Modeling and Dynamics of Infectious Diseases (Series in Contemporary Applied Mathematics)

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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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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89

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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92

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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Julien Arino

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

Diseases in Metapopulations Bp

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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98

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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99

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.

Diseases in Metapopulations

101

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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Julien Arino

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

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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,

Julien Arino

108

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

Diseases in Metapopulations

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.

Julien Arino

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.

Infectious Diseases of Humans.

[4] J. Arino, J.R Davis, D. Hartley, R Jordan, J.M. Miller, and P. van den Driessche. A multi-species epidemic model with spatial dynamics. Mathematical Medicine and Biology, 22(2): 129-142, 2005. [5] J. Arino, R Jordan, and P. van den Driessche. Quarantine in a multi-species epidemic model with spatial dynamics. Math. Biosci., 2006.

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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.

Vertically Transmitted Diseases.

[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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volume 126 of IMA Series on Mathematics and its Applications. Springer, 200l. [19] C. Castillo-Chavez and H. Thieme. Asymptotically autonomous epidemic models. In O. Arino, D. Axelrod, M. Kimmel, and M. Langlais, editors. Mathematical Population Dynamics: Analysis of Heterogeneity, 33-49, Wuerz, Winnipeg, 1995. [20] D. Clancy. Carrier-borne epidemic models incorporating population mobility. Math. Biosci., 132: 185-204, 1996. [21] O. Diekmann and J.A.P. Heesterbeek. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley, 2000. [22] M.J. Faddy. A note on the behavior of deterministic spatial epidemics. Math. Biosci., 80: 19-22, 1986. [23] M. Fiedler. Special Matrices and their Applications in Numerical Mathematics. Martinus Nijhoff Publishers, 1986. [24] E. Fromont, D. Pontier, and M. Langlais. Disease propagation in connected host populations with density-dependent dynamics: the case of the Feline Leukemia Virus. J. Theor. Biol., 223: 465-475, 2003. [25] G.R. Fulford, M.G. Roberts, and J.A.P. Heesterbeek. The metapopulation dynamics of an infectious disease: tuberculosis in possums. Theoretical Population Biology, 61: 15-29, 2002. [26] R.F. Grais, J.H. Ellis, and G.E. Glass. Assessing the impact of airline travel on the geographic spread of pandemic influenza. European Journal of Epidemiology, 18: 1065-1072, 2003. [27] B.T. Grenfell and A.P. Dobson, editors. Ecology of Infectious Diseases in Natural Populations. Cambridge University Press, Cambridge, 1995. [28] 1. A. Hanski and M.E. Gilpin. Metapopulation Biology: Ecology, Genetics, and Evolution. Academic Press, 1997. [29] H.W. Haskey. Stochastic cross-infection between two otherwise isolated groups. Biometrika, 44(1/2): 193-204, 1957. [30] H. W. Hethcote. The mathematics of infectious diseases. SIAM Review, 42(4): 599-653, 2000. [31] H. W. Hethcote and H. Thieme. Stability of the endemic equilibrium in epidemic models with subpopulations. Math. Biosci., 75: 205227,1985. [32] H. W. Hethcote and J.A. Yorke. Gonorrhea Transmission Dynamics and Control, volume 56 of Lecture Notes in Biomathematics. Springer-Verlag, 1984.

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[33] H.W. Hethcote. An immunization model for a heterogeneous population. Math. Biosci., 14: 338-349, 1978. [34] J.M. Hyman and T. LaForce. Modeling the spread of influenza among cities. In H. Banks and C. Castillo-Chavez, editors, Bioterrorism, 215-240. SIAM, 2003. [35] V. Isham and G. Medley, editors. Models for Infectious Human Diseases: Their Structure and Relation to Data. Cambridge University Press, Cambridge, 1996. [36] W.O. Kermack and A.G. McKendrick. A contribution to the mathematical theory of epidemics. Proc. Roy. Soc. London, Ser. A, ll5: 700-721, 1927. [37] A. Lajmanovich and J.A. Yorke. A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci., 28: 221-236, 1976. [38] V. Lakshmikantham, S. Leela, and A. Martynyuk. Stability Analysis of Nonlinear Systems. Marcel Dekker, 1989. [39] S. A. Levin, T.M. Powell, and J.H. Steele, editors. Patch Dynamics, volume 96 of Lecture Notes in Biomathematics. Springer-Verlag, 1993. [40] LM. Longini. A mathematical model for predicting the geographic spread of new infectious agents. Math. Biosci., 90: 367-383, 1988. [41] M. Markus and H. Minc. A Survey of Matrix Theory and Matrix Inequalities. Dover, 1992. [42] H. Minc. Nonnegative Matrices. Wiley Interscience, 1988. [43] D. Mollison, editor. Epidemic Models: Their Structure and Relation to Data. Cambridge University Press, Cambridge, 1995. [44] D.J. Rodriguez and L. Torres-Sorando. Models of infectious diseases in spatially heterogeneous environments. Bulletin of Mathematical Biology, 63: 547-571, 200l. [45] S. Ruan, W. Wang, and S.A. Levin. The effect of global travel on the spread of SARS. Mathematical Biosciences and Engineering, 3(1): 205-218,2006. [46] S.M. Rump. Bounds for the componentwise distance to the nearest singular matrix. SIAM J. Matrix Anal. Appl., 18(1), 1997. [47] S. Rushton and A.J. Mautner. The deterministic model of a simple epidemic for more than one community. Biometrika, 42(2): 126-132, 1955. [48] L.A. Rvachev and LM. Longini. A mathematical model for the global spread of influenza. Math. Biosci., 75:3:22, 1985.

