to the most important special case of a differential mortality model in Section 6. Finally we compare differential
Epidemic Models with Reservoirs
255
mortality with case fatality in Section 7.
2
SIR model in a given population
Consider the classical epidemic model with a reservoir '" ;?: 0 as an additional feature,
S = -{381 j = {381
",8,
+ ",8 -
aI,
R=aI
(2.1)
and the corresponding two-dimensional system
S = -{381 - ",8, j = (381 + ",8 - aI.
(2.2)
The case", = 0: The system (2.2) has a continuum of uninfected stationary points (8,0). We are interested in the unstable manifold of the point (8, I) = (1,0) which describes the time course of an outbreak in an uninfected population with the normalized total population size l. The system (2.2) has an invariant of motion a
V(8,I) = 8 -131og8 + I.
(2.3)
Along a trajectory the number of infected is a function of the number of susceptible, hence we have the differential equation
dI a 1 = -1+-d8 {38
-
(2.4)
and the formula for the trajectory (which is another version of (2.3), this time for (8(0),1(0)) = (1,0)), I
a
=-
{3
log 8 - 8
+ l.
(2.5)
If Ro = {3/a > 1 then an orbit (the unstable manifold) connects the uninfected point (1,0) to the final state (800 ,0) whereby the number of remaining susceptible 8 00 < 1/ Ro can be obtained from the equation
(2.6) If Ro < 1 then the point (1,0) is (weakly) stable. There is no unstable manifold.
K.P. Hadeler
256
The case K> 0: Now the picture is very different. The function V as defined in (2.3) is no more an invariant of motion since .
V =
a
73K.
The point (8,1) = (0,0) is the only stationary point and this point is This claim follows immediately: The total number globally stable in of non-immune Q = 8 + 1 satisfies Q = -al. Hence 1 goes to zero and Q goes to a limit Qoo. Then the first equation in (2.2) implies Qoo = o. The Jacobian matrix at the point (0,0) is
JR.!.
(
-K K
0 ) . -a
Hence the point (0,0) is a node with incoming directions depending on the relative size of the parameters K and a. In applications we think of K « a. The eigenvectors are (0, If with eigenvalue -a and (a-K, Kf with eigenvalue -K. In the case K = 0 the time course of the epidemic is described by the unstable manifold of (1,0) which connects the points (1,0) and (800,0). For K > 0 the time course of the epidemic is described by the trajectory which passes through the point (1,0) at non-zero speed. Hence we cannot invoke continuous dependence on initial data to show that the two trajectories for K > 0 and K = 0 are in some sense close. However, we can use continuous dependence in the (8, I)-plane away from the line 1 = o. Consider any point (8, J) with 8 > 0 and J > 0 and the solution which passes through this point at time t = O. For the system with K = the trajectory lies on the curve given by the invariant of motion a - a (2.7) 1 = 73 log S - S + S - 73 log S + 1.
°
For t -> +00 and also for t -> -00 this trajectory approaches the line 1 = 0. If we introduce K> then for finite times, i.e., away from 1 = 0, nothing really changes. But along the line 1 = all trajectories pass this line with dl/ dS = -1 independently of K. Hence for K > the trajectory arrives at the line 1 = from S > > 0, 1 < < 0, passes through the line 1 = transversally and then converges towards the point (0,0). Now we follow, for K > 0, the trajectory through (8(0),1(0)) = (1, 0) forward in time. The function S (t) decreases from S (0) = 1 to for -> +00 while the function l(t) first increases to a maximum and then decreases to 0. Let Imax be the maximum of l(t) and let Smax be the corresponding value of 8. The point (Smax,lmax ) is located on the hyperbola
°
°
°
°
°
°
Epidemic Models with Reservoirs
1=
257
",S (3S
(2.8)
0'. -
which has a pole at S = 1/ Ro. We call this hyperbola the maximum prevalence curve. The qualitative behavior of the trajectory through (1,0) is the same for Ro ~ 1 and for Ro > 1. It appears that for this problem Ro has no qualitative meaning any more, except when considering the limiting case", -+ O. In Figure 2.1 the time course of the epidemic is depicted for", = 0 and for '" > 0, for Ro > 1 and for Ro < 1. Although the chosen value of '" is small, the effect on maximal prevalence is remarkably strong.
0.3
0.25 0.2
0.2
0.4
0.6
0.8 S
0.3
Figure 2.1: Time course of the epidemic for", = 0 (upper row) and for '" = 0.1 (lower row). 0'. = 1, left column: {3 = 2, right column: {3 = 0.5. If Ro > 1 then'" > 0 produces a higher prevalence, there are no remaining susceptible. If Ro < 1 there is no outbreak with", = 0 while there is an outbreak with", > o.
258
3
K.P. Hadeler
Singular perturbation
The system contains the small parameter /1,. For /1, = 0 the system has a one-dimensional manifold of stationary points (the line I = 0) which is hyperbolic (as a manifold) except at the point (8,1) = (1/ R o, 0). We expect that this manifold persists in some sense. Therefore we cast the problem into the standard form of a singular perturbation problem. Again we use the non-immune Q = 8 + I and Q = -aI. Then we put I = /1,J, use the equations for Q and for I, and 8 = Q - I, finally we divide the equation for J by /1,. The idea is that 1= 0(1) is small while J is not small. Then we get
Q= j
=
-/1,aJ, {3(Q - /1,J)J + (Q - /1,J) - aJ.
(3.1)
Then we scale time by T = /1,t and denote the derivative with respect to by'. When T runs from 0 to 1 then t runs from 0 to very large values. Afterwards we divide the Q' equation by /1,. Then we get
T
Q' = -aJ, /1,J'
= {3(Q - /1,J)J + (Q - /1,J) - aJ.
(3.2)
Now we have essentially a standard singular perturbation problem where the small parameter occurs also on the right hand side (which does not preclude the application of Fenichel's theory). If we put /1, = 0 then we get the slow manifold as J=
Q
(3.3)
a - (3Q which looks similar to (2.8). Indeed, it is the same expression if we identify I = /1,J and 8 = Q - /1,J = Q + 0(/1,). Hence we have found the following proposition. Proposition 1. For 8 < 1/ Ro and small /1, the hyperbola (2.8) is the slow manifold along which trajectories enter the stationary point (0,0). Of course convergence to the slow manifold is not uniform, and the manifold is disrupted at 8 = 1/ Ro because hyperbolicity is lost.
4
SIR model with demographic renewal
Here we endow the previous model (2.1) with demographic turnover but for the time being we exclude differential fertility or mortality,
S = p,- p,8 - /1,8 - {38I, j = -p,I + /1,8 + {381 - aI, R = -p,R+aI.
(4.1)
Epidemic Models with Reservoirs
259
R!
The positive orthant is positively invariant. All non-negative solutions are bounded in view of (djdt)(8+1 +R) = f.L(1-(8+1 +R)). Hence the existence problem is trivial. It suffices to study the two-dimensional system
s = f.L j
=
f.L8 - ",8 - (381,
-f.LI + ",8 + (381 - aI.
(4.2)
The Bendixson-Dulac criterion with weight function 1/81 excludes periodic orbits. For the discussion of stationary states it suffices to consider (4.2),
o = f.L -
f.L8 - ",8 - (381,
0= -f.LI + ",8 + (381 - aI.
We write these two equations as 1=
f.L-f.L8 -",8 (38 '
1=
",8 a+f.L-(38'
(4.3)
equate these two expressions for I and obtain a quadratic equation for S,
¢(8) == (3f.L8 2
-
f.L((3
We define
+ a + f.L)8 -
"'(f.L
+ a)8 + f.L(a + f.L) = O.
(4.4)
a+f.L
S= -(3-' The quadratic equation has always two real positive roots 8 1 < 8 2 , We find that
Hence S1
< min(l, S) ~ max(l, S) < S2·
(4.6)
From the second equation (4.3) it follows that at the stationary point (8 2 , I) the component I is negative. At the stationary point (81 ,!) the I component is positive. Hence we get, with 8 = S1, the following proposition. Proposition 2. For all choices of the parameters with", > 0 there is a unique feasible stationary point (s,1). The component S is the smaller solution of the quadratic equation (4.4), the component I is given by any of the two expressions (4.3) with S = S. The point (8,1) is globally stable in
lRt.
K.P. Hadeler
260
Proof: The system is dissipative, there are no periodic orbits and there is a single stationary point. By the Poincare-Bendixson theorem this point is a global attractor in IR? D Now we explore the local character of the point (8, J). Proposition 3. The point (8, J) is linearly stable.
Proof The Jacobian at the stationary point is (4.7) and hence, using (4.3), /1 tr J
+ 0: -
{38
> 0,
= -(/1 + '" + (3J) - (/1 + 0: -
(38)
< 0,
det J = (/1 + '" + (3J)(/1 + 0: - (38) + (38(", + J) >
o.
0
Proposition 4. The trajectory through (1, 0) starts with dI/ d8 = -1..
Either the function I(t) increases to maximum prevalence J at the stationary point or it first increases to maximal prevalence and then approaches the equilibrium prevalence 1. Proof The stationary point is the intersection of the isoclines given by (4.3). Either the trajectory through (1,0) enters the stationary point before crossing the j = 0 isocline or it crosses it a first time before eventually converging to the stationary point. This intersection of the trajectory and the isocline gives the maximal prevalence along the trajectory. D Now we discuss how the case with small '" > 0 is related to the classical case", = o. We introduce the basic reproduction number as
Ro =
_(3_.
(4.8)
0:+/1
If '" = 0 then (1,0) is a stationary point which describes the uninfected population. If Ro =J- 1 then there is a second stationary point A
/1
1
1= - ( 1 - -). /1+0: Ro
(4.9)
This point is feasible if and only if Ro > 1. Now we assume Ro =J- 1 is fixed and '" > 0 is small. Then again we have two scenarios. i) If Ro < 1 then there are a feasible stationary point (8,1) close to (8,1) = (1,0) and a non-feasible stationary point close to (8,1), away from (1,0).
Epidemic Models with Reservoirs
261
ii) If Ro > 1 then there are a feasible stationary point (8, I) close to (8,1) and a non-feasible stationary point close to (1,0). Hence, as long as 0:, {3 are kept fixed and I'>, is sufficiently small, the cases Ro < 1 and Ro > 1 are well distinguished. The differences disappear if I'>, gets large. Next we further explore the effect of the reservoir. Following an idea of Thieme we distinguish infected individuals by the source of their infection. Let h denote those which became infected by the reservoir, and let 12 be those who became infected by contact with another infected individual. Then the system becomes A
s = p, - p,8 jl
j2
1'>,8 - {381,
+ 1'>,8 - o:h, = -p,12 + {381 - o:h
=
-p,h
R = -p,R+o:l with 1 = h + h of the infected.
At the equilibrium
(4.10)
(8, I) we find the following partition
Proposition 5. At equilibrium let II be the number of infected by contact with the reservoir and 12 the number of infected by contact with other infected. Then -
12
{3J2
= ----. I'>,
The proportion v
=
•
v
+ {31
(4.11)
hi 1 satisfies +- {31 = 8I'>, 1
(I'>, I'>,
+ (31
)
- v .
(4.12)
Hence the proportion of infected by the reservoir is initially 1 and then decreases to the equilibrium proportion v = 1'>,/(1'>, + (3I).
5
Homogeneous demographic model
In models with demographic replacement and differential mortality it makes sense to assume that the transmission term is homogeneous of degree 1. Since recovery, death and birth are modeled by linear terms, the resulting models are homogeneous systems of differential equations [3]. Such models typically do not have stationary point (except the origin). Like in linear systems the solutions of primary interest are exponential solutions. For the stability theory of exponential solutions see
K.P. Hadeler
262
[7]. Here we introduce a reservoir into a model from [3], .
I
S = b1S + b2I + b3R - I-£ I S - ",S - (3S p' .
I
1= -1-£21 + ",S + (3S p - 0'.1, (5.1)
R= -1-£3R+aI with P
=
S
+ I + R. We assume
and we introduce The basic reproduction number for the homogenous problem is (3
Rhomog _
o
- b1 - 1-£1
(5.2)
+ 1-£2 + 0'.
As usual the numerator contains those rates which favor infection (only (3) and the denominator contains those rates which work against infection, mortality of infected 1-£2 (not just differential mortality), recovery 0'., and washout by demographic growth b1 - 1-£1. Following [3] we can look at this problem in two ways, we can either discuss the homogeneous system (5.1) or we can project it to a twodimensional set. The projection
S x = p'
Y=
carries the system (5.1) to the set {x, Y ~
I
P a: x + Y ~
1},
b1x + b2y + b3(1 - x - y) - (3xy - ",x - X(blX + b2y + b3(1- x - y)), iJ = -1-£2Y + (3xy + ",x - ay - y(b 1x + b2y + b3(1- x - y)) (5.3)
i; =
while I
7] = -
R
carries it to the set {(~, 7]) : ~, 7] ~ a},
(5.4)
Epidemic Models with Reservoirs
263
In the first approach we must discuss the non-linear eigenvalue problem
= blS + b21 + b3R - J.lIS - /'i,S - j3SI/ P, AI = -J.l21 + /'i,S + j3SI/P - aI, )"R = -J.l3R + aI. )"S
(5.5)
It seems that the approach of [3] does not work for /'i, > O. Therefore we base our analysis entirely on the projected systems. In [3] the following has been shown for what in the present context is a limiting case, /'i, = O.
Proposition 6. Let /'i, = O. If R~omog ::;; 1 then the only feasible exponential solution of (5.1) is the uninfected solution with exponent bl - J.lI and the eigenvector (1,0, O)T. If R~omog > 1 then there is a second feasible exponential solution with exponent 5. ::;; bl - J.lI and eigenvector (S, 1, Ii) with 1 > O. This infected solution attracts all solutions with I > 0 in the sense of homogeneous systems. For the case
/'i,
> 0 we find the following.
Proposition 7. Let /'i, > O. Then: i) The system (5.3) has no limit sets which intersect the boundary part defined by x + y = 1. ii) All trajectories of the system (5.4) stay bounded. iii) The system (5.4) has no periodic orbits and no closed chains of saddle-saddle connections. The system has a feasible stationary point. iv) Every trajectory of the system (5.4) converges to a stationary point. v) The system (5.1) has at least one feasible exponential solution. All solutions converge to exponential solutions in the sense of homogeneous systems. Proof i) Denote z
i
=
=
1 - x - y. Then
-(bIx + b2 y + (b 3
-
b3 (x + y)))z + ay.
Hence, along x + y = 1 the vector field points strictly inward, except at the point (1,0). At (1,0) we have x = -/'i, < O. ii) The line x + y = 1 corresponds to R = O. The systems (5.3) and (5.4) are equivalent except for the line x + y = 1 which corresponds to (~, 7]) at infinity. Since limit sets of (5.3) stay away from that line, trajectories of (5.4) stay eventually away from 00 and have (compact) limit sets. iii) To (5.4) we use the multiplier 1/(~7]) and apply the criterion of Dulac. The divergence is b
b3
~2
~27]
- -2 - -
/'i,
a
7]2
~
- - - - < O.
264
K.P. Hadeler
Hence there are no periodic orbits and no closed chains. Then the theorem of Poincare-Bendixson yields the existence of at least one stationary point. iv) There are only finitely many stationary points since the right hand side is rational and non-degenerate. v) The statement is just a reformulation of the previous statements in terms of homogeneous systems. 0
6
Special homogeneous system
We consider the special case with constant rates and differential mortality b1 = b2 = b3 = b, /11 = /13 = /1, /12 = /1 + 8. This system is best suited to be compared to a case fatality model [10], [11],
S = bP -
/1S - ",S - j3SI/P,
j = -/11 - M - aI + ",S + j3SI/ P,
R = -/1R+aI.
(6.1)
The projected system becomes
(6.2) The reproduction number (5.2) becomes j3 -b+a+8
Rhomog _
o
(6.3)
= o. If R~omog ~ 1 then there is only the uninfected exponential solution. If R~omog > 1 then there is, in addition, a unique infected exponential solution. ii) Let", > o. Then the system (6.2) has a unique stationary point and the system (6.1) has, up to a positive factor, a unique exponential solution.
Proposition 8. i) Let",
Proof i) This statement has been proven in [3]. ii) For the stationary states we have
o = b(1 + ~ + 1]) - "'~ o=
"'~
+
j3~1] 1+~+1]
-
j3~1]
1+~+1]
a1](1
-
a~1]
+ 1]) - 81] .
, (6.4)
Epidemic Models with Reservoirs
265
Adding these equations leads to
(b - a1])(1
+ ~ + 1]) = 81].
(6.5)
We solve for ~ and insert the expression for ~ into the second equation of (6.4). The resulting equation can be written as follows, (6.6) The left hand side of (6.6) is positive for small 1] and increases to +00 when 1] goes to the pole b/a. The right hand side of equation (6.6) has a unique pole at p E (0, b/a), and it is negative for 1] < p. If 1] runs from p to infinity then the right hand side decreases from +00 to 1. Since p < b/ a there is a unique intersection. D
7
Case fatality
Traditionally in epidemic modeling additional mortality caused by the infectious disease is modeled by differential mortality as in (6.1). If one looks closely at the stochastic interpretation then infected stay infectious all the time and eventually recover with the rate a or die with the rate f-L + 8. Hence death and recovery are considered as competing risks. In [5] (see also [1]) the original idea of Daniel Bernoulli has been followed according to which infected exit from the infected state with some rate , > 0 and then either die with some "case fatality" c or recover with some probability 1 - c. Then the system assumes the form
S=
bN - f-LS - ""S - (lSI/P,
j
""S + (lSI/ P - f-LI - ,I,
=
R=
(1 - chI - f-LR.
