DETERMINISTIC AND STOCHASTIC EPIDEMIC MODELS

WITH MULTIPLE PATHOGENS

by

NADARAJAH KIRUPAHARAN, B.E., M.S.

A DISSERTATION

IN

MATHEMATICS

Submitted to the Graduate Faculty

of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Chairpefson of the Committee

Accepted

Dean of the Graduate School

August, 2003

ACKN()\\'LI':I)(;MENTS

1 would like \o express my apprccialioii to Dr. Linda J.S. Allen for her patience

and guidance throughout this research and the wilting (jf this dissert at ion, I am

grateful for all of the time and work she has given on my helialf. I would also like to

thank the other members of the eoniiuittee Dr. Edward Allen, Dr. Roger Barnard and

Dr. Rob Paige, for their time and advice. I would also like to thank the Mathematics

and Statistics Department of the Texas Tech University for providing an excellent

atmosphere for graduate study. Finally, I would like to thank my wonderful wife

Mythely, m\- sweet little daughter Nila, my loving mother Parathathevi, and my late

father Xallatliamby Nadarajah for their emotional support throughout my academic

career.

This work was partially supported by the Texas Higher Education Coordinating

Board ARP grant # 003644-0193 and the National Science Foundation grant # DMS-

0201105.

This dissertation is dedicated to my mother Parathathevi Nadarajah and my

father, the late Nallathamby Nadarajah.

11

CONTENTS

ACKNOWLEDGMENTS ii

ABSTRACT v

LIST OF FIGURES vii

I INTRODUCTION 1

II DETERMINISTIC SIS EPIDEMIC MODELS WITH MULTIPLE

PATHOGEN STRAINS 3

2.1 Introduction 3

2.2 Single Patch 4

2.2.1 Model 4

2.2.2 Analytical Results 7

2.3 Two Patches 13

2.3.1 Model 13

2.3.2 Analytical Results 16

2.3.3 Infective Dispersal 21

2.3.4 Susceptible Dispersal 22

2.4 Numerical Examples 23

2.5 Summary 31

III DISCRETE-TIME MARKOV CHAIN SIS EPIDEMIC MODELS WITH

MULTIPLE PATHOGENS 33

3.1 Introduction 33

3.2 Single Patch 35

3.2.1 Stochastic Model 35

3.2.2 Probability of Extinction 36

3.2.3 Quasistationary Distribution 38

3.3 Two Patches 39

3.3.1 Stochastic Model 39

3.3.2 Probability of Extinction 40

iii

3.3.3 Quasistationary Distribution 41

3.4 Numerical Examples 42

3.5 Summary 57

IV STOCHASTIC DIFFERENTIAL EQUATION SIS AND SIR EPI­

DEMIC MODELS WITH MULTIPLE PATHOGENS 59

4.1 Introduction 59

4.2 Derivation of Stochastic Differential Equations 61

4.3 SIS and SIR Epidemic Models 65

4.3.1 Ordinary Differential Equations 65

4.3.2 Stochastic Differential Equations 68

4.4 Probability Distribution of the SDEs 72

4.5 Numerical Examples 73

4.5.1 Competitive Exclusion 74

4.5.2 A Study of Coexistence 83

4.6 Summary 87

V CONCLUSIONS AND FUTURE DIRECTIONS 89

BIBLIOGRAPHY 91

IV

ABSTRAC;r

Conipetiti\-e exclusion and coexistence of multiple pathogens in deterministic and

stochastic epitlemie models are invest igaicnl in this dissertation which consists (jf three

parts. In the first part, the persistence and extinction dynamics of multiple pathogen

strains for a diserete-time SIS epidemic model in a single patch and in two patches are

studied. It is shown for the single patch model that the basic rej^roduction number

determines which strain dominates and persists. Howe\'er, in the two-patch epidemic

model, both the dispersal probabilities and the basic reproduction numbers for each

strain determine whether a strain persists. With two patches, there is a greater chance

that more than one strain will coexist.

In the second part, the stochastic spatial epidemic models with multiple pathogen

strains for the above deterministic models are formulated as discrete-time Markov

chain models and analyzed for coexistence and competitive exclusion. When infected

individuals disperse between two patches, coexistence may occur in the stochastic

model. However, in the stochastic model, eventually disease extinction occurs but it

will take a long time. An estimate for the probability of disease extinction is obtained

for the stochastic model. The distribution conditioned on non-extinction is compared

to the solution of the deterministic model.

In the third part, the dynamics of continuous-time stochastic SIS and SIR epidemic

models with multiple pathogen strains and density-dependent mortality are studied

using stochastic differential equation models. The dynamics of these stochastic models

are then compared to the analogous deterministic models. In the deterministic model,

there can be competitive exclusion, where only one strain, the dominant one, persists

or there can be coexistence, persistence of more than one strain. In the stochastic

model, all strains will eventually be eliminated because the disease-free state is an

absorbing state. Generally, it will take a long time until all strains are eUminated.

Numerical examples show that coexistence cases predicted in the deterministic models

may not occur in the stochastic models.