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[49] M. Salmani. A model for disease transmission in a patchy environment. Master's thesis, University of Victoria, 2005. [50] M. Salmani and P. van den Driessche. A model for disease transmission in a patchy environment. Discrete and Continuous Dynamical Systems B, 2006. [51] L. Sattenspiel and K. Dietz. A structured epidemic model incorporating geographic mobility among regions. Math. Biosci., 128: 71-91, 1995. [52] L. Sattenspiel and D. A. Herring. Structured epidemic models and the spread of influenza in the central Canadian subartic. Human Biology, 70(1): 91-115,1998. [53] L. Sattenspiel and D.A. Herring. Simulating the effect of quarantine on the spread of the 1918-19 flu in central Canada. Bull. Math. Biol., 65: 1-26, 2003. [54] 1. Sattenspiel, A. Mobarry, and D.A. Herring. Modeling the influence of settlement strucure on the spread of influenza among communities. Am. J. Human Biol., 12: 736-748,2000. [55] L. Sattenspiel and C.P. Simon. The spread and persistence of infectious diseases in structured populations. Math. Biosci., 90: 341-366, 1988. [56] H.L. Smith. Periodic solutions of periodic competitive and cooperative systems. SIAM J. Math. Anal., 17(6): 1289-1318,1986. [57] H.L. Smith and P. Waltman. The Theory of the Chemostat Dynamics of Microbial Competition. Cambridge University Press, 1995. [58] H.R Thieme. Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equation. J. Math. Biol., 30: 755-763,1992. [59] H.R Thieme. Mathematics in Population Biology. Princeton Series in Theoretical and Computational Biology. Princeton University Press, 2003. [60] P. van den Driessche and J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180: 29-48, 2002. [61] RS. Varga. Gersgorin and His Circles. Springer, 2004. [62] W. Wang, P. Fergola, and C. Tenneriello. Innovation diffusion model in patch environment. Applied Mathematics and Computation, 134: 51-67, 2003. [63] W. Wang and G. Mulone. Threshold of disease transmission in a patch environment. J. Math. Anal. Appl., 285: 321-335,2003.

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[64] W. Wang and X.Q. Zhao. An epidemic model in a patchy environment. Math. Biosci., 190(1): 97-112,2004. [65] W. Wang and X.Q. Zhao. An age-structured epidemic model in a patchy environment. SIAM J. Appl. Math., 65(5): 1597-1614,2005.

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

133

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).

136

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· . .

137

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

138

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

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141

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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143

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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Troy Day

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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146

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

Reb) for the EDM, and

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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

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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.

167

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

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0.6

~

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t

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10

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50

40

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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

-~(a)¢(a)]

[Q:v(a, t)

W (a, t),

ax aa

+ ax = A (a, t) W (a, t)

ay aa

+ ay

az aa

+ 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

,""- ...... ,

"

'\

," ,/

\\

II

0.1

§

'-e

",-

"\ \

\\ \,

/

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'- /

0.08

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'x.

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.:; 0.06

\

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0.04

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'-.. 22.5

(b) - - - - - - -- - - - - -::--:..=-=- -..=.a/'

, .. ,,,,,,."

/'

0.8

/

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, /

~ 0.6

i

:"'u

,

0.4 0.2

--

,,

,, ,, ,,

,,

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,

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o --7.5

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12.5 15 17.5 Age (mos or yrs)

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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

'

I /

,

I

,

I

,

I

\

"

0.08

\/-"

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.~

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0.02

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--~-~--=-

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40

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Age (years) (b)

- - - - - - - - -:-- =-= -=-= - - -.::. ./

/

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/

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/

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0.2

I

/

/ /

/

/ /

20

25

30

35

40

45

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.

187

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.

188

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).

192

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)

198

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}.

Persistence of Vertically Transmitted Parasite Strains which· . .

207

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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208

(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

Persistence of Vertically Transmitted Parasite Strains which· . .

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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,

210

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

Persistence of Vertically 'Ifansmitted Parasite Strains which· . .

211

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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212

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.

Persistence of Vertically Transmitted Parasite Strains which· . .

213

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.

288

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-

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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)

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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].

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306

Let S = {