(7.1)
Although the parameters a,8 and "c are connected by some simple equations,
8 = C/, a = (1 - ch, , = a + 8, c = 8/ (a + 8),
(7.2)
and hence the systems (6.1) and (7.1) are equivalent in the sense that each case fatality system corresponds to one differential mortality system and conversely (except for c = I), the interpretations are quite different because a,8 are rates while c is a probability. The system (7.1), for varying parameters, in particular for the limiting case c = 1 (where
KP. Radeler
266
the correspondence (7.2) breaks down), has been studied in [10], [I1J for Ii = O. The tools developed there can be used to study the relation between (7.1) and (6.1) also for the case Ii > O.
Acknowledgement The author thanks Christina Kuttler and Johannes Muller for useful suggestions.
References [lJ Andreasen, V., Disease regulation in age-structured host populations. Theor. Population BioI. 36, 214-239 (1989)
[2] Brauer, F., The Kermack-McKendrick epidemic model revisited. Math. Biosc. 198, 119-131 (2005) [3J Busenberg, S., Radeler, KP., Demography and epidemics. Math. Biosci. 101,63-74 (1990) [4J Diekmann, 0., Reesterbeek, J.A.P., Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. Wiley 2000 [5J Dietz, K, Reesterbeek, J.A.P., Daniel Bernoulli's epidemiological model revisited. Math. Biosci. 180, 1-21 (2002)
[6] Esteva, L., Radeler, KP., Maximal prevalence and the basic reproduction number in simple epidemics. In: C. Castillo-Chavez et al. (eds), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Springer-Verlag, IMA Vol. Math. Appl. 126, 31-44 (2002) [7J Radeler, K.P., Periodic solution of homogeneous equations. J. Diff. Equ. 95, 183-202 (1992) [8] Rethcote, R.W., The mathematics of infectious diseases. SIAM Review 42, 599-653 (2000) [9] Rethcote, R.W., Wang, W., Li, Y., Species coexistence and periodicity in host-host-pathogen models. J. Math. BioI. 51, 629-660 (2005) [10J Safan, M., Spread of infectious diseases: Impact on demography, and the eradication effort in models with backward bifurcation. Dissertation Dept. Math., University of Tubingen (2006) [11] Safan, M., Radeler, KP., Dietz, K, Effects of case fatality on demography. submitted.
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[12] Singh, S., Chandra, P., Shukla, J.B., Modelling and analysis of the spread of carrier dependent diseases with environmental effects. J. Biological Systems 11, 325-325 (2003) [13] Sengar, Shikha, Modelling and analysis of the spread of infectious diseases: Effects of environmental and ecological factors. Ph.D. Thesis. Department of Mathematics, Indian Institute of Technology Kanpur (2005) [14] Thieme, H.R., Mathematics in Population Biology. Princeton University Press (2003)
268
Global Stability in Multigroup Epidemic Models Hongbin Guo, Michael Y. Li, Zhisheng Shuai Department of Mathematical and Statistical Sciences, University of Alberta Edmonton, Alberta, T6G 2Gl, Canada Email: [email protected]@math.ualberta.ca. zshuai@math. ualberta. ca
Abstract The question of the uniqueness and global stability of endemic equilibria of multigroup epidemic models of SEIR type is revisited. After a brief review of the literature, we prove that, for a general class of n-group models with bilinear incidence, the endemic equilibrium is unique and globally stable when the basic reproduction number is greater than 1. Our proof utilizes a global Lyapunov function well-known in the literature. The key to our proof is a complete description of the complex patterns exhibited in the derivatives if the Lyapunov function uses results from graph theory.
1
Introduction
Multigroup models have been proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations. Heterogeneity can result from many factors. Individual hosts can be divided into groups according to different contact patterns such as those among children and adults for Measles and Mumps, or to distinct number of sexual partners for sexually transmitted diseases including HIV / AIDS. Groups can be geographical such as communities, cities, and countries, or epidemiological to incorporate differential infectivity or co-infection of multiple strains of the disease agent. Multigroup models can also be used to investigate infectious diseases with multiple hosts such as West-Nile virus and vector borne diseases such as Malaria. For a recent survey of multigroup models, we refer the reader to [27]. A multigroup model is, in general, formulated by dividing the population of size N(t) into n distinct groups. For 1 ~ k ~ n, the k-th group
Global Stability in Multigroup Epidemic Models
269
is further partitioned into four compartments: the susceptibles, the exposed (latent), infectious, and recovered, whose numbers of individuals at time t are denoted by Sk(t), Ek(t), h(t), and Rk(t), respectively. For 1 ~ i, j ~ n, the disease transmission coefficient between compartments Si and Ij is denoted by fJij ? 0, and fJij = 0 if there is no transmission of the disease between the two compartments. The new infection occurs in the k-th group is given by n
LfJkj SkIj. j=1
(1.1 )
The form of incidence in (1.1) is bilinear. Other incidence forms have been used in the literature, depending on the assumptions made about the mixing among different groups. Within the k-th group, it is assumed that death occurs in Sk, E k , h, and Rk compartments with rate constants d~ , df, di" and df, respectively. These rates may include death due to natural causes and due to the disease. The influx of individuals into the k-th group is given by a constant A k , which are assumed to be susceptible. After infection, an individual remains in the latent class before becoming infectious. We assume that the transfer rate from Ek to Ik is fk so that l/fk is the mean latent period for the k-th group. We assume that individuals in h recover with a rate constant "(k, so that l/"(k is the mean infectious period for the k-th group, and once recovered they remain permanently immune for the disease. Based on these assumptions, the following system of differential equations can be derived n
S~ = Ak - d~Sk - LfJkjSk1j, j=1 n
E~
=
I~ =
R~
LfJkjSkIj - (df + fk)Ek, j=1 fkEk - (di, + "(k)h,
k = 1,2, ... , n.
(1.2)
= "(h - dfRk,
All parameters are non-negative constants, and we assume fk, d~, df, di" df > 0 for all k. For each k, adding all equations in (1.2) gives
+ Ek + h + Rk)' =
(Sk
Ak - d~ Sk - df Ek - di,h + df Rk + Ek + Ik + Rk),
~ Ak - d'k(Sk
where d*
Rk)
k
Hence limsuPt---+oo(Sk + Ek + h Similarly, from the Sk equation we obtain lim SUPt---> 00 Sk
= min{d~, df, di" df} > O.
~ ~.
+ ~
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
270
~. Observe that the variable Rk does not appear in the first three
e~uations of (1.2). This allows us to consider first the following reduced system for Sk, Ek and h n
S~ = Ak - d~Sk - 'Lf3kj S k1j, j=1 n
E~ = 'Lf3kjSk1j -(d~ + Ek)Ek , j=1 I~ = EkEk - (d{ + "Ik)Ik,
(1.3)
in the following region of the non-negative cone of ]R.3n
(1.4)
The behaviours of Rk can then be determined from the last equation in (1.2). It can be verified that region r is positively invariant. System (1.3) always has the trivial equilibrium Po = (S~, 0, 0, ... , S~, 0, 0), where = ~, 1 ~ k ~ n, is the equilibrium of the Sk popula-
s2
k
tion in the absence of disease (Ek = Ik = 0, 1 ~ k ~ n). For this reason, Po is called the disease-free equilibrium. An equilibrium P* = o
(Si, Ei ,Ii , ... , S~, E~,I~) in the interior r of r, namely, St" Et,,Tk > 0, 1 ~ k ~ n, is called an endemic equilibrium. The matrix B = (f3ij) encodes the patterns of contacts and transmission among groups that are built into the model. Throughout the paper, we assume that B is irreducible (see Section 2 for definition of irreducibility). Biologically, this is equivalent to assuming that any two groups have a direct or indirect route of transmission. Set
Ro
=
p(Mo),
(1.5)
where
f3ij Ei S P
M _ ( 0-
(df
)
+ Ei)(d{ + "Ii) 1~i,j~n'
(1.6)
and p denotes the spectral radius. The parameter Ro is a key threshold parameter for (1.3), and is referred to as the basic reproduction number (see, e.g., [28]). Its biological significance is that if Ro < 1 then the disease dies out, and that if Ro > 1 then the disease becomes endemic (see, e.g., [27, 28]). A long-standing open question in mathematical epidemiology is that whether a mUlti-group model such as system (1.3)
Global Stability in Multigroup Epidemic Models
271
has a unique endemic equilibrium P* when Ro > 1, and that whether P* is globally asymptotically stable when it is unique. We prove the following theorem, which settles this open problem for system (1.3), and any other model that can be converted to the same form.
Theorem 1.1. Assume that B = (/3ij) is irreducible. If Ro > 1, then system (1.3) has a unique endemic equilibrium P*, and P* is globally o
asymptotically stable in
r.
One of the earliest work on multi group models is the seminal paper by Lajmanovich and Yorke [17] on a class of SIS multigroup models for the transmission dynamics of Gonorrhea. The global stability of the unique endemic equilibrium is proved using a quadratic global Lyapunov function. Subsequently, more complicated multigroup models have been proposed and analyzed, see e.g., [1, 9, 10, 11, 12, 13, 22, 24, 26, 27] and references therein. Hethcote [10] proved global stability of the endemic equilibrium for multigroup SIR model without vital dynamics. Beretta and Capasso [1] derived sufficient conditions for global stability of the endemic equilibrium for multigroup SIR model with constant population in each group. Thieme [26J proved global stability of the endemic equilibrium of multigroup SEIRS models under certain restrictions. The most recent result on global stability is Lin and So [22] for a class of SIRS models with constant group sizes, in which they proved that the endemic equilibrium is globally asymptotically stable if the cross-group contact rates /3ij, i -=I- j, are small or if the recovery rates in each group are small. On the other hand, results in the opposite direction also exist in the literature. For a class of n-group SIR models with proportionate incidence, uniqueness of endemic equilibria may not hold when Ro > 1 (see [13, 27]). In light of these results, complete determination of the global dynamics of these models is essential for their application and further development. In the case when n = 1, system (1.2) is the single population SEIR model, whose global dynamics have been completely determined [15, 20, 21]. The basic reproduction number in (1.5) becomes
/3EA
Ro where
/3
=
/3u
=
(E + d) b
+ d) d '
and all subindices are suppressed. When Ro :::;; 1, the o
disease-free equilibrium is globally stable in the feasible region r, while if Ro > 1, Po becomes unstable, and a unique endemic equilibrium P* s globally stable in r. The global stability of P* was first established in [20] (also see [19, 21]) using a geometric approach to global stability of Li and Muldowney. Another proof was given in [15J using a global Lyapunov function.
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
272
Our proof of Theorem 1.1 uses the following global Lyapunov function n
V
=
L Vk [(Sk -
Si" lnSk)
+ (Ek -
k=l
~
Ei" lnEk) + k f+ Ek (Ik - Ii" lnlk)]. k
(1.7) We note that V is a linear combination of functions of form
which were used in [15] for the single group SEIR models. The form of Lyapunov functions in (1.7) has long been known in the literature of ecological models. Its introduction can be traced to a paper of Goh [7] in which basic properties of this class of functions are discussed. Applications to other ecological and epidemic models can be found in [5, 22] and references therein. The key to our analysis is finding the right choice of coefficients Vk in (1.7) such that the derivative V'is nonnegative. This is made possible by a complete description of the complex patterns exhibited in V', using graph theory. As structured models are being used to describe more and more complicated biological problems, we expect this form of Lyapunov functions and our graph theoretical analysis will have much wider applicability. In the next section, we recall some definitions and results from graph theory, and prove a preliminary result regarding system (1.3). In Section 3, we present the proof of our main result, Theorem 1.1.
2
Preliminaries
Let E = (eij)nXn, F = (fij)nxn be nonnegative matrices, namely, all of their entries are nonnegative. We write E ~ F if ekj ~ Aj for all k, j, and E > F if E ~ F and E =I- F. For n > 1, an n x n matrix F is reducible if, for some permutation matrix Q,
and F l , F3 are square matrices. Otherwise, F is irreducible. The following properties of nonnegative matrices are standard (e.g., see [3]). PI. If F is nonnegative, then the spectral radius p(F) of F is an eigenvalue, and F has a nonnegative eigenvector corresponding to p(F). P2. If F is nonnegative and irreducible, then p(F) is a simple eigenvalue, and F has a positive eigenvector x corresponding to p(F). P3. If 0 ~ E ~ F, then peE) ~ p(F). Moreover, if 0 ~ E E + F is irreducible, then peE) < p(F).
< F and
Global Stability in Multigroup Epidemic Models
273
P4. If F is nonnegative and irreducible, and E is diagonal and positive (namely, all of its entries are positive), then FE is irreducible. Irreducibility of matrices can be easily tested using the associated directed graphs. A directed graph G n is a set of n vertices and a set of directed arcs joining two vertices. It is strongly connected if any two distinct vertices are joined by an oriented path. The directed graph G(F) associated with an n x n nonnegative matrix F is a directed graph of n vertices, 1,2"" ,n, such that, there exists an arc (j, k) leading from j to k if and only if Aj i= 0. We have the following property. P5. Matrix F is irreducible if and only if G(F) is strongly connected. An oriented cycle in a directed graph is a simple oriented path from a vertex to itself. A directed tree is a connected directed graph with no cycles. A directed tree is said to be 'rOoted at a vertex, called root, if every path between a non-root vertex and the root is oriented towards the root. We refer the reader to [23] for more details. Consider the linear system (2.1)
Bv =0, where 2:1#1/311 - /312
-/321 2:1#2 /321 ...
(2.2)
and /3ij ;;: 0, 1 ~ i, j ~ n. (.
Lemma 2.1. Assume that the matrix (/3ij)nxn is irreducible and n ;;: 2. Then the followings hold.
(1) The solution space of system (2.1) has dimension 1. (2) A basis of the solution space is given by (V1,V2,'"
,vn ) = (0 11 ,022 ,'" ,Onn),
where Okk denotes the cofactor of the k-th diagonal entry of B, 1 ~ k ~ n.
(3) For 1 ~ k
~
n,
Okk =
L
II
/3jh > 0,
(2.3)
TE'lrk (j,h)EE(T)
where 'lI'k is the set of all directed trees of n vertices rooted at the k-th vertex, and E(T) denotes the set of directed arcs in a directed tree T.
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
274
Proof. Since the sum of each column in B equals zero, we have
(2.4) where C jk denotes the cofactor of the (j, k) entry of B. Since B is singular, we know that (C11 , C I2 ,··· ,C1n ) is a solution of system (2.1). Therefore, by (2.4), (C 11 , C 22 ,··· ,Cnn ) is also a solution of system (2.1). For 1 ~ k ~ n, in the k-th column of B, the diagonal entry, Ll# 13kl, equals the negative of the sum of nondiagonal entries. By a result on directed graphs in [23, p. 47, Theorem 5.5], we obtain Ckk
L
=
II
13j h.
TElI'k (j,h)EE(T)
Since (/3ij) is irreducible, its associated directed graph is strongly conI1 /3jh nected, by P5. Thus, for each k, at least one term in L TElI'k (j,h)EE(T)
is positive. Therefore, Ckk > 0 for k = 1,2,··· ,n. Since C 11 is a (n -1) minor of B, we know rank(B) = n - 1, and the solution space of (2.1) has dimension 1, completing the proof of Lemma 2.1. 0 As an illustration of (2.3), let n = 3 and '][\ be the set of all directed trees rooted at the first vertex. Then, as shown in Figure 1, '][\ = {Tl, Tn, and E(Tn = {(3, 2), (2, I)}, E(Tf) = {(2, 1), (3, I)}, E(Tt) = {(2,3), (3, I)}. Therefore,
Tr,
C11
=
L
II
T{E1I'l (j,h)EE(T{)
3
2
2
3
,
T.'
Figure 1. All directed trees with three vertices and rooted at 1. A unicyclic graph is a directed graph that is obtained from a collection of directed rooted trees by joining their roots and contains a unique
Global Stability in Multigroup Epidemic Models
275
oriented cycle. For 1 :::; l :::; n, let D(n, l) denote the number of unicyclic graphs with n vertices whose cycle has length l. Then (2.5) and
n
I: D(n, l)l.
nn =
(2.6)
l=l
For proofs of these relations, we refer the reader to [2, Chapter 2]). In a directed rooted tree, if we add a directed arc from the root to any non-root vertex, we obtain a unicyclic graph, see Figure 2.
.i
"-
"- ......
k
Figure 2. A unicyclic graph is formed from a directed tree rooted at vertex k by adding a directed arc from k to j. Let Ro be defined in (1.5). We first establish the following result for system (1.3). Results like Proposition 2.2 are known in the literature, at least for some special classes of model (1.3) (see [10, 22, 27]). We provide a proof for completeness and to demonstrate our derivation of
Ro· Proposition 2.2. Assume B hold.
=
((3ij)
is irreducible.
Then followings
(1) If Ro :::; 1, then Po is the unique equilibrium and it is globally stable in
r.
(2) If Ro > 1, then Po is unstable and system (1.3) is uniformly pero
sistent in
r.
Proof. Let 8 = (81 ,'" ,8n ), 8 0 = (8~,··· ,8~), and
M(8)
= (
E
(d i
jE
(3i i8;
+ Ei)(di + 'Yi)
)
l";i,j~n
.
For 1 :::; k :::; n, since 0 :::; 8k :::; 8Z, we have 0:::; M(8) :::; M(80 ) = M o, and if 8 =I- So, then M(S) < Mo. Since B is irreducible, we know M(8)
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
276
is irreducible for 0 < 8 ~ 8 0 . Therefore, by P3, p(M(8)) < p(Mo) Ro ~ 1 provided 8 i= 8 0 . This implies that
=
M(8) I = I has only the trivial solution I = (h,'" ,In)t = 0, and that Po is the only equilibrium of system (1.3) in r if Ro ~ 1. Let (Wl,W2,'" ,wn ) be a left eigenvector of Mo corresponding to p(Mo), i.e., (Wl,W2,'"
,wn)p(Mo) =
(Wl,W2,'"
Since Mo is irreducible, we know, by P2,
Wk
,wn)Mo.
> 0 for k = 1,2",' ,n. Set
Then n
L'
=
~ (df + fk~(dr + ')'k) [fkE~ + (df + fk)I~] ,wn ) [M(8) I - I]
= (Wl,W2,'"
~ (Wl,W2,'" =
,wn ) [Mo I - I] [p(Mo) -1](Wl,W2,'" ,wn ) I ~ 0,
Furthermore, if Ro L' = 0 implies
= p(Mo) < 1, then L' = 0 {=} I = O.
(Wl,W2,'"
If 8
i= 8
0
,
if Ro
,wn)M(8)I =
~
1.
If Ro
(Wl,W2,'"
,wn)I.
,wn)Mo =
(Wl,W2,'"
= 1, then (2.7)
then
,wn)M(8) <
(Wl,W2,'"
(Wl,W2,'"
,wn),
and (2.7) has only the trivial solution I = O. Therefore, L' = 0 {=} 1= 0 or 8 = 8 0 provided Ro ~ 1. It can be verified that the only compact invariant subset of the set where L' = 0 is the singleton {Po}. By LaSalle's Invariance Principle, Po is globally stable in f if Ro ~ 1. If Ro = p(Mo) > 1 and I i= 0, we know that (Wl,W2,'"
,wn )MO -(Wl,W2,'" ,wn ) = [p(Mo)-I](Wl,W2,'" ,wn ) > 0,
and thus L' o
=
(Wl,W2,'"
,wn )[M(8)I - I] > 0 in a neighborhood of
Po in f, by continuity. This implies Po is unstable. Using a uniform persistence result from [6] and a similar argument as in the proof of Proposition 3.3 of [19], we can show that, when Ro > 1, the instability of Po implies the uniform persistence of (1.3). This completes the proof of Proposition 2.2. 0
Global Stability in Multigroup Epidemic Models
277
Uniform persistence of (1.3), together with uniform boundedness of o
solutions in r, implies the existence of an equilibrium of (1.3) in Theorem D.3 in [25] or Theorem 2.8.6 in [4]).
0
r
(see
Corollary 2.3. Assume B = ((3ij) is irreducible. If Ro > 1, then (1.3) has at least one endemic equilibrium.
3
Proof of Theorem 1.1
Denote an endemic equilibrium, whose existence is established in Corollary 2.3, by P* = (S;, E;,1;,'" ,S~, E~,1~),
Sk' E k, Ik > 0 for k = 1, 2, ... ,n. In this section, we prove that P* is globally asymptotically stable when Ro > 1. In particular, this implies o
that the endemic equilibrium is unique in the region r when it exists. Set (3.1) i3ij = (3ij S; Ij , 1 ~ i,j ~ n, n ~ 2, and
2::1# 131/ -/321 -1312 2::1#21321 '"
-i3nl -i3n2
B=
(3.2)
-i31n
-i32n
2::1#n i3nl
Then, by Lemma 2.1, a basis for the solution space of the linear system (3.3)
Bv =0 can be written as
(3.4) where C kk denotes the cofactor of the k-th diagonal entry of B, 1 ~ k ~ n. By the irreducibility of B, we know that (i3ij) is irreducible and Vk = C kk > 0, k = 1"" ,n, by Lemma 2.1. For n = 1, i.e., the case of single group SEIR model, Theorem 1.1 is well know (e.g. see (15]). We only consider the case n ~ 2. Let VI,'" ,vn be chosen as in (3.4). Set n
V
=
L Vk [(Sk k=1
Si., lnSk)
+ (Ek -
Ei., lnEk)
+
~
k E: Ek (h - Ii., lnh) ]. (3.5)
278
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
Differentiating V and using the equilibrium equations n
Ak = drS'k
+ L(3kj S 'k 1j,
(3.6)
j=l n
(df + Ek)E'k = L(3kj S 'k 1j,
(3.7)
j=l
(3.8) and
(3.9) which follows from (3.7) and (3.8), we obtain
(3.10)
Global Stability in Multigroup Epidemic Models
279
since
S'k Sk
Sk
+ S'k
- 2;;:: 0,
and the last equal sign holds if and only if Sk = S'k. In the above derivation, we have substituted the two incidences of Ak using (3.6). Next, for V1,'" ,Vn as in (3.4), we claim
for all (h,'" ,In) E lR+.. To see this, we note that n
n
n
LVk L{3kj S 'k I j k=1 j=1
n
n
n
= LVj L{3jk S ;h = L (L{3jk S ;Vj)h. (3.12) j=1
k=l
k=1
j=1
It suffices to show k = 1,2"" ,n.
(3.13)
In fact, by (3.1)-(3.3) and (3.9), we have
[
{3ll~i Ii
.... :
{3nl~~Ii
{31nSi I~ .. ,
{3nnS~I~
1
and this leads to (3.13). Using inequality (3.10), notation 13kj in (3.1), and identity (3.11), we have
Denote
(3.14)
280
Hongbin Guo, Michael Y. Li, Zhisheng Shuai o
In the following we show Hn ~ 0 for all (81, E 1, h,'" ,8n , EnIn) E r. Since the proof for the general case uses a graph theoretic approach and is quite abstract, we choose to illustrate the main ideas first with detailed proofs for cases n = 2 and n = 3. Then major steps of the proof for arbitrary n ~ 2 will be given. Case n = 2: In this case, we obtain from (3.3) V1 = /321 and V2 Substituting V1, V2 into (3.28) and expanding H2 we have
- - (
8i -
h81Ei Ii 8 i E 1 Iz8 1Ei I;8iE1 -
3 - 82
-
I182E; E2I;) Ii 8:;' E2 - E:;' Iz
8; 3 - 82
-
Iz 8 2 E; E2I; ) I;8;E2 - E:;'I2 .
H2 = i321i3n 3 - 8 1
- - ( 8i + i32d312 3 - 8 1
- - (
+ i312i321 -
-
+ i312i322
(
-
8;
= /312,
E1Ii) Eih EIIi) Eih (3.15)
Note that the middle two terms have the same coefficients /321/312' We show that this is not a coincidence and can be seen as follows: write the subindices of /3ij'S in each coefficient in the form (3.16) respectively. Each expression in (3.16) defines a transformation from row 1 to row 2 and possesses a unique cycle of length 1 or 2. Moreover, the two coefficients in (3.15) corresponding to the transformations with a 2cycle (a cycle with length 2) are the same, since they arise from rotations of the same 2-cycle. Therefore, the terms in H2 can be naturally grouped according to the length of cycles appearing in their coefficients.
Note that
Global Stability in Multigroup Epidemic Models
281
and
/32Ii312 (6 -
Si _
I 2S 1Ei _ E1Ii _ S2 _ hS2E2 _ E2I2) , :::: 0 I 2 S i E 1 Eih S2 Ii S2 E2 E2I2 '" .
Sl
We thus obtain H 2(Sl, E 1, h, S2, E 2, 12) ~ 0, and thus V' ~ 0, for all o
(Sl,E1,h,S2,E2,I2) E r. From (3.10), we know V' = 0 if and only if Sk = Sic, k = 1,2, and H2 = O. Moreover, irreducibility of matrix B, or equiv~le~tly, the strong connectedness of the directed graph G(B), implies (321(312 > O. Consequently, we obtain from (3.17) V'=O
Sk=Sic, Ek=aE'k, h=aI'k,
{=:::>
k=1,2,
where a is an arbitrary positive number. Case n - 3:
In this case H 3
= ~ 6
k,j=l
-(3 _ S'kS _
v (3k kJ
k
SkIjE'k _ EkI'k) S* 1* E E* I . k J k k k
(3.18)
From (3.4) we obtain V1
= /332/321 + /331/321 + /323/331,
V2 = V3 =
+ /313/332 + /312/332, /312/323 + /321/313 + /313/323.
/331/312
(3.19)
Substituting these expressions of Vk into H 3 , we observe that H3 is the sum of 33 = 27 terms of forms
(3.20) or - - - ( Sk SkIjEk EkIk) (3rk(31k(3kj 3 - -S - S* 1* E - E* I ' k kj k kk
(3.21 )
where {r, l, k} is a permutation of {I, 2, 3}, and 1 ~ j ~ 3. Write the subindices of /3ij'S in (3.20) and (3.21) in the form of transformations
l k}
r { l k j
and
rlk} {k kj ,
(3.22)
respectively. When j = k, lor r, both transformations in (3.22) possesses cycles of length 1, 2 or 3. The terms in H3 will be grouped together according to the length of cycles appearing in (3.22).
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
282 When j
= k,
both transformations in (3.22) have a I-cycle {: :
~},
and accordingly, the terms in (3.20) and (3.21) satisfy
- - - ( Sk SklkEk Eklk) /3rl/3lk/3kk 3 - Sk - Sk1k Ek - Ekh :(: 0, and
- - - ( Sk Sk1jEk Eklk) /3rk/3lk/3kk 3 - Sk - SkI; Ek - Ekh :(: O. When j
= r,
the first transformation in (3.22) produces two distinct
3-cycle patterns
!
{~ ~}
{~ 7;}.
and
There are 6 terms in H3 of a
3-cycle form, three of them correspond to each cycle pattern, and thus have the same coefficients i3rli3lki3kr or i3rki3kli3lr' These six terms can be divided into two groups and each has a sum of form si., SkIrEi., Ek I ;) + /3- /3- /3- (3 /3-rl /3-lk /3-kr (3 - Sk - S*k1* Ek - E*k Ik lk kr rl r /3/3(3 §L SlIkEi ElIi) +/3kr rl lk - Sl - Si I; El - Ei Il
s;
SrItE;
S r - S*r1* Er l
-
ErI; )
E*r T I
(3.23) When j = r, the second transformation in (3.22) has a 2-cycle
~ : ; }. Also, when j = l, both transformations in (3.22) have a 2-cycle * k l . There are altogether 12 terms in H3 corresponding to 2-cycle { *lk}
{
patterns. Each 2-cycle pattern corresponds to 2 terms in H3 with the same coefficients. These 12 terms can be grouped into 6 pairs and each has a sum of form
/3-rk /3-lk /3-kr (3 -
§1. - SkIrE'k - Ek I ;) Sk
-- /3-rk /3-lk /3-kr (6 -
S*k1*rEk
E*kIk
+ /3-kr /3-lk /3-rk (3 - !E:.. S -
§1. - SkIrE'k - EkI; Sk
S*k1* Ek r
E*k Ik -
r
!E:.. S r
-
SrIkE; S*r1* E k r
SrhE; S*I*E rkT
ErI;)
E*r T I
0
:(:,
(3.24)
or
= /3-rl /3-lk /3-kl (6 -
Err) =-::..z:.E*I rT
S'k _ SkIlE'k _ EkI'k _ Sk S*I*E E*I kl k kk
§I _ SlhEi _ ElIi) Sl
S*I*E l k l
E*I II
:(:
O. (3.25)
Global Stability in Multigroup Epidemic Models
283
In summary, each coefficient in H3 corresponds to a transformation in (3.22) which possesses a unique cycle of length 1,2, or 3. By property (3) of Lemma 2.1, the number of transformations in (3.22) with an 1cycle is given by D(n, I) x I, I = 1,2,3. In particular, by (2.5), the number of I-cycles in (3.22) is D(3, 1) x 1 = 9, the number of 2-cycles is D(3,2) x 2 = 12, and the number of 3-cycles is D(3,3) x 3 = 6. Therefore, by (2.6), 33
= 27 = D(3, 1) x 1 + D(3, 2) x 2 + D(3, 3) x 3.
This shows that all terms in H3 are accounted for in our grouping according to cycle patterns and lengths in (3.22). Therefore, we have shown o
0 for all (Sl,E 1 ,h,S2,E2,I2,S3,E3,h) E f, and thus V' ~ O. From (3.10), we know V' = 0 if and only if Sk = S'k, k = 1,2,3, and H3 = O. We claim that if Sk = S'k, k = 1,2,3, then
H3
~
(3.26) where a is an arbitrary positive number. It suffices to show that H3 = 0 implies 1
~
k,r
~
3.
(3.27)
If f3kr = 0, for some k #- r. Then, by the irreducibility of B = (f3ij) , or equivalently, the strong connectedness of the G(B), we necessarily have - f3klf3lr
#- 0, for I #- k, r. Therefore, either a 3-cycle {klr} Irk exists or both
2-cycles {
7~ :} and {~ ~ :} exist. In either case, (3.27) follows from
H3 = 0, and from relations (3.23), (3.24), and (3.25). If all f3ij #- 0, i #- j, then i3kli3lr #- 0, for I #- k, r, and the same argument shows that (3.27) holds. We thus obtain Vi = 0
{==:}
Sk = S'k, Ek = aE'k, h = aI'k,
k = 1,2,3,
where a is an arbitrary positive number. Case n ;;::: 2:
We have
(3.28) Here Vk = Ckk as given in (2.3) is a sum of nn-2 terms, each of which is a product of (n - 1) i3i/S whose subindices can be represented by
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
284
all arcs in a directed tree T rooted at the k-th vertex, by Lemma 2.1(3). Therefore, subindices of Vk 13kj, which is a product of n 13i/S, are represented by all directed arcs of a unicyclic graph Q obtained by adding an oriented arc (k, j) from the root k to the vertex j to the directed tree T, as shown in Figure 2. Unicyclic graph Q has a unique cycle CQ of length 1 ~ I ~ n. Furthermore, all l rotations of the I-cycle CQ give rise to l terms in Hn with the same coefficients, and these I terms are naturally grouped together. We can show, as in the cases of n = 2,3, the sum ofthese I terms is nonpositive. More specifically, using (2.3), we can first group all terms in Hn according to the cycle length present in their coefficients, then further group the terms of the same cycle length according to their cycle patterns represented in the unicyclic graph Q, as shown in the following H
n
=
=
~ v
{3- .
L k kJ k,j=1
t
1=1
(3 _ 8"k _ 8k 1j E "k 8 8* 1* E k
k
J
_ EkI;') E* I k k k
[L L ( II !3jh) QED(n,l) (1',m)EE(CQ) (j,h)EE(Q)
( 3 _ 8; _ 8 1' 1m E ; _ E1'I;)] 81' 8;I;"E1' E;I1'
(3.29)
n
=L
1=1
[QED(n,l) L ((j,h)EE(Q) II !3jh)
L
(3 _ 8; _ S1'1m E ; 81'
(1',m)EE(CQ)
8;I;"E1'
_ E1'I;)] E;I1' '
where V(n, l) presents the set of all unicyclic graphs of n vertices with an oriented cycle of length l, CQ is the oriented cycle of length l in a unicyclic graph Q E V(n, l), and E(CQ), E(Q) represent the sets of arcs in CQ, Q, respectively. Since the cardinality of E(Q) is n, the coefficient of each term in (3.29), TI(j,h)EE(Q) 13jh, is a product of n 13i/S. The cardinality of the set V(n, l) is (3.30) and the cardinality of E(CQ) is the length l of the cycle CQ. By the identity (2.6) n
nn =
L D(n, l)l, l=1
(3.31)
Global Stability in Multigroup Epidemic Models
285
we see that all terms in Hn are accounted for in our grouping (3.29). For the oriented cycle CQ in any Q E D(n, l), we have
(3.32) By (3.29) and (3.32), we know H n (Sl, E 1 , h,··· ,Sn, En'!n) ~ 0 for all o
(Sl, E 1 , h,··· ,Sn, Enln) E r. Therefore, we have V' ~ we claim that if Sk = S'k, 1 ~ k ~ n, then
o. Furthermore, (3.33)
where a is an arbitrary positive number. It suffices to show that
when i3kr =f. By the irreducibility of (fJij), there exist 1 ~ m1, m2, ... , ms ~ n, 0 ~ s ~ n - 2 such that k, r, m1,··· ,m s are distinct, and
o.
the product i3kri3rm li3mlm2 ... i3m s k =f. o. Furthermore, there exists a unicyclic graph Q E D(n, l) such that CQ = {(k, r), (r, md,··· ,(m s , k)} and IT i3j ,h =f. O. Therefore, from (3.29) and (3.32), we know (j,h)EE(Q)
Ek - b.. E; and lJ.. Ik -- Ix. I; 1·f H n -- 0 . From (3.10) and (3.33), we know that V' ~ 0 for all (Sl, E,I,· .. ,Sn,
§Jr.. -
o
r,
and V' = 0 {o} Sk = S'k,Ek = aE'k,h = al'k,k = 1,2,··· ,no Substituting Sk = S'k, Ek = aE'k, and h = alk into the first equation of system (1.3), we obtain
En'!,) E
n
0= Ak - d~S'k - a LfJkjS'klj. j=l
(3.34)
Since the right-hand-side of (3.34) is strictly decreasing in a, we know, by (3.6), that (3.34) holds if and only if a = 1, namely at P*. Therefore, the only compact invariant subset of the set where V' = 0 is the singleton {P*}. By LaSalle's Invariance Principle, P* is global asymptotically o
stable in
r
if Ro
> 1. This completes the proof of Theorem 1.1.
Acknowledgments This research is supported in part by grants from the Natural Science and Engineering Research Council of Canada (NSERC) and Canada Foundation for Innovation (CFI). Both HG and ZS acknowledge the support
286
Hongbin Guo, Michael Y. Li, Zhisheng Shuai
of the Josephine Mitchell Graduate Scholarships from the Department of Mathematical and Statistical Sciences at the University of Alberta. The authors also acknowledge the financial support from NCE-MITACS Project "Transmission Dynamics and Spatial Spread of Infectious Diseases: Modelling, Prediction and Control".
References [1] E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in: T.G. Hallam, L.J. Gross, and S.A. Levin (Eds.), Mathematical Ecology, Singapore World Scientific, Teaneck, NJ, 1986,317-342. [2] F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge University Press, Cambridge, 1998. [3] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. [4] N. P. Bhatia and G. P. Szego, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, Vol. 35, Springer, Berlin, 1967. [5] H.I. Freedman and J.W.-H. So, Global stability and persistence of simple food chains, Math. Biosci. 76 (1985), 69-86. [6] H.I. Freedman, M.X. Tang, and S.G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat. 6 (1994), 583-600. [7] B.S.Goh, Global stability in many-species systems, Am. Nat. 111 (1977), 135-143. [8] H. Guo and M.Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng. 3 (2006), 513525. [9] K.P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Math. Biosci. 146 (1997), 15-35. [10] H.W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol. 14 (1978), 338-349. [11] H.W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2000), 559-653. [12] H.W. Hethcote and H.R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci. 75 (1985), 205-227.
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[13] W. Huang, K.L. Cooke, and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV / AIDS transmission, SIAM J. Appl. Math. 52 (1992), 835854. [14] A. Korobeinikov, A Lyapunov function for Leslie-Gower predatorprey models, Appl. Math. Lett. 14 (2001), 697-699. [15] A. Korobinikov and P.K. Maini, A Lyapunov function and some properties for SEIR, SIS Epidemic models, Math. Biosci. Eng. 1 (2004), 157-160. [16] A. Korobeinikov and G.C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett. 15 (2002), 955-960. (17] A. Lajmanovich and J.A. York, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), 221-236. (18] J.P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [19) M.Y. Li, J.R. Graef, L. Wang, and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci. 160 (1999), 191-213. [20) M.Y. Li and J.S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci. 125 (1995), 155-164. [21] M.Y. Li and L. Wang, Global stability in some SEIR epidemic models, in: C. Castillo-Chavez et. al. (Eds.), Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, The IMA Volumes in Mathematics and Its Applications, Vol. 126, Springer, New York, 2002, 295-312. [22) X. Lin and J.W.-H. So, Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations, J. Austral. Math. Soc. Ser. B 34 (1993), 282-295. [23) J.W. Moon, Counting Labelled Trees, William Clowes and Sons, London, 1970. [24] L. Rass and J. Radcliffe, Global asymptotic convergence results for multitype models, Int. J. Appl. Math. and Comp. Sci. 10 (2000), 63-79. [25] H.L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
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Hongbin Guo, Michael Y. Li, Zhisheng Shuai
[26] H.R. Thieme, Local stability in epidemic models for heterogeneous populations, in: V. Capasso, E. Grosso, and S.L. PaveriFontana (Eds.), Mathematics in Biology and Medicine, Lecture Notes in Biomathematics 57, Springer 1985, 185-189. [27] H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [28] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29-48.
289
Epidemic Models with Time Delays* Wendi Wang School of Mathematics and Statistics, Southwest University Chongqing 400715, China E-mail: [email protected]
Abstract This article overviews mathematical approaches for analysis of epidemic models with time delays. We start by introducing the methodology of modeling epidemic diseases with time delays and illustrating the D-subdivision method for characteristic equations. Then we show the application of the Liapunov functional method. Lastly, we present the persistence techniques to determine conditions under which diseases are permanent in the population.
1
Introd uction
Since the pioneer work of Kermack and McKendrick [35], mathematical models of differential equations have become important tools in analyzing the spread and control of infectious diseases [1, 6, 7, 15, 16, 17, 21, 19,30,31,33,37,48,49]. Although models of ordinary differential equations play fundamental roles, time delays are common and important in modeling epidemic diseases. First, the incubation periods of infectious diseases, the infection periods of infective members and the periods of recovered individuals with immunity can be represented by time delays. More importantly, these delays can not be neglected in most of cases. This is not only because the lengthes of delays may be long, for example, the incubation time for HIV could reach 10 years, but also because quite different dynamical behaviors of mathematical models can be induced by time delays. The objective of present work is to survey recent advances of delayed epidemic models. This paper is organized as the follows: L Mathematical models with time delays 2. The methods of stability analysis 3. Persistence of diseases 4. Summary *The author is partly supported by NSF of China (No. 10571143).
290
2
Wendi Wang
Mathematical modeling
As any other mathematical modeling, we must begin from selecting model's structures to construct epidemic models. Basically, we follow the methodology of compartments from Kermack and McKendrick [35]. Then we branch according to the durations of infectious diseases. If the disease spreads quickly and is not fatal, we can ignore the birth and death of populations. On the other hand, we should consider demographic structures if an infectious disease persists a long time or the disease-induced mortality rate is high. For illustration purpose, we present several typical epidemic models to show ingredients of models ignoring birth and death processes or models with demographic structures, and ways of introducing time delays. We always adopt the following nomenclature: t: time;
N: population density or population number;
S: the number or density of susceptible individuals; E: the number or density of exposed individuals;
I: the number or density of infectious individuals; R: the number or density of recovered individuals.
2.1
Models without demographic structure
For an SIR type of disease, if r is the latent period of the disease, we have
S' (t) = -(3S(t)I(t), / (t)
=
(3S(t - r)I(t - r) - "(I(t),
R' (t) = "(I(t),
where {3 is the contact coefficient and "( is the recovery rate (Ma et al., [33]). Cooke and Yorke [12] proposed the following model for gonorrhea epidemics:
dS dt = -(3S(t)I(t)
+ (3S(t -
r)I(t - r),
dI dt = (3S(t)I(t) - (3S(t - r)I(t - r), where r is the infection period of the disease.
Epidemic Models with Time Delays
2.2
291
Model with vital dynamics
General demographic structure of a population in the absence of disease takes the form:
~ = B(N)N -
D(N)N,
(2.1)
where N is the population size, B(N) is the per capita birth rate and D(N) is the per capita death rate. Some structures frequently used in literature are listed below:
(1) B(N) = D(N) = /-1, where /-1 is a constant. This means that the birth and death are balanced so that the population size is a constant, which simplifies mathematical analysis (see, for example, Ma et al., [33], Wang [39]); (2) The net birth rate of population is a constant A, and the per capita death rate of population is a constant d. Then vital dynamics is given by dN (2.2) ill =A-dN. One advantage of this population dynamics is that the population size is variant and there is a saturation effect for population growth. Further, mathematical analysis for epidemic models from this demographic structure may be much easier. But this seems reasonable only when population size is approximately a constant or there is recruitment from outside; (3) The population growth is simulated by the logistic equation. A typical differentiation of the logistic growth into birth and death process is given by B(N) = b - arNjK and D(N) = d + (1a)r N j K with r = b - d (see Gao and Hethcote [18]).
(4) B(N) = be-aN,D(N) = /-1 with a > O,b > 0,/-1 > 0. This means that population grows according to the Ricker law [13]; ~,D(N) = /-1 with p > O,q > O,m > 0,/-1 > 0. This type of vital dynamics was proposed by Mackey and Glass [34] and used by Jin and Wang [24];
(5) B(N) =
= ~ + L,D(N) = /-1 with A > O,L > 0,/-1 > 0. This structure represents a constant immigration rate A together with a linear birth term LN, which is used by Cooke, van den Driessche and Zou [13], Wang and Zhao [45, 46, 47].
(6) B(N)
On the basis of the population dynamics (2.2) and the standard incidence, Hethcote and van den Driessche [22] proposed the following SIS
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292
epidemic model:
l
(t) = 'x(l- ~\tl))l(t) - ,Xa(l - ~~t-=-~))l(t - w) - (d + f)l(t),
N' (t)
=
(2.3)
A - dN(t) - EI(t),
where w is the infectious period and a = exp( - (d + f)W), which gives the survival fraction of infectives through the infection period. Beretta and Kuang [2] proposed the following model for bacteriophage infection:
~~
=
as(t)(l- S(t)
~ l(t)) _ KS(t)P(t),
dl = KS(t)P(t) _ Ke-Jl-i TS(t - T)l(t - T) - /-Lil(t), dt dP = (3 - KS(t)P(t) dt
(2.4)
+ bKe-Jl-iTS(t - T)l(t - T) - /-LpP(t),
where S is the density of susceptible bacteria, I is the density of infected bacteria, P is the density of viruses or bacteriophages, and T is the survival fraction of infected bacteria through the infection period. In this model, only susceptible bacteria S are capable of reproducing by cellular division according to the logistic growth, whereas the infected bacteria, under the genetic control of virulent phages, replicate phages inside themselves up to the death by lysis after a infection period. However the infected bacteria I still compete with susceptible bacteria S for common resources. Cooke and van den Driessche [9] studied an SEl RS model with two time delays: latency period and the period of temporary immunity:
dS dt
=
bN( ) _ bS( ) _ 'xS(t)l(t) t t N(t)
l( _ ) -bT
+ "( t T e
E(t) = it 'xS(u)l(u) e-b(t-U)du t-w N(u) , dl dt
=
'xS(t - w)l(t - w) _ (b )l( ) N(t-w) +"( t,
R(t) =
,
(2.5)
l~T "(l(u)e-b(t-U)du,
where w is the incubation period, T is the period of temporary immunity, b is the per capita birth rate and the per capita death rate, ,X is the valid contact rate. In the study of HIV models, it is interesting to introducing distributive delays. Culshaw, Ruan and Webb in [14J incorporated continuous
Epidemic Models with Time Delays
293
delays into a model of ceU-to-cell spread of HIV-1:
~~
=
reG(t)
dI dt = kr I
jt
-00
(1 - G(t~:I(t)) - krG(t)I(t), (2.6)
G(u)J(u)F(t - u)du - f.LrI(t),
where G(t) represent the concentration of healthy cells, J(t) is the concentration of infected cells, re is the effective reproductive rate of healthy cells, GM is the effective carrying capacity of the system, kr represents the infection of healthy cells by the infected cells in a well-mixed system k~/kr is the fraction of cells surviving the incubation period, f.LI is th~ death rate of the infected cells, and F is the kernel function. In paper [44], another time delay that mimics the density regulation effects of cells is also incorporated:
4ft- = reC(t)(l - f~oo f(t-~~(s)+I(s)) )ds { dft = k~ J~oo G(u)I(u)F(t - u)du - f.LrI(t),
krG(t)J(t), (2.7)
where f is the kernel function. If a population can be split into juvenile group and adult group, and an epidemic disease propagates only in the adult community, then a maturation delay can be introduced into the epidemic model. In [13], Cooke, van den Driessche and Zou investigated the following model:
~~ = dN dt
=
)..(N(t) - J(t)) ~~l)
-
(d + E + 'Y)J(t) ,
B(N(t _ T))N(t _ T)e- d1T
-
dN(t) - El(t),
where S is the density of susceptible adults, J is the density of infected adults, N = S + J, T is the maturation time of juveniles, d 1 is the per capita death rate of juveniles, A is the valid contact rate, and E is the disease-induced death rate. Summarizing above discussions, we have models that can be split into two types: one with coefficients independent of time delays (type I), the other one with delay-dependent coefficients (type II). The stability analysis of models is distinguished according to these types.
3
Analysis of local stability
The basic approach for the local stability analysis of a delayed system is to linearize it at an equilibrium and then consider the characteristic
Wendi Wang
294
equation of the linearized system. For a linear system (3.1) where L is a linear operator from C([ -T, 0], Rn) into Rn. If we write
(3.2) where ry((}) is an n x n matrix whose elements are of bounded variation on [-T,O], then the characteristic equation of (3.1) is
where I is an identity matrix. By the stability theory of delayed differential equations [20, 27, 43], we have Theorem 3.1. The equilibrium is stable if all characteristic roots of (3.3) have negative real real parts, and is unstable if there is one characteristic roots has the positive real part. If there is only one delay, the characteristic equation takes the form: ~(>') := P(>.)
where
+ Q(>.)e- AT
n
P(>.)
= 2:::: ak>.k, k=O
=
(3.4)
0,
m
Q(>.)
= 2:::: bk>.k,
(3.5)
k=O
where ak and bj are constants. For a system of type I, ak and bj are independent of the delay, while some of them are the functions of the delay for the system of type II.
3.1
Models with constant coefficients
Typical characteristic equations of type I systems in epidemiology are
(>. + a) + be-AT = 0, (>.2 + al>' + ao) + (b l >' + bo)e- AT
=
O.
(3.6) (3.7)
The basic approach of stability analysis for such equations is the Dsubdivision method (see, for example, the book of Kolmanovskii and Myshkis [26]). For a fixed delay T, we consider the equation ~(iy) = 0 with a variable y. As y varies in IR = (-00,00), the graphs of the equation split the parameter space of ai and bj into a number of domains
Epidemic Models with Time Delays
295
in each of which, the stability is unchanged. The exact stability information can be drawn with the aid of the direction of the real parts of characteristic roots as a point crosses the curves. We illustrate this by (3.6). In the a-b plane, the curves of A(iy) = 0 are given by
a = -ycot(yr),
b = y/ sin(yr)
(3.8)
and a
+ b = o.
(3.9)
These curves divide the plane into two parts: domain U that lies in the left of (3.9) and above (3.8), and the domain V that lies in the right of (3.9) and below (3.8). U is an unstable domain because a point in the negative a-axis corresponds to an characteristic root A = -a > O. Similarly, V is a stable region. Furthermore, since the curve of (3.8) lies above the line of b = a with a ~ 0 for any r > 0, the domain of Ibl < a is absolutely stable, i.e., a system with the characteristic equation (3.8) is stable irrespective of the length of the delay. Advanced techniques for the applications of the D-subdivision method can be found in Ma's book [32J.
b
a
Figure 3.1: The solid curve is the graph of (3.8), the dashed line is the graph of b = a with a ~ 0 and the solid line is the graph of a + b = o. Generally speaking, an equilibrium of epidemic models is stable when r = O. Basically, this means that all the characteristic roots have negative real parts. As the time delay r increases, these roots continuously move in the complex plane. The equilibrium becomes unstable if one root crosses the pure imaginary root from the left to the right. If this root stays in the right of the pure imaginary root afterwards, the equilibrium remains unstable. if it goes back to the left and all the other roots always stay in the left, the equilibrium becomes stable again. This phe-
Wendi Wang
296
nomenon of stability transitions is called as stability switches. Multiple stability switches can occur in epidemic models. For (3.4) with ak and bj independent of the delay, Cooke and van den Driessche [10] proved that there may be only finite stability switches and eventually unstable state may occur as the delay increases. Criteria for stability switches are given in [27, page 83]: Theorem 3.2. Suppose that P(>.) and Q(>.) are analytic functions in Re>. > 0 and satisfy the following conditions:
(i) P(>.) and Q(>.) have no common imaginary root; (ii) P(-iy) = P(iy),Q(-iy) = Q(iy) for real y; (iii) P(O)
+ Q(O) i- 0;
(iv) limsup{IQ(>')/ P(>')I : 1>'1 ~ := IP(iy)1 2
(v) F(y) of real zeros.
00,
Re>. ?: O}
< 1,-
-IQ(iy)12 for real y has at most a finite number
Then the following statements are true:
(a) If F(y) = 0 has no positive roots, then no stability switch occur; (b) If F(y) = 0 has at least one positive root and each of them is simple, then a finite number of stability switches may occur and eventually, the equation becomes unstable.
3.2
Models with delay-dependent coefficients
Stability analysis for models with delay-dependent coefficients may be much more complicated because components of an endemic equilibrium could be the functions of the delay, and therefore, the endemic equilibrium does not exist when the delay is larger. This means that the delay should be confined to an interval to discuss its stability analysis. As a consequence, stability switches of models with delay-dependent coefficients may be quite different from those of models with delayindependent coefficients. Beretta and Kuang [3] proposed a geometric procedure to determine stability switches for models with delay-independent coefficients, illustrated that an equilibrium can be eventually stable after transitions of several stability switches. We present the method by the following characteristic equation: (3.10)
Here, we assume that the coefficients a, band c are real smooth functions of T, have continuous derivatives in T and satisfy
(3.11)
Epidemic Models with Time Delays
297
°
(3.11) means that>. = is not a root of (3.10). Then, we want to find conditions such that pure imaginary roots occur. Set>. = ±iw with w > 0. Without loss of generality, we consider only>. = iw. Substituting it into (3.11), we have
P(iw, r)
+ Q(iw, r)e-
iWT
= 0,
(3.12)
where
P(iw, r) = b(r) + iwa(r),
Q(iw, r) = c(r).
It follows from (3.12) that
b(r) cos(wr) = - c(r) ,
. (
) _ w(r)a(r) c(r) .
(3.13)
+ b2 -
(3.14)
sm wr -
As a consequence, we obtain
F(w, r)
:=
IP(iw, rW - IQ(iw, r)1
=
w2 a2
c2 = 0.
Therefore, the necessary conditions for the occurrence of pure imaginary roots are Ic(r)1 > Ib(r)l,
a(r)=;fO, {
(3.15)
b2
2
w(r) = (~)1/2. 2 a
Let (3.15) hold. In order to obtain sufficient conditions for the occurrence of pure imaginary roots, we set O(r) = w(r)r. Then O(r) satisfies cos(O(r))
b(r)
= - c(r) ,
. (O( )) r
sm
=
w(r)a(r) c(r).
(3.16)
Evidently, if O(r) is a solution of (3.16), so is O(r) +2mr. Hence, we have
rw(r) = O(r) + 2mr, which leads to r=
If
rn :=
O(r) + 2mr w(r) O(r) + 2mr w(r)
and (3.15) holds, we obtain the sufficient condition for the occurrence of pure imaginary roots: (3.17)
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298
By Beretta and Kuang [3], the direction of the pure imaginary roots crosses the imaginary axis as T increases can be computed by the formula:
dReA
sgn ~ I.>-=iw = sgnR(T), where
R(T)
:=
a2(T)w(T)W' (T)(a(T)b(T)
(3.18)
+ C2T)
+w 2(T)a 2(T)(a' (T)b(T) - a(T)b' (T)
(3.19)
+ C2(T)).
Example 1. We consider an extended Bass model with an evaluation stage:
dA(t) dt
-- =
p('y + AA(t - T))(O - A(t - T)) - aA(t),
(3.20)
where A(t) is the number of adopters of a new product, A is the valid contact rate of adopters of the product with potential adopters, 0 is the capacity of potential adopters of the product, T is the average time for an individual to evaluate the product, a = 8 + ZJ and p = e-(Hp)T. Here, 8 is the birth rate and the death rate of a population, "( is the intensity of advertisement of product, ZJ is the discontinuance rate of adopters of the product, and p is the rate that individuals leave the evaluation class since they have decided not to buy the product. See paper [40] for more details. This model admits a unique positive equilibrium
A*
= -
P"( + PA 0 - a
+ J (p"( -
PA 0
+ a)2 + 4 p2 A "( 0
2PA
.
Its characteristic equation is:
where
q=p(AO-2AA*-"(). By (3.15), we obtain
F(W,T) := w 2 If a 2
+ a 2 -l.
< q2, this equation admits a positive solution (3.21 )
Thus, (3.22)
Epidemic Models with Time Delays
299
Assume that 8(T) E (0,27l") satisfies (3.22). Then we have
Tn =
8(T) +2n7l" W(T) ,
Sn(T) = T - Tn·
(3.23)
Since it is difficult to obtain the zeros of Sn(T) analytically, we fix the parameters and use numerical calculations. We take C = 20, a = 0.2,.A = 0.1, I = O.l,p = e- 0 .2T . Then numerical calculations show that W(T) exists when 0 :( T :( 6.5047 and
W(T) = p' (1.9 _ 0.1 19.0 P - 2.0 + J 44~.0 p' - 76.0 p + 4.0) _ 0.04 2
}1/2
{
Furthermore, q(T)
< 0 when T varies in the interval. Thus, we define 8(T) = 7l" + arctan(- W(T)), a
_ 8(T) +2n7l" Tn (T) W(T) . By numerical calculations, we see that only So (T) = 0 has two roots TOl = 1.50164 and T02 = 4.41394. Furthermore, by (3.19) we obtain R(Tod = 1 and R( T02) = -1. Hence, the equilibrium is stable when 0 :( T < 1.50164 and 4.41394 < T, is unstable if 1.50164 < T < 4.41394. Further analysis
Figure 3.2: two roots of SO(T)
Wendi Wang
300
shows that there is a stable periodic solution in 1.50164 < r < 4.41394 (see [40]). Li and Ma extended the geometric method of paper [3] in ([28, 29]). We consider D()", r) := P()", r) + Q().., r)e- AT = 0, (3.24) where
P()", T)
=)..2
+ a(T) .. + e(T), Q().., T) = beT)>' + d(T)
with T ~ O. We make the following assumptions: (AI) a( T), b( r), e( T) and d( r) are continuously differentiable in R+o
=
[0, +(0);
i= 0 for any T E R+o; P(iy, T) + Q(iy, T) i= 0 for any T E R+o;
(A2) e( T) + d( r) (A3)
(A4) Any pure imaginary root of (3.24) is simple; (A5) All roots of (3.24) have negative real parts when
T
= O.
(AI) ensures that the solutions of (3.24) are continuous in T. (A2) means that>. = 0 is not a root of (3.24). (A3) implies that P(>., T) = 0 and Q(>', T) = 0 do not have a common pure imaginary root. (A4) guarantees the transection of pure imaginary roots with the imaginary axis. (A5) indicates that (3.24) is stable when T = O. We begin from conditions for the occurrence of pure imaginary roots of (3.24). Substituting).. = iy into (3.24) and separating its real part and imaginary part, we obtain
b(T)ysinyr + d(r) COSyT = -[e(r) _ y2], { d(T) sin yT - b(T)y cos yT = a(T)y,
(3.25)
It follows that
F(y,T) where
:=
y4 - hCT)y2
+ hCr) = 0,
h(T) = b2(T) + 2c(r) - a2(r), h(T) = C2(T) - d2(T), h(T) = N(r) - 412(T).
To determine positive solutions of (3.26) easily, we assume
C3.26)
Epidemic Models with Time Delays (A6) Each of !i(T) any T E R+.
= 0 (j = 1,2,3)
When h(T) > 0, if F(y,T)
y! =
~ [!leT) + Jh(T)]
has at most one positive root for
= 0 has roots y~ '
301
and y~, we have
y~ = ~ [!leT) -
Jh(T)].
(3.27)
Assume that yeT) is a positive root of (3.26). In order to make iY(T) a pure imaginary root of (3.24), Y(7) must satisfy (3.25). From (3.25), we have
. _ -b(T)Y(C(T) - y2) + a(T)d(T)Y smYT b2(T)y2 + d2(T) , d(T)(C(T) _ y2) + a(T)b(T)y2 { COSyT = b2(T)y2 + d2(T) .
(3.28)
If T = 7* satisfies (3.28), then iY(T*) is a pure imaginary root of (3.24). Now, we replace y(T)T in (3.28) by OCT) to obtain
. {)( ) _ -b(T)y(C(T) - y2) + a(T)d(T)y ~ ( ) T b2(T)y2 + d2(T) - 'P y, T ,
sm
{
cos O( T)
= _
d(T)(C(T) - y2) + a(T)b(T)y2 ~ .1,( ) b2(T)y2+d2(T) 'l-'Y,T.
(3.29)
(3.29) determine a function: 'P arctan -;j' 7r
if sin{} > 0, cos{} > 0; if sin{}
= 1, cosO = 0;
'P 7r + arctan -;j ,
ifcosO
< 0;
37r 2 '
ifsinO
=
2'
{)( T) =
'P 27r + arctan -;j'
(3.30)
-1, cosO
ifsin{} < 0, cos{}
= 0;
> O.
If T = T* satisfies (3.28), it follows from (3.28) and (3.29) that there exists kENo = {O, 1,2,3, ... } such that
y(T*)T*
=
O(T*)
+ 2k7r.
(3.31 )
Now, we define
8(T) ~ y(T)T - {}(T). 27r For convenience, we assume also
(3.32)
Wendi Wang
302
(A7) There exist at most finite solutions in equation S(7) = k for any kENo" and every solution of it is simple. Let (AI )-(A 7) hold. By the above discussions, we see that the necessary and sufficient condition that ±iy(7*) are roots of (3.24) is that 7* is a root of y(7)7-0(7) = k that is, S(7) = k, 21f
'
where 0(7) is defined by (3.30), and y(7)(7 E (0:, fJ)) is a positive root of (3.26). Now, we consider the direction that a pair of pure imaginary roots crosses the imaginary axis as 7 varies [28]. Theorem 3.3. Let (Al)-(A7) hold. If 7* E (0:, fJ) such that A = iY(7*) is a root of (3.24), the direction of A( 7), when 7 increases in a neighborhood of 7 = 7*, depends upon
v = sgn {tl (7*)} sgn { d~~7) IT=TJ . (1) If V = 1, when 7 passes through 7 = 7*, then A = A(7) passes through the imaginary axis from the left to the right;
(2) If V = -1, when 7 passes through 7 = 7*, then A = A( 7) passes through the imaginary axis from the right to the left.
The following two theorems [28, 29] give criteria for the stability and ultimate stability of delay-dependent equations. Theorem 3.4. Let (Al)-(A7) hold. If there is no positive solution in equation F(y,7) = 0 for any 7 E (0,00), or there is no positive solution in equation S(7) = k for any kENo, then (3.24) is stable for any
7 E [0, +00). Theorem 3.5. Let (Al)-(A7) hold and (3.24) have pure imaginary roots for some 7 E (0, +00). Suppose that the equation
has a unique positive root y+(7). Then we have
(i) (3.24) is ultimately stable if Y+(7) is defined on a finite interval; (ii) Assume that Y+(7) is defined in an infinite interval. Then (3.24) is ultimately stable ijlimsupS+(7) if lim sup S+(T)
T---?+oo
> O.
T---?+oo
< 0, and is ultimately unstable
Epidemic Models with Time Delays
Example 2.
Let us consider an SElS Model [28]:
dS dt = (b - d)S(t) - (3S(t)I(t) dE
dt
=
303
+ I'I(t),
(3S(t)I(t) - (3S(t - T)I(t - T)e- dy - dE(t),
(3.33)
dI dy dt = {3S(t - T)I(t - T)e- - (d + CY + I')I(t), where b is the birth rate of the population, d is the death rate of the population, {3 is the disease transmission coefficient, T is the incubation period, CY is the disease-induced death rate and I' is the recovery rate of infected individuals. When 9 = b - d > 0 and w := d + a + I' > I'e- dY , (3.33) has a unique endemic equilibrium P*. Its characteristic equation is:
(oX + 9 + f(T))(oX + w) - w(oX + g(l(T) - l))e->'Y = 0, where
(3.34)
I'e- dY f(T) = w-I'e- dY ·
It is easy to see that P* is stable when
F(y, T)
:=
T
= O. Set
y4 + g2 f2(T)y2 - w2g2[1 - 2f(T)]
(3.35)
= O.
When w ~ 31', (3.35) has a unique positive root y(T), which is defined in T E (0, +(0). If I' < w < 31', (3.35) has a unique positive root y(T), defined in T E ((In ~)/d, +(0). Thus, the existence interval for Y(T) is infinite in each case. Since limy--->oo f(T) = 0, we have limy--->oo Y(T) = vwg. Note that
It follows that lim S(T)
=
y--->oo
+00.
Consequently, Theorem 3.5 shows that P* is ultimately unstable. For another type of characteristic equation:
D(oX, T)
:=
P(oX, T)
+ Q(oX, T)e->'y + R(oX, T)e- 2>.y
where n
P(oX,T) = I>k(T)oX k , k=O m
k=O I
R(oX,T)
=
LTk(T)oXk, k=O
n > max{m,l},
= 0,
304
Wendi Wang
which is important in the analysis of epidemic models, Li and Ma [29] present excellent criteria for stability and stability switches.
4
Liapunov direct methods
The basic method for stability analysis of an equilibrium is to consider its characteristic roots. The advantage of this approach is that sharp conditions may be found in some cases. However, it is difficulty to apply when the number of equations is more than two or when there are several delays. However, Liapunov direct method could be a good option in these cases. We introduce the Liapunov functional method for the case of finite delay. Let G([-T, 0], Rn) denote the set of continuous functions mapping [-T, 0] into Rn. With linear operations and the norm defined by Ilepll = max lep(e)1 for any ep E G([-T, 0], R n), G([-T, 0], Rn) becomes (lE[-r,O]
a Banach space. If x(t) is continuous in [to - T, to + A) with A > 0, for to ~ t < A, we define Xt = x(t + e), e E [-T, 0]. Then a system of an autonomous delay-differential equations can be written as (4.1) Let R+ = [0,00). Suppose that x = 0 is an equilibrium of (4.1). Then we have the following fundamental stability theorems [20, 27]. Theorem 4.1. Suppose f : G -+ R n takes bounded sets of G into bounded sets of Rn, and WI, W2 : R+ -+ R+ are continuous nondecreasing functions, satisfying WI (0) = W2 (0) = 0, lim WI (r) = +00 and r-+oo
WI(r) > 0'W2(r) > 0 for r > O. If there is a continuous functional V : G -+ R such that
Then x = 0 is stable, and every solution is bounded. If we also have w2(r) > 0 for r > 0, then x = 0 is globally stable. We say V : G -+ R is a Liapunov function on a set G in G if V is continuous on G, the closure of G, and V ~ O. Let
S = {¢ E G: V(¢) = O}, M = largest invariant set in S which is invariant with respect to (4.1). Theorem 4.2. If V is a Liapunov function on G and Xt(¢) is a bounded solution that remains in G, then Xt(¢) tends to M as t -+ 00.
Epidemic Models with Time Delays
305
We now apply the Liapunov method to consider the stability of an
SEIRS model with two delays [38]. s'
= b - As(t)i(t) + j3i(t - r) - bs(t)
i'
= Aas(t - w)i(t - w) - b + b)i(t),
(4.2)
where s is the fraction of susceptible individuals, i is the fraction of infectious individuals, b is the birth rate and death rate of the population, A is the average number of adequate contacts of an infectious individual per unit time, w is the latent period of the disease, r is the immune period of the population, 'Y is the recovery rate of infectious individuals, a = exp(-bw) and (3 = 'Ye-bT. (4.2) is a simplification of (2.5). Indeed, the population N in (2.5) is a constant. If e(t) is the fraction of exposed individuals and ret) is the fraction of recovered individuals, wee obtain (4.2) and set) + e(t) + i(t) + ret) = 1. The model has two time delays and delay-dependent coefficients. Set T = max{r,w}. Due to our background, we will consider (4.2) in the set
It is easy to show that D is positively invariant for (4.2). It is clear that (1,0) is the disease free equilibrium of (4.2).
Theorem 4.3. If Ro (1,0) is globally stable.
=
Aa/ ('Y
+ b)
~ 1,
the disease free equilibrium
Proof. Note that the second equation can be rewritten as
i' = Aas(t)i(t) - b + b)i(t) + Aa[s(t -
w)i(t - w) - s(t)i(t)]
t
=
i(t) [Aas(t) -
b + b)]- Aa
!J
(s(u)i(u))du.
t-w
If
Xt = (s(t + e), e(t + e), i(t + e), ret + e)), e E [-T,O],
we define
t
Va(Xt) = i(t)
+ Aa
J
s(e)i(e)de.
t-w
The derivative of Va along solutions is V~ (Xt)
= i(t) [Aas(t) - b + b)] = b + b)i(t) [Ros(t) - 1].
Wendi Wang
306
Let S = {
+ b)(Ro - l)i(t) If Ro = 1, we have
V; (Xt) :::;; (r It follows that M
= (1,0).
V; (Xt) = (r
:::;;
o.
+ b)i(t) [s(t) - 1].
It follows that M = (1,0). We conclude from Theorem 4.1 and Theorem 4.2 that (1,0) is globally stable. D Let us now apply the Liapunov functional method to a model of the vector-transmitted disease. Malaria and West Nile disease are among the list of vector-transmitted diseases. Ronald Ross got the Nobel prize in physiology and medicine in 1902 for his great work of mathematical modeling of malaria. An SIR model was proposed by Cooke [8] for epidemics which are spread in a human population via a vector (such as mosquitos). It is assumed that when a susceptible vector is infected by an infectious person, there is a time T > 0 during which the infectious agents develop in the vector and it is only after that time that the infected vector becomes itself infectious. If Set) is the number of human susceptible population and I(t) is the number of human infective population, the infection force at time t is f3S(t)I(t - T). Beretta and Takeuchi [4, 5] improved the infection force by distributed delays. Jin and Ma [25] considered the case where there are direct infectious contacts among individuals in human population and there are vectortransmitted infections. The model is:
S'(t)
= -f31S(t) Joh f(s)I(t - s)ds - f32S(t)I(t) - p,S(t) + b,
I'(t)
= f3 1S(t) Joh f(s)I(t - s)ds + f3 2S(t)I(t) - (p, + A + c)I(t),
R'(t)
= M(t) - p,R(t).
{
(4.3) The term f3 1S(t) f(s)I(t - s)ds describes the disease transmission rate because of vectors, while f3 2S(t)I(t) gives the transmission rate from direct contacts between susceptible individuals and infectious individuals h in the human population. We assume Jo f(T)dT = l. When b 1 -(f31 + (32) + A > 1,
J:
p,
P,
+c
(4.3) admits a unique endemic equilibrium
E
=
+
(S* I* R*) "
=
(p, + A + c (b - p,S*) A(b - p,S*) ) f31 + f32 '(f31 + (32)S* ' p,(f31 + (32)S* .
Epidemic Models with Time Delays
307
Although the model looks complicated, we can show that the equilibrium is stable whenever it exists [25]. Theorem 4.4. E+ is locally asymptotically stable when it exists.
Proof. By means of transformations UI = S - S*, U2 = I - 1* and U3 = R - R*, we obtain
U~ = -/3I(UI + S*) Jo f(S)U2(t - s)ds - /3I(UI + S*)1* h
-/32(UI + S*)(U2 + 1*) - P,UI - p,S* + b, u~
=
h
/3I(UI + S*) Jo f(S)U2(t - s)ds + /3I(UI + S*)I* +/32(UI + S*)(U2 + 1*) - (p, + >. + C)(U2 + 1*),
u~
= >'(U2 + 1*) - P,(U3 + R*).
Its linearized system is
ui
=
h
-((/31 + (32)1* + P,dUI - /3IS* Jo f(S)U2(t - s)ds - /32 S *U2,
U~ = (/31 + (32)1*UI + /3IS* Jo f(S)U2(t - s)ds + /32S*U2 - (p, + >. + C)U2, h
{
u3 = >'U2 - P,U3' (4.4) We consider a Liapunov functional defined by
1 1 2 + -2W3U3 1 2 1 * V(Ut) = -WI(UI + U2) 2 + -U + -/3IS 2 2 2 2 where Wi > 0 (i Note that
lh it f(s)
0
t-s
2 u2(v)dvds,
= 1,2,3) are to be determined later. 1 2 1 2 1 2 V(ut) ~ "2WI(UI + U2) + "2U2 + "2W3U3'
Calculating the derivative of V along the solutions of (4.4), we obtain
V'(Ut) = WI(UI + U2)[-p,UI - (>. + P, + C)U2] + (/31 + (32)1*UIU2
-(>. + P, + C- /32S*)U~ + /3IS*U2 Jo f(S)U2(t - s)ds h
+W3U3U; + ~/3IS*U~ - ~/3IS* Jo f(s)u~(t - s)ds. h
By simplifications, we obtain
V' (Ut) = -wIP,uI - [WI (>. + P, + c) + (>. + p, + C- /32S*)]U~ +[-WIP, - WI (>. + P, + c) + (/31 + (32)I*]UIU2
+ W3>'U2 U3 - W3p,U~
+/31S*U2 Joh f(S)U2(t - s)ds + ~/3IS*U~ - ~/3IS* J: f(s)u~(t - s)ds. (4.5)
Wendi Wang
308 Choose
Wl
> 0 such that
Note that For any U2, we have
/3S*U2
t
io
U2(t - s)ds
~ ~/3S*u~ + ~/3S* t 2 2 io
f(s)u 2(t - s)ds.
It follows from (4.5) that
V'(ut} ~ -wlp,ui -
Wl().
+ P, + c)u~ -
W3P,U~
+ W3).U2 U3·
(4.6)
If we choose W3 such that
i.e.,
4(). + p, + C)p,(/31 ).2().
+ /32)1* + 2p, + c)
it follows that the right hand side of (4.6) is negative definite. Conse0 quently, the equilibrium is stable. For the application of Liapunov functional method to the global stability of an endemic equilibrium, readers can refer to [41].
5
Conditions of disease persistence
For epidemic models, it is important to know conditions for the stability of endemic equilibria. That the endemic equilibrium is stable means the spread of infectious diseases. However, it could be very hard to perform mathematical analysis for endemic equilibria for many models. In such cases, we turn to the persistence theory of dynamical systems. A good way for this is to analyze the limit set of flows on the diseasefree subspace to see if it repels positive solutions (see paper [36] for the general theory, and papers [47, 50] for the applications). Another alternative is to adopt persistence functionals [23] Let us illustrate the second one by SE1S model (4.2) [38].
Ro = ).a/b + b) > 1. Then there is a positive constant E such that each positive solution (s(t),i(t)) of (4.2) satisfies i(t) ;? E ift is large. Theorem 5.1. Suppose
Epidemic Models with Time Delays
309
Proof. Let us consider a positive solution (s(t), i(t)) of (4.2). According to this solution, we define t
V1 (t)
=
i(t)
J
+ AC¥
s(())i(())d().
t-w
Then we have
v; (t) = i(t) [AC¥S(t) -
b + b)] = b + b)i(t)(Ros(t) -
1).
(5.1)
J
Since Ro > 1, we have h := ~(1- o ) > o. Claim: For any to > 0, it is impossible that i(t) :::;; h/2 for all t ;? to. Suppose the contrary. Then there is a to > 0 such that i(t) :::;; h/2 for all t ;? to. The first equation of (4.2) is:
s' = b - As(t)i(t)
+ (3i(t -
Then, for t ;? to,
s' (t) > b - (Ah/2
T) - bs(t).
+ b)s(t)
which implies
J t
s(t) > e-(AI1 /2+b)(t-t o)[s(to)
+ b e(Ah/2+b)(O-to)d()] (5.2)
to
> where 0
b (1 _ e-(AIl/2+b)(t-to)) Ah/2+b '
< s(to) is used. Since AI)2+b =
2
Ro +1' we have
s(t) > _2_(1 _ e-(Ah/2+b)(t-t o)). Ro + 1
(5.3)
Choose T1 > 0 such that
~(1 - ~) = e-(Ah/2+bm. 4 Ro
(5.4)
Then (5.3) implies
3Ro + 1 s(t) > 2Ro(Ro + 1) It is easy to see
R>
Jo.
6.
=
R,
£ or t
>t T 7 0 + 1·
(5.5)
Then, by (5.1) we have
v; (t) > b + b)i(t)(RRo -
1),
for t;? to
+ T1 ·
(5.6)
Wendi Wang
310 Set
i=
min i(to
liE[-w,O]
+ TI + w + B).
Statement: i( t) ): i for all t ): to + TI. Suppose the contrary. Then there is a T2 ): 0 such that
i(t) ): i, for to + TI ~ t i(to + TI + w + T2) = i,
~
to
+ TI + w + T2,
i' (to + TI + W + T2) ~ o. However, the second equation of the model is:
i' = Aas(t (5.5) implies that for t
=
to
w)i(t - w) - (-y + b)i(t).
+ TI + w + T 2, we have
i' (t) ): [Aas(t - w) -
(-y + b)]i. > (-y + b) [RoR - 1]i. > O.
This is a contradiction. Thus, i(t) ): i for all t ): to quence, (5.6) leads to
v; (t) > (-y + b)i.(RoR -
1)
for t): to
+ TI .
As a conse-
+ TI,
which implies that as t -+ 00, VI (t) -+ 00. This contradicts VI (t) ~ 1 + Aaw. Hence, the claim is proved. By the claim, we are left to consider two possibilities. First, i(t) ): h/2 for all large t. Secondly, i(t) oscillates about h/2 for all large t. Define
(5.7) We hope to show that i(t) ): 12 for all large t. The conclusion is evident in the first case. For the second case, let hand t2 satisfy
i(tI) = i(t2) = h/2
i(t) < h/2
for tl < t < t2'
If t2 - tl ~ TI + w, since i' (t) > -(-y + b)i(t) and i(tI) = h/2, it is obvious that i(t) ): 12 for h < t < t2' If t2 - tl ): TI + w, by the second equation of the model i = Aas(t - w)i(t - w) - (-y
we obtain i'
> -(-y + b)i(t).
+ b)i(t),
Epidemic Models with Time Delays
311
It leads to i(t) ~ 12 for t E [h,tl +Tl +wJ. For tl +Tl +w::;;; t::;;; t2, we have
Set i*
= 8E[-w,O] min i(h + Tl + w + e)
~ [2.
Proceeding exactly as the proof for above claim, we see that i(t) ~ i* ~ ~ 12 for t E [h, t2J. Since this kind of interval [h, t2J is chosen in an arbitrary way (we only need tl and t2 are large), we conclude that i(t) ~ h for all large t in the second case. In view of our above discussions, the choices of Tl and 12 are independent of the positive solution, we have actually proved that any positive solution of (4.2) satisfies i(t) ~ 12 for all large t. The proof is complete. D
h for tl + Tl + w ::;;; t ::;;; t 2. Consequently, i(t)
6
Summary
In this paper, we have presented the approaches of mathematical modeling for epidemic models with time delays. It has been illustrated that the incubation period, infection period and immune-lasting period could be represented by delays. We have shown that the stability of some equations can be analyzed by the D-subdivision method. Then criteria to test stability switches for systems of delay-independent coefficients and for systems of delay-dependent coefficients are introduced, respectively. The Liapunov functional method is also given to show its power when there are more than two equations or more than two delays. Finally, we have shown the way to prove epidemic disease is persistent under suitable conditions. In all, our aim is to present basic mathematical techniques for analysis of epidemic models with time delays.
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A Simulation Approach to Analysis of Antiviral Stockpile Sizes for Influenza Pandemic* Shenghai Zhang Centre for Infectious Disease Prevention and Control Public Health Agency of Canada E-mail: [email protected]
Abstract The model described here is developed for simulating an influenza pandemic. It intends to explain the manner in which the pandemic develops in a specific community. It shows how the longitudinal cases occurring pattern is built from inception and how this pattern is affected by various sizes of stockpiles of antiviral agents combining some preventing strategies. It assumes that potential inter-household and household-school contacts play very important role in the disease transmission. It uses statistical data about characteristics of households in the community and the parameters of disease transmission to generate repeated random trials of possible outcomes following the introduction of infective individuals into that community, then to track the statistically determined pathway of the infection and average the outcome results. It aims to describe the dynamics of pandemic itself, then the affection of antiviral treatment based on the available sizes of stockpile of the drugs can be obtained.
1
Introduction
Avian flu is unprecedented in its scope as an animal disease and it poses a greater challenge to the world than any previous infectious disease [1]. The virus has spread to birds in many countries in Africa, Asia, Europe *The author is grateful to Dr. Ping Van and his colleagues for their valuable suggestions. This is only a personal point of view and it does not represent any official views of the organization. The research was done before the middle of May, 2006 and presented in the Canada-China Workshop of Infectious Disease Modeling" at Xi'an, China.
316
Shenghai Zhang
and the Middle East [2]. The cases of human infection with the H5N1 avian influenza virus are increasing. The transmissibility of the virus among humans could lead to a global influenza pandemic, although the effect of the transmissibility of the virus, either from birds to humans or from one person to another, is not fully understood. At the beginning of a pandemic, the possible public health measures we have so far is to give antiviral medicine to those infected with flu, to quarantine areas and to isolate people. Even if a pandemic cannot be stopped, it is said that such measures can buy time for health authorities to improve their response strategies and stave off the disease until a pandemic vaccine can be produced. Therefore, one of the pressing public health questions is whether and how we stockpile antiviral drugs which can be used for treatment in the early phases of a flu pandemic. A discrete-time stochastic simulation model of influenza spread is used to estimate the distribution of infective people, the number of deaths and the duration of the pandemic of using the different antiviral stockpile sizes to treat infection. The first case of an influenza pandemic could occur in any of a variety of locations and households. The subsequent spread of the pandemic might be very different according to the availability of anti-flu drugs, for instance, for susceptible population. And the spread of the epidemic might also be different according to the presence or absence of school-age children, for instance, in the households first affected [5]. It is assumed that the schools and households are most important in evolution of the pandemic. The model described here was developed for simulating an influenza pandemic. It intends to explain the manner in which the pandemic develops in a specific community. It shows how the longitudinal cases occurring pattern is built from inception and how this pattern is affected by various sizes of stockpiles of antiviral agents combining some preventing strategies. The information about possible social contacts in the community and the parameters possibly describing transmission of the disease are used to determine the probability that a susceptible person will be infected in a day (also see [6], [7] and [5]). It uses statistical data about characteristics of households in the community and the probabilities of disease transmission to generate repeated random trials of possible outcomes following the introduction of infective individuals into that community, to track the statistically determined pathway of the infection and then to average the outcome results. It aims to describe the dynamics of pandemic itself, then the affection of antiviral treatment based on the available sizes of stockpile of the drugs can be obtained. The paper is organized as follows. In section 2 the network of potential inter-household and household-school contacts by which the disease can be transmitted is described for a community of 3200 people. Assumptions about transmission of the disease and transmission probabilities between contacts in the social network are given in section 3.
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 317 Following the main results summarized in section 4, we will discuss the potential application of the methodology provided in this article and its limitations.
2
Methods
An outbreak of influenza pandemic in 1918 generated more than 400 million cases worldwide and the number of deaths was more than 20 million, based on the estimation by Cunha [4]. The schools as well as households were important in the evolution of the pandemic. So, the simulation will be based on information from data about the size, composition and immune status of households and the details of school classes and other components. Two separate groups of parameters are used: The first group of parameters defines the components involved in community (such as sizes of households and schools) and the structure of their inter-connections. These parameters define the social and related knowledge about the community. This knowledge, then used with a second group of parameters which describe how the infection moves between individuals and between households and the other sites. Motivated by the idea in the paper by Longini et al. [6], a community of population of 3200 people is stochastically generated by the age distribution and approximated household size published by Statistics Canada (see Table 1). To generate typical families for the simulation,
Table 1· Proportion of the population by sex and age group Age group U. .4 5 .. 9 10 .. 14 15 .. 19 20 .. 24 25 .. 29 30 .. 34 35 .. 39 40 . .44 45 . .49 50 .. 54 55 .. 59 60 .. 64 65 .. 69 70 .. 74 75 .. 79 80 .. 84 85 .. 89 90 and over 'lotal
Ganada (%)
rate %)
Female (%)
5~
5.b
b.l
6.0 6.6 6.7 7.0 6.8 7.0 7.5 8.6 8.0 7.0 6.0 4.6 3.7 3.3 2.7 1.9 1.0 0.5 lOU.U
6.2 6.9 6.9 7.2 6.9 7.1 7.7 8.7 8.1 7.0 6.0 4.5 3.6 3.1 2.3 1.5 0.6 0.3 1uu.u
5.8 6.4 6.4 6.7 6.6 6.8 7.4 8.5 8.0 7.0 6.0 4.6 3.8 3.4 3.0 2.3 1.3 0.7 100,0
Note: Population as of July 1, 2004. Source: Statistics Canada, CANSIM, table 051-0001. Last modified: 2005-02-10.
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318
information on household composition in Canada is needed. In the following, a child means a person with age less than 19 years old; a young child is a child who is less than 5 years old. Values of parameters about families structure can be obtained from Canada 2001 Census population data released from Statistics Canada [3]. Among all families, 36.55% of them are ones without any child at home; 63.45% of them are families with one child or more children. The 43.02% of families with at least one child have only one child; 39.30% of them have two children; 17.68% of them are ones with three or more children at home. Among the people who are 65 or older, 25.4% of them are living alone; 48.4% of them are living with their spouses or partners; 26.2% of them are living with their children or living with other arrangements. The proportion of lone parents who are 65 or older among the families which have 65 or older people is 5.43%. The 75.32% of families with at least one child are couple families. Among these couple families, 37.08% of them have only one child; 42.90% of them have two children; and the families with three or more children at home are 20.02%. The 36.55% of all families are couple families without any child at home. The 24.68% of families with one child or children at home are lone-parent families. The 61.14% of lone parent families are ones with one child; 28.30% of them have two children; The percentage of ones with three or more children is 10.56%. These parameters determine the hierarchy of decomposition for households. For example, the households with three people are generated, based on the hierarchy of decomposition shown in the Figure 2.1.
~ g, g-
0
'" '< 0
~
...,o
~ 0 8
8
g-
"@
8
i
@
u. I
.g
e:+
i
@
'"
e:+
Figure 2.1: An example of household decomposition
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 319 Table 2 shows the living arrangement for senior people in Canada. In general, the household sizes as shown in Table 3. Table 2: Living arrangements of seniors aged 65 and over by sex and age group, Canada, 2001 Sex
Living alone
Males Females
(%) 16.0 34.8
Living with spouse or partner (%) 61.4 35.4
Living with children
Living in health care into (%) 4.9 9.2
(%) 13.3 12.1
Other
(%) 4.4 8.4
Table 3: Household size
#
Persons % of households
The community with a population of 3200 is arranged by four neighborhoods with the age structure and the families introduced above. Each neighborhood will have two small day-care centres and a large day-care centre. There are two elementary schools and one high school in the community. The community decomposition is shown in Figure 2.2. The "S", community (3200)
neighbourhood I
/ t \
family···
~ +
1+\
family
neighbourhood 2
neighbourhood 3
neighbourhood 4
/ t \
/ t \ ""tl
family···
family...
family family
~+I
I
I children I I \
I
family
/ t \ ~ +I
I children I \
I
m11\
family
ay care
/
J~ \
[88[8[8 G [8 1------1-1
High School
1--1-------'
Figure 2.2: The structure of the community
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320
"L" and "E" in the figure are "small day-care centre", "large day-care centre" and "Elementary school" , respectively.
3
Model
As an infectious disease, influenza follows SIR time line shown in Figure 3.1. It has incubation period, latent period, infectious period and recovery period. The onsets of symptoms are fever, respiratory symptoms, nasal discharges, cough, headache and sore throat. However, there is a time period between the start of infection and the onset of symptoms, which is incubation period. The length of incubation period is roughly the same as the time between being infected and becoming infectious that is, capable of passing on the infection to others, which is latent period. Then, infectious period follows. Once the infectious period ends, the recovery period starts, during which the infected individual is no more transmitting the infection to others. Based on the research done by Stiver [11], the influenza antiviral agents can modify the severity of illness (relative reduction in influenza complication rate) about 50%, and reduce the duration of illness about 1.5 to 2.5 days. The reduction of time to resume normal activity is 1.5 day to 3 days. However, the treatment is most effective when given within 48 hours after the onset of illness; the earlier the antiviral is started after the onset of symptoms of influenza-like illness, the better the treatments. The work here aims at statements about the distribution of infective people, the number of deaths and the duration of a pandemic in a typical community in Canada at certain sizes of stockpile for antiviral drugs under given scenarios. Symptoms
Viral infection
No infectious
Recovering
Incubation
Latent E
:>
~
Infectious E
~
Figure 3.1: SIR time line Transmission probabilities are important parameters for the simulation model. It is obvious that there is no such parameters for an unknown
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 321 pandemic. However, statistical models have been used for the analysis of infectious disease data from family studies in a community (see [9] and [10]), and the household transmission probabilities have been provided by analyzing data from influenza A(H3N2) [6]. The transmission probabilities within small groups, large day-care centers, elementary schools, middle schools, and high schools are 0.04, 0.015, 0.0145, 0.0125 and 0.0105, respectively. The probabilities within a family are 0.08 (between a child to a child), 0.03 (between a child to an adult (vise versa)), and 0.04 (between an adult to an adult (vise versa)). The transmission probability that a preschool child is infected from a infective person in his neighborhood (other than his/her group or his/her family) is 0.00004; the probability for a school child can be 0.00012; the probability for an adult is increased to 0.00016. The probability that a pre-school child is infected from an infective person in the community (other than his/her neighborhood) is 0.00001; the probability for a school child is 0.00003; and the probability for an adult is 0.00004. However, to describe the transmission accurately, the process may be decomposed into contact process and infection process. Let Pcontact is the probability that a contact is a sufficient contact for transmission of influenza. The Pcontact depends on the age groups and is given by Longini et al. [12]. The calculation of P, the probability that a susceptible is infected for possible contacts with infectives, is shown as the following. P = min(1)PcontactNcontact, 1),
where 1> is determined by baseline attack rate; the Ncontact is a number of contacts in a day. The 1> for an unknown pandemic has to be determined by a simulation study. The 1> used in this article is determined by the baseline age specific attack rates given by Longini et al. (see Table S4 in "Supporting Online Material" for the paper by Longini et al. [12]). For example, the 1> = 0.1 corresponds to the situation that the overall illness attack rate is 33%. It is assumed that the efficacy of the antiviral drug for symptomatic disease given infection is variable as the time for giving the drug, based on the discussion by Stiver [11]. If a person becomes ill and takes antiviral drugs within 48 hours, the duration of illness will be reduced by 1.52.5 days (uniform distribution) [11]. In this paper, it is assumed that antiviral drugs are only for therapeutic use, excepting for high risk group people. The antiviral efficacy for infectiousness is 0.80 ; and the antiviral efficacy for symptomatic disease given infection is 0.60, which are the same as ones in the article by Longini et al. [12]. The antiviral efficacy for symptomatic disease given infection is how much an antiviral agent will reduce the probability that an infected person will develop influenza symptoms compared with an infected person who is not using an antiviral
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Shenghai Zhang
agent. The antiviral efficacy for infectiousness is how much an antiviral agent will reduce the probability that an infected person will transmit influenza to others compared with an infected person who is not using an antiviral agent. An asymptomatic infection is assumed to be 50% as infectious as a symptomatic infection, based on some studies. It means that the clinical attack rate is the half of infection rate (serologic attack rate). The distribution of incubating days was estimated by Longini et. al. [12J: 30% of infected persons have an incubation of one day; 52% infected persons have two days and 18% of infected persons have three days. The distribution of infectious days is as follows: 30% for three days; 40% for four days; 20% for five days and 10% for six days. Now, the calculation of the probability that a susceptible person is infected on a day can be shown as the following example. Considering an elementary school child, suppose he is a susceptible and he is exposed to a number of infective children and adults in his household, his neighborhood cluster, his school and his social groups. Because infective people may have different infectious forces for infecting a susceptible person, the" numbers" of infective people in each group is given by the following models: suppose the number of new infective people in a group is y(t) on the day t, a total of Y(t) infective people in this group is based on the new infective people in the past five days and Y(t) = L~~~(5,t) y(t- j)'I1 j. Here, \[J j describes the force of an infected person who is still infectious j days after initially contracting the diseases. Table 4 gives the force of an infective person who takes antiviral drugs in time, based on the results provided by Daley and Gani [13J. Let Ync(t), Yna(t), Yes (t) , YsI(t) and Y s2 (t) are the total of infective people in each group, respectively; and let Pnce, Pnae, Pes, PsI, and Ps2 are transmission probabilities from a infective person in a corresponding group to the child. Then, the probability P(t) that the child becomes a new infective person on day t is as the following. P(t) = 1- (1- Pnce)Ync (t)(1- Pnac)Yna(t)(l- Pes)Yes(t)
x (1 - PsdYsdt) (1 - Ps2)Y 2(t). S
Figure 3.2 shows how the probabilities that a susceptible child is infected on a day is calculated.
A Simulation Approach to Analysis of Antiviral Stockpile Sizes ... 323 A susceptible child
Infected? Yes Prob.~ 1- II., (I-p.l'
Infected
)~E_ _Ye_s_---11
Use antiviral drugs
No
It-_NO__~ ~su~sc;ep~tib§ley----.J
~~ Figure 3.2: Calculation of the probability
It is also interesting to know the probability, Pn(t), that there are certain number of infective people in the community at time t, where Pn (t) =Prob{ there are n infective people in a certain population at time t}, where t = 0.1, ... , refers to time in day. The calculation of this probability is based on the dynamics of disease in population level. In this case, it is assumed that the community is open, it means that patients are allowed to arrive from outside of the community and spread the disease to other people. Four forces are considered: the first is the immigration force, bring new cases; the second force is the infectious force, spreading the disease; the third force is the death force or immunity force and the removing patient force: emigration force. This scenario is described by the Immigration-infectious- cure (immunity)-Emigration model studied by Daley and Gani [13]. The important assumption in this model is that each of these four forces acts independently of the other three. An arrival patient occurs with probability >"6t; a patient could leave the system with probability f-t6t; given n - 1 patients in the system at time t, the probability of a new case could occur in the time interval (t, t + 6t) is (n -1)1I6t; given n + 1 patients in the system at time t, the probability
Shenghai Zhang
324
of a patient being recovered or death is (n the following equation:
+1)w6t.
The Pn(t) satisfies
Through the generating function: 00
l: sn Pn(t)
Gt(s) =
n=O
the solution can be obtained (if there are no patients with illness in the time 0):
Pn(t)
= [exp(-(v -
w)t)(v -
min{n+.tc,no+.tc} ~
W
~
] (no+e+j-l) [v-wexp(-(v-W)t) -(v - w)t) j
J=O
no + ~
w)]~-~ [ v-wexp / - (v-w )t)]no+~
.)
( n+~-J
v(exp(
1)
[w(l - exp( -(v - w)t))C+:-
[vexp(-(v-w)t)-w]
0-
j
J
+)
The parameters in the formula can be obtained from doing the simulation on the individual level.
4
Results
In the simulations, it is assumed that an initial person infected with the newly emergent influenza strain is randomly introduced to the community of 3200 people. Then, there are three possible outcomes: there is a large epidemic (more than four cases in the community); there is a small epidemic (more than one but less than or equal to four cases in the community); no further people are infected except for the original case. Also, it is assumed that the first case is a symptomatic infection. It is known that the transmissibility of the new strain depends on the illness attack rates, which are unknown. The only situation of the overall illness attack rate is of 33% is discussed in this paper. The effect on the number of infections of using different antiviral stockpile sizes to treat infection is estimated. It is estimated that the probability of controlling the infection to at most four cases in the community is 0.77, when the stockpiles cover 30% of the population.
A Simulation Approach to Analysis of Antiviral Stockpile Sizes . " 325 Suppose there is no antiviral stockpile, the possible intervention is household quarantine and the probabilities of withdrawing ill people to their homes are 0.8, 0.75 and 0.50 for preschool children, school child and adults, respectively. The chance that there is no large epidemic exists in this scenario. However, this chance is small, the probability of the outbreak of larger epidemic is greater. The simulations are run to calculate the number of infection for each day. Figure 4.1 shows the average numbers of infective adults and children respectively on each day, when there is no antiviral stockpile. In Figure 4.1, horizontal axis represents time (in day); the vertical axis represents the numbers of infective adults (upper curve) and children (lower curve). 140 120 100 80 60 40 20 0
0
20
40
60
80
100
120
Figure 4.1: Numbers of infective adults and children in each day without antiviral stockpile Suppose the antiviral stockpile covers 30% of the population. It means that we have enough antiviral stockpile to be dispensed to 960 ill persons in the community of 3200 people. Based on the simulations, it is estimated the probability that the number of cases can be controlled under four in the community is 0.77. This is based on the scenario that the original case can not be identified in a very early day of the infection and the only prevention is the limited quarantine: withdrawing ill people to their homes. However, there is a 23% chance for a large epidemic, that is, the infected rate will be bigger than 15/10000. It is interesting that once a large epidemic happen, the majority of people in the community will be infected under this scenario. The maximum averaged number
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326
of infective people in a day is 74 when the worst cases is considered. It will be about 30th day after the initial case was introduced into the community. The dynamic change of infective people can be seen in Figure 4.2. In Figure 4.2, horizontal axis represents time (in day); the vertical axis represents the numbers of infected adults (upper curve) and children (lower curve). Here, it is assumed that patients who are given antiviral drugs can get the drugs in time, that is, the treatment can be obtained within 48 hours without delay. 45 40 35 30 25 20 15 10 5 0
0
20
60
80
100
120
Figure 4.2: Numbers of infective adults and children in each day with antiviral drug treatment in time If antiviral stock pile is available for 30% population, but only 80% of patients who get antiviral drugs can get treatment in time, then the dynamic is shown by Figure 4.3. It is interesting that there will be a strong chance of curing all infected people in about 20 days after the initial case is introduced in the community. However, once it becomes a larger epidemic, it needs more than two or three months to cure all infected people and extinguish the infection sources in the same community. The point here is that the epidemic will be controlled within 20 days or it will need more that 60 days. The frequency of the duration of the epidemic is shown in Figure 4.4, based on 100 simulations. The vertical axis represents the numbers of simulations corresponds to the duration (in days represented by horizontal axis) of the epidemic per 100 simulations. The death could happen. The death rate for the worst case is shown in Figure 4.5, which is an accumulative death. For example, the accu-
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 327 45 40 35 30 25 20 15 10 5 20
40
60
80
100
120
140
Figure 4.3: Numbers of infective adults and children in each day with 80% of antiviral drug treatment in time
70 60 50
o
7
17
27
37
47
57
67
77
87
97
Figure 4.4: The frequency of duration of the pandemic among 100 simulations
mulative death reaches the highest in the 60th day, which is 17/100000 with antiviral stockpile covering 30% population.
Shenghai Zhang
328 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0
10
20
30
40
50
60
70
Figure 4.5: The accumulative death rate per 2000 people
5
Conclusion and discussion
The goal of this article is to build a simulation framework to answer the following question: What is the probability that we can control the number of infected people in a community under certain level if there is only limited antiviral drug stockpile. It is estimated that the probability of controlling the infection to at most 4 cases in the community of 3200 people is 0.77 if the stockpiles can cover 30% of the population and patients who get medical treatment can get anti-viral drugs in time. The results were obtained based on some naive scenarios. In practice, the situation may be more complex. For example, although diagnostic tests for influenza are available for all patients, the tests may take extra time for some patients because office-based tests may be not available for all clinicians. Also, it was estimated by Stiver [ll])that once an influenza outbreak in a community, the 30%-50% individuals with ILl (influenza like illness) who may not have influenza will accept antiviral drugs treatments, if anti-influenza drug therapy is prescribed to all people with ILL Considering this situation, the size of stockpile for antiviral drugs should be inflated. The discussion in this paper is under the assumption of the use of antiviral drugs for treatment of patients. The potential use of antiviral agents for prophylaxis has been investigated in many articles (see, for example, [14] and [6]) and the use of antiviral agents for containing an emerging influenza pandemic was studied by Ferguson [15] and Longini et al. [12].
A Simulation Approach to Analysis of Antiviral Stockpile Sizes· .. 329 The future extension of this work could be into more complex situation, where, for example, the disease progression considering the status of vaccination of the population and the use of antiviral drugs for prophylaxis could be important components of the model. It could be modeling to compare the effectiveness of a combination of antiviral treatment and the quarantine intervention strategies against a new strain of influenza.
References [1] http:j jwww.who.intjcsrjdiseasejinfluenzajenjindex.html.onMarch 28,2006. [2] http:j jwww.MercuryNews.com on March 6, 2006.
[3] http:j jwwwI2.statcan.cajenglishjcensusOl on November 6, 2005. [4] B.A. Cunha, Influenza: historical aspects of epidemics and pandemics. Infect. Dis. Clin. North Am. 18 (2004), 141-155. [5] B.M. Sayers and J.J. Angulo, A new explanatory model of an SIR disease epidemic: A knowledge-based, probabilistic approach to epidemic analysis. Scandinavian Journal of Infectious Disease 37 (2005), 55-60. [6] I.M. Longini Jr., E.M. Halloran, A. Nizam, and Y. Yang, Containing pandemic influenza with antiviral agents, American Jouranl of Epidemiology, 159 (2004), 623-633. [7] J.C. Hudson, Geographical diffusion theory: Studies in Geography, Vo1.19, Northwestern University, Evanston, Illinosis 1972. [8] F.T. Cadham, The use of a vaccine in the recent epidemic of influenza, Canadian Medical Association Journal, 11 (1919), 519-527. [9] I.M. Longini, J.S. Koopman, M. Haber, and G.A. Cotsonis, Statistical inference for infectious diseases, American Journal of Epidemiology, 128 (1988), 845-859. [10] D.L. Addy, I.M. Longini, and M.S. Haber, A generalized stochastic model for the analysis of infectious disease final size data, Biometrics, 47 (1991), 961-974. [11] G. Stiver, The treatment of influenza with antiviral drugs, Canadian Medical Assocation Journal, 168 (2003), 49-57. [12] I.M. Longini Jr., A. Nizam, and S. Xu, et al., Containing Pandemic Influenza at the Source, Science, 309 (2005), 1083-1087. [13] D.J. Daley, and J. Gani, Epidemic modelling: an introduction, Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, United Kingdom 1999.
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[14] R.D. Balicer, M. Huerta, and I. Grotto, Trackling the next influenza pandemic, BMJ, 328 (2004), 1391-1392. [15] N.M. Ferguson, D.A. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn, and D. S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature, 437 (2005), 209-214.
331
Modeling and Simulation Studies of West Nile Virus in Southern Ontario Canada Peter BuckA , Rongsong Liu B , Jiangping Shuai A , Jianhong Wu c , Huaiping Zhu C A Foodborne,
Waterborne and Zoonotic Infections Division Centre for Infectious Disease Prevention and Control Public Health Agency of Canada 255 Woodlawn Road West Unit 120 Guelph, Ontario NiH 8Jl, Canada B Department of Mathematics, Purdue University 150 N. University Street, West Lafayette, IN 47907-2067, USA C Department of Mathematics and Statistics, York University Toronto, Ontario, M3J lP3, Canada
Abstract
We carry out a preliminary mathematical modelling study of the West Nile Virus in the Peel region, Ontario Canada. The surveillance data from this region is used to show that the noncrow family birds which are susceptible to West Nile Virus are one of the key factors responsible for the endemic of the virus in the region. The density of dead crow is no longer an accurate indicator for the virus after the virus has sustained in the region, while the density of mosquitoes can give us better quantification of the risk level of the West Nile virus.
1
Introduction
Although West Nile Virus (WNV) was isolated in the West Nile district of Uganda in 1937[21J, and WNV in the Eastern Hemisphere has been maintained in an enzootic cycle involving culicine mosquitoes and birds[1l,12J, WNV activities in North America were not recorded until August of 1999 in the borough of Queens, New York City[7,16 J. Despite this long delay of invasion into North America, the virus has rapidly expanded spatially within the subsequently several years and evidences seem to indicate that WNV becomes a permanent fixture of the North America medical landscape[18J. It is this endemic nature of WNV that motives this research.
332
Peter Buck, Rongsong Liu, Jiangping Shuai, ...
Through past successful WNV surveillance program in Southern Ontario, it is confirmed that the virus has caused serious problems in certain bird populations in North America. For the last several years it has been seen the drop of the number of avian species (such as American crows and blue jays) in southern Ontario. In 1999, in the New York area, the crow population crashed by about 90 per cent in a few months. In Canada, American Crow population also crashed in 2002. Since birds are the host and reservoir of WNV, the decline of some bird populations will likely affect both the virus spread pattern and risk levels, and the effectiveness of the existing surveillance systems. For example, it was previously believed that Canada would face a high-risk WNV year in 2005. However, WNV risk was reported relatively low in the year. This might be explained by the decline of American Crow population in the region. It is therefore important for us to explore how the virus might affect the population of major bird species. This study can then give us guidelines for surveillance program, surveillance focuses, and dead bird collecting and testing strategies. There have been some modeling studies on the transmission dynamics of West Nile Virus, and most of the models are autonomous system of ordinary differential equations or discrete systems. The work[14J summarized the available models and made comparative studies of two continuous and discrete-time West Nile Virus models. In the previous modeling studies, the avian species was considered as one family. In the paper, we will develop a mathematical model by classifying the bird species into crow-family and non-crow family. The density of mosquitoes is modeled by time-dependent Gaussian functions. The focus is to study the impact of WNV on the ecology of birds. By using the surveillance data of the major avian species in Southern Ontario, numerical simulations allow us to investigate if the WNV is responsible for the decline of the number of certain species of birds in the region, and to study how surveillance program can be made more effective to alert the possible outbreak and recurrence of the WNV.
2
The model formulation
WNV is transmitted from birds to birds by mosquitoes. When a female mosquito feeds on an infected bird, it picks up the virus and transmit it to other uninfected susceptible [l,6,19J. Occasionally, infected mosquitoes will feed on mammals such as horses, dogs, cats, and humans, and transmit the virus to them. Mammals are the dead-end hosts, however, and they do not contribute to the transmission cycle. Most birds do not become ill when they got infected with this virus, but crow family birds (including crows, ravens, magpies, blue jays, gray jays and Stellar's jays[6J)
Modeling and Simulation Studies of West Nile Virus in . . .
333
are susceptible to the WNV and often die when they are infected, due to inflammation of many organs including the brain (encephalitis) caused by the virus[81. For this reason, crow family birds have been chosen as an indicator species for the presence of WNV. The crow family birds may not contribute significantly to the spread of the virus because of the higher mortality rate, and they die from the infection in a short period. Therefore, we will divide all birds susceptible to the virus into two classes: crow family and non-crow family, based on whether they show symptoms of illness or not and we will denote their populations sizes by B 1 (t) (crow family birds) and B 2 (t) (the remaining birds under consideration). We shall investigate the impact of the second class of birds for the ongoing outbreak and spread of the virus. For non crow family birds, we divide them into susceptible, infected and recovered, denoted by B 2s , B 2i , B 2r , respectively. While for crow family birds, we divide them into susceptible and infected, denoted by B 1s , B 1i . Due to the higher mortality rate, we ignore the recovered class for crow family birds since most infected crow family birds will die due to the disease. And we will abuse notations and use, for example, B 1s to denote the number of susceptible crow family birds at time t. Let Nb = N 1b + N 2b, where N 1b = B 1s + B1i and N 2b = B 2s + B2i + B 2r are the total number of crow family birds and non crow family birds. Figure 1 gives the flow chart of the model. . Based on the assumptions and the flow chat, we have the following equations to model the avian species: dB 1s
&
dBli
dt
= h 1(B1s ) =
Bls
b1{31 Nb Mi - d 1i
dB2s & = h2(N2b ) dB2i
B1s
b1{3l Nb Mi - dlsB ls , B 1i,
B 2s
b2{32 Nb Mi -
B 2s Nb
- - = b2{32--Mi - d 2B 2i -
dt
r
d B 2 2s, B
(1)
2i,
dB2r
- - = r B2i - d 2B2n
dt
where h 1 (.) and h2(.) are birth functions of the Bl and B2 class, bi , i = 1, 2 are the biting rate of a mosquito on ith kinds of birds, Mi(t) is the number of infected adult mosquitoes. Cross-infection between birds and mosquitoes is modeled using mass-action normalized by the bird density, {31 and {32 are the probability of transmission from infected mosquitoes to crow family and non crow family susceptible birds, respectively, d1s and d 1i are the mortality rates of susceptible and infected crow birds respectively, r is the recover rate of infected non crow family birds, and d 2 is the mortality rate of the non-crow family birds B 2 ·
334
Peter Buck, Rongsong Liu, Jiangping Shuai, ...
~
~dUCeddeath
~ irth
B2s
- .......- - - t L-_,--_...J
death
Figure 1: Flow chart of the model. Let Nm(t) be the total number of (adult female) mosquitoes, divided into infected mosquitoes Mi (t) and susceptible mosquitoes N m (t)Mi(t). Death and reproduction of mosquitoes are assumed not to depend on their infection status, and so the number of infected adult mosquitoes Mi(t) are assumed to obey
dMi _ -d M. dt -
m,
+
(N _ M.)b1a1B 1i + b2a 2B 2i m,
Nb
'
(2)
where al and a2 are the probability of transmission from susceptible mosquitoes to crow family and non-crow family infected birds, respectively, dm is the natural death rate of mosquitoes which varies over time.
3
Surveillance data and numerical simulations
Since WNV is particularly virulent in crow family birds, the dead American crows and blue jays were usually used as the indicator of the arrival of WNV in a geographic area. In [4], for example, the data of the number of WNV human disease cases and the density of dead crows in New York State from 2001 to 2003 are used to develop a threshold value of 0.1 dead crows per square mile (0.04 dead crows per square km) as a risk indicator for WNV in New York State. Our purpose in this paper is to validate this threshold value as the indicator and a corresponding similar threshold for the establishment phase of the virus. We will carry out the
Modeling and Simulation Studies of West Nile Virus in . . .
335
study for the Peel Region of Southern Ontario, and use the surveillance data from this area and model based simulations of the transmission dynamics.
3.1
Birds
CDC[6j listed 284 species of bird which have been reported to CDC's West Nile Virus avian mortality database from 1999-present. Using the information for North American birds[lOj, we assume that the region under consideration has two crow family birds (American crows and blue jays) and 41 other species which are susceptible to WNV (these include American Robin, Red-winged Blackbird among others). By the breeding bird census (BBC)[lOj, we estimate the density (number of birds/lOO ha (1 square km)) of crow family birds in the range [35,110] and the density of non crow family birds in the range [465,1024]. Based on those estimation, in our simulations we will assume that the initial value for the susceptible crow family birds B ls = 50 and susceptible non crow family birds B2s = 650.
3.2
Mosquitoes
From the adult mosquito surveillance data of the Peel region, we note that the distribution of mosquitoes is quite heterogeneous. We concentrate on the trap of mosquitoes, we call it TX (trap x), mainly because this tra~ has relatively complete data over the mosquito season, and we will consider this trap as a representative sample in the surrounding region by ignoring the spatial heterogeneity. We recall that there is a standard method[17j to calculate mosquito population density: take the total number of captured mosquitoes, by species, and divide it by the number of trap nights. Figure 2 gives the density of mosquitoes in the trap TX from the year 2002 to 2004. Observation of the density of mosquitoes for the year 2002 to 2004 in Figure 2 suggests that a) The shape of the density of mosquitoes for each year has binormal distribution; b) The density of mosquitoes achieves its first peak around July 15, and then reaches its second peak at the end of August. Therefore, we use the following combination of Gaussian functions to simulate the total number of mosquitoes n
Nm(t)
=
L
2 [Ail exp( -(t.- (i - 1) * 365 - aid / bil)
i=l
+Ai2 exp( -(t - (i - 1) * 365 - ai2)2/bi2)] ,
(3)
r<
r-3
i:r
('!)
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M-
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M-
f!J,
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10,:)
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20/7/2004"
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~. ('l)
719/2004
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0 ...,
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(I)
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density
22/07/2003
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14110/2003
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g .g &' 05/08/2003
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'1:J
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Modeling and Simulation Studies of West Nile Virus in . . .
337
where the time unit is day and t = 0 corresponds to May 1 of 2001, i = 1 represents the year of 2001 aij, (j = 1, 2) represents the time when the density of mosquitoes reaches its first and second peak of the ith year respectively, and bij , j = 1, 2, determines the length of the corresponding peak. Figure 3 gives the simulated density of mosquitoes, in comparison with the reported density of mosquitoes in Trap TX. In the figure, i = 2 represents the year of 2002. All the parameters in 3 are taken as the following: A21 = 1100, A22 = 1100, A31 = 500, A32 = 450, A41 = 800, A42 = 450, a21 = 75, a22 = 120, a31 = 85, a32 = 115, a41 = 80, a42 = 110 and b 21 = 0.003, b 22 = 0.007, b31 = 0.0015, b 3 2 = 0.3, b 41 = 0.005, b 42 = 0.009. density of mosquito
1400 r;::::=::::;=c:;:::::::;::;::=;::'c::::;::=::::;:::=:!=:;-----,------,~
I
1-- simulated density of mosquito
I
1-
... -
real data of the density of mosquito
1200 1000
400 200
(- 2004
~2002
Ii
I~
A I
I II I II I II II I
i
I .1
! r
1111
o.l'lh. o
J 200
400
I
II
~
I
~ ~ 800
~ time
,
2003 600
1000
Figure 3: Comparison of the simulated and reported densities of mosquitoes in the trap TX. The lifespan of mosquitoes is closely related to the temperature of the region. Here we again employ a linear combination of Gaussian functions to construct a periodic function with period 365 to describe the death rate of mosquitoes. Namely,
dm(t) = 1 -
Cl
* [exp{ -S((t - ([t]- 1)365) - td 2)} + exp{ -S((t - ([t] - 1)365 - t2)2}],
where [t] = t( mod 365), and we use tl S = 0.0009 in our simulations.
=
60, t2
=
105,
Cl
=
0.7 and
338
3.3
Peter Buck, Rongsong Liu, Jiangping Shuai, ...
Bird ecology in the absence of WNV
In the following, we assume that the birth functions for crow family are given by h1(X) = d 1x and h2(X) = d 2x. Namely, we assume that the natural birth rate of birds equals natural death rate of birds. Therefore, without the presence of WNV, the density of birds remains unchanged. This seems to be a good approximation for the crow family birds from the Christmas Bird Count (CBC) [5]. Since there is no mosquitoes in winter season, we will focus only on the transmission cycle of WNV during the summer season. We also assume that the non-crow family birds do not die of the WNV, most of the infected non-crow family of birds survive the infections. We do not have the surveillance data of adult mosquitoes for the year 2001, but according to Environment Canada, the summer of 2001 was the third warmest for the season (but not as warm as the summer of 2002)[3,22]. Therefore, we will use the following parameter values in the simulations: An = 800, A12 = 800, an = 75, a12 = 120 and bn = 0.003, b12 = 0.01. Recall from section 3.1 that we have B 1s (0) = 50, B 2s (0) = 650 and t = 0 corresponds to May 1 of 2001.
3.4
Bird ecology with WNV
Because crow family birds die of WNV within only several days, their contribution to the spread of the virus is limited. Thus we assume B1i(O) = O. We also assume B2i(0) = 30: so on May 1 of 2001, there was 30 infected non-crow family birds per square kilometer. We use (3i = 0.88, C¥1 = 0.5, C¥2 = 0.3, d 1i = 0.143, d2 = 1/(4 * 365)[2,13,15,23]. A critical parameter is the biting rate of the mosquitoes. We assume that the host are bitten in proportion to their abundance with host preference ¢: ¢ = 1 means mosquitoes biting birds without preference, while ¢ > 1 means that mosquitoes are preferential to feeding on crow family birds. The biting rate is defined by the days between blood meals on all birds (
¢(B1s
b = I
b = 2
+ B li )
~
+ B 1i ) + (B 2s + B2i + B 2r )
¢(Bls
In our simulations, we take
Modeling and Simulation Studies of West Nile Virus in . . .
339
Simulation results, with the best fit non crow family birds recover rate r 1/365 There seems to be no scientific data for the recover rate of uon-crow family birds, and we are thus forced to use the model to find the this parameter so that the simulated results agree best with the l'or",,"'''''' data, and such a process gives r 1/365. Note that we do not have the data of adult mosquitoes and birds in 2005 at this while according to the temperature and humidity of summer of 2005, we take 1000, 800, a51 75, a52 115, b51 0.003, b52 0.01 which are 4 defined in the function of total number of mosquitoes. surveillance and simulated densities of dead crow family birds in the area of Trap B2 from 2002-2005. The fact that our simulation numbers are higher than the reported data seems to be justified first of not all dead crow family were picked up, and secondly, the health units of Region of Peel only counted the dead American crow in 2002 and 2003, and then began to add dead blue jays in 2004. density of dead crow in area trap TX
12 10 ~
8<.> 8
al
.g "-' 0
6
.~ 0>
""
4
2 0
2002
2005
Figure 4: Simulations and the surveillance data about the density of dead crow in the surrounding area of the trap TX. Figure 5 gives the simulated weekly density of dead crow from 20022005. From the graph and considering the error of collecting dead crow, we observe that the indicator was accurate (for example, in the years 2002 and 2003) when the disease first arrived in a region. But after it becomes endemic, this indicator loses its efficacy. Figure 6 the simulated results of the ratio of infected mosquitoes with the total mosquitoes. The amplitude of the peak of 2005 in Figure 6 heavily depends OIl the
Peter Buck, Rongsong Liu, Jiangping Shuai, ...
340 rlAT1"'T"
of mosquitoes in this region. 1.8,------;----,.-----;----,------,
5
25
20
week
Figure 5: Weekly density of dead crows in area of the trap TX. 0.18 .~ 0.16
IIl 0.14
1] 0.12 .Jl 0.1
<- 2002
100.08 .~ ~
0.06
~
'
..g 0.02 !l
00
<- 2003
~2001 200
\<- 2004
\<-2005
.\ 400
600
800 1000 1200 1400 1600 1800 time (day)
Figure 6: Percentage of infected mosquitoes. We do not have the data of adult mosquitoes and birds in 2005 at this stage, but three dead birds found in Peel had tested positive for WNV. Two of the birds are crows and one is a blue jay. The first crow was picked up on July 24 from the Central Parkway East and Bloor Street area in Mississauga; a blue jay was found on July 25 in the QEW and
Modeling and Simulation Studies of West Nile Virus in . . .
341
Hwy. 410 area also in Mississauga. The second crow was picked up on July 25 in Caledon East. This seems to indicate that WNV is already enzootic in Southern Ontario.
3.6
Remarks on the preference parameter
Although there is no scientific data for the recover rate of non-crow family birds, Rappole etc.l 20 ] pointed out that migratory birds might be the main reason for spreading WNV to the Western Hemisphere. And there are evidences that hawk died of WNV in February, 2006[9]: a strong evidence that the virus can sustain in the body of non-crow family birds at least more than 8 months and could be transmitted. The simulated results agree best with the reported data, and such a process gives the average recover rate of non crow family birds r = 1/365. While r is increased to the level of 0.01, the disease will die out within three years which obviously conflict with the real situation. If r is decreased, we find the amplitude of each outbreak would be too high and does not agree well with the surveillance data.
4
Discussions
In summary, we have observed through model-based simulations that the number of dead crow family birds is down while the percentage of infected non crow family birds sustains a relatively high level. This indicated the endemic status of WNV in the region, and the number of dead crows is no longer a very good indicate of the level of WNV infection. On the other hand, the density of mosquitoes plays a key role in the transmission dynamics and seems also to give a very good indication of the level of risk and infection: if the density of mosquitoes is large, the ratio of infected mosquitoes will increase dramatically, and so will the percentage of non crow family birds. It seems that for the control and prevention, the surveillance of mosquitoes should be kept and improved. We are not in a position to suggest how this improvement can be achieved, however, it seems that traps shall be properly maintained and baited appropriately, accurate records should be maintained. A threshold defined in terms of the density of mosquitoes may be useful for the management of possible future outbreaks.
Acknowledgement This work was partially supported by Mathematics for Information Technology and Complex Systems (MITACS), by Natural Sciences and Engineering Research Council of Canada (NSERC), by Canadian Foundation
342
Peter Buck, Rongsong Liu, Jiangping Shuai, ...
of Innovation (CFI) and Ontario Innovation Trust (OIT), by Canada Research Chairs Program, by Public Health Agency of Canada (PHAC), Ontario Ministry of Health and Long-Term Care, Peel, Toronto and Chatham-Kent health units.
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[13] M. Lewis, J. Renclawowicz, and P. van den Driessche, Traveling waves and spread rates for a west nile virus model, Bull. Math. Biol., in press. [14] M. Lewis, J. Renclawowicz, P. van den Driessche, M. Wonham, A comparison of continuous and discrete-time West Nile virus models. Bull. Math. Biol. 68 (2006), no.3, 491-509. [15] C.C. Lord and J.F. Day, Simulation studies of st. louis encephalitis and west nile virues: the impact of bird mortality, Vector Borne and Zoonotic Diseases 1 (2001), noA, 317-329. [16] D. Nash, F. Mostashari, A. Fine, et al., The outbreak of west nile virus infection in the new york city area in 1999, N. Eng. J. Med. 344 (2001), 1807-1814. [17] Colorado Department of Public Health and Environment, 2005 mosquito surveillance plan, (2005), http://www.cdphe.state.co.us/dc/zoonosis/wnv/05SurvPlan.pdf. [18] L.R. Petersen and A.M. Anthony, West nile virus: A primer for the clinician, Ann. Intern. Med. 137 (2002), 173-179. [19] L.R. Petersen, A.A. Marfin, and D.J. Gubler, West nile virus, JAMA 290 (2003), noA, 524-528. [20] J.H. Rappole, S.R. Derrickson, and Z. Hubalek, Migratory birds and spread of west nile virus in the western hemisphere, Emerging Infectious Diseases 6 (2000), 319-328. [21] K.C. Smithburn, T.P. Hughes, A.W. Burke, and J.H. Paul, A neurotropic virus isolated from the blood of a native of uganda, Am. J. Trop. Med. 20 (1940),471-492. [22] L.A. Vincent, B.R. BonsaI, X. Zhang, and W.D. Hogg, Homogenization of daily temperatures over canada. journal of climate, Journal of Climate 15 (2002), 1322-1334. [23] M.J. Wonham, T. de Camino-Beck, and M. Lewis, An epidemiological model for west nile virus: Invasion analysis and control applications, Proceedings of the Royal Society of London, Series B 271 (2004), no.1538, 501-507.