LIST OF FIGURES

2.1 SIS epidemic model 4

2.2 Dispersal between two patches 14

2.3 Solutions are graphed in phase plane 24

2.4 The stable equilibria are identified in pik - p2k parameter space . . . 27

2.5 Susceptible and infective dispersal 28

2.6 Solutions to model (2.11) 29

2.7 Solutions to model (2.11) graphed in the In - I21 phase plane . . . . 30

2.8 The stable equilibria are identified in pi — p2 parameter space . . . . 31

3.1 Frequency histograms for single patch 44

3.2 The solution to the deterministic model 45

3.3 Two sample paths for the stochastic model 46

3.4 The frequency distributions for elimination 47

3.5 Frequency histograms of the stochastic model with one strain 49

3.6 The solution to the deterministic model 50

3.7 Frequency histograms of the stochastic model with two strains . . . . 51

3.8 The solutions to the deterministic model 52

3.9 The frequency distributions for elimination 53

3.10 The stochastic model when infectives disperse 55

3.11 Three different sample paths 56

3.12 Elimination in both patches 57

4.1 Stochastic mean conditioned on nonextinction 74

4.2 Frequency histograms for SDE SIS model 76

4.3 Two sample paths for the SDE SIS model 77

4.4 Elimination in SDE SIS epidemic model 79

4.5 The solution to the SIR deterministic model 80

4.6 Frequency histograms for SIR stochastic model 81

VI

4.7 Elimination in the stochastic SIR epidemic model 83

4.8 Coexistence of strain 1 and strain 2 84

4.9 Frequency histograms for the SIR SDEs 85

4.10 The coexistence solution to the SIR deterministic model 86

4.11 Disease elimination in SDE SIR model 87

Vll

CHAPTER I

INTRODUCTION

The dynamics of discrete-time and continuous-time, deterministic and stochastic

epidemic models are studied in three chapters of this dissertation. In Chapter II,

the dynamics of discrete-time SIS epidemic models with multiple pathogen strains

are studied. The population infected with these strains may be confined to one

geographic region or patch or may disperse between two patches. The models are

systems of difference equations. It is the purpose of this investigation to study the

persistence and extinction dynamics of multiple pathogen strains in a single patch and

in two patches. It is shown for the single-patch model that the basic reproduction

number determines which strain dominates and persists. The strain with the largest

basic reproduction number is the one that persists and outcompetes all other strains

provided its magnitude is greater than one. However, in the two-patch epidemic

model, both the dispersal probabilities and the basic reproduction numbers for each

strain determine whether a strain persists. With two patches, there is a greater chance

that more than one strain will coexist. Analytical results are complemented with

numerical simulations to help illustrate both competitive exclusion and coexistence

of pathogens strains within the host population.

In Chapter III, the coexistence and competitive exclusion in stochastic SIS epi­

demic models with multiple pathogen strains are investigated. Either the population

is restricted to one spatial location-a single patch or the population may disperse

between two patches. In the two-patch model, it is assumed that infected and sus­

ceptible individuals disperse between the two patches. The stochastic models are

discrete-time Markov chain models. They are based on Chapter II deterministic for­

mulations which can be expressed in terms of systems of difference equations. In the

underlying deterministic models, a competitive exclusion principle applies when the

population is restricted to a single patch. The pathogen strain with the maximal basic

reproduction number is the one that persists and outcompetes all other strains pro-

vided its magnitude is greater than one. When infected individuals disperse between

two patches, coexistence may occur. However, in the stochastic model, eventually

disease extinction occurs. The probability of disease extinction approaches one in the

limit as / ^^ oo. Because the convergence to one may be very slow, the probability of

disease extinction may remain relatively constant. An estimate for this probability

of disease extinction will be obtained. In addition, the frequency distribution of the

population conditioned on nonextinction, known as the quasistationary distribution,

will be compared to the solution of the deterministic model. Through extensive nu­

merical simulations and estimates based on the theory of branching processes, the

quasistationary distribution and the extinction dynamics of the stochastic model are

studied.

In Chapter IV, the dynamics of continuous-time stochastic SIS and SIR epidemic

models with multiple pathogen strains are studied using stochastic differential equa­

tion models. Stochastic differential equations are derived from the corresponding

system of ordinary differential equations assuming there is demographic stochastic-

ity. The dynamics of these stochastic models are then compared to the analogous

deterministic models. In the deterministic model, there can be competitive exclusion,

where only one strain, the dominant one, persists or there can be coexistence, persis­

tence of more than one strain. The magnitude of the basic reproduction numbers are

important as well as the magnitude of the transmission rate, birth rate, disease-related

death rate, and recovery rate. In the stochastic model, all strains will eventually be

eliminated because the disease-free state is an absorbing state. However, if the popu­

lation size is sufficiently large, it may take a long time until all strains are eliminated.

The dynamics of the deterministic model are compared to the stochastic model. Nu­

merical examples show that coexistence cases exhibited by the deterministic models

may not occur in the stochastic models. The stochastic numerical examples in these

models predict competitive exclusion.

Chapter V summarizes the findings in Chapters II, III, and IV, and describes some

future research work.

CHAPTER II

DETERMINISTIC SIS EPIDEMIC MODELS WITH MULTIPLE

PATHOGEN STRAINS

2.1 Introduction

The principle of competitive exclusion is well-known in the ecological literature. It

states that two or more competing species cannot coexist on a single limiting resource

[46]. The classical model used to illustrate the principle of competitive exclusion is the

Lotka-Volterra competition model. The principle of competitive exclusion has been

demonstrated in epidemic models as well. It has been shown in some continuous-time,

deterministic epidemic models with multiple pathogen strains that the strain with the

largest basic reproduction number outcompetes all others [1, 15, 16, 18, 27, 37, 47].

Howe^'er, there are many exceptions to this principle in multi-species competition

models and in multi-pathogen epidemic models. For example, epidemic models which

include density-dependent host mortality, partial cross immunity, coinfection, or su­

perinfection may lead to coexistence of several strains in a single host population

(e.g., [1, 8, 10, 11, 16, 17, 18, 27, 37, 47, 49, 53, 54, 55, 58, 62, 66, 71]). In addition,

models with two or more species competing in a spatial environment have demon­

strated very complex behavior, e.g., coexistence, extinction, periodicity, and chaos

(e.g., [46, 48, 52, 63, 65, 68, 72]).

The principle of competitive exclusion has not been investigated in discrete-time

epidemic models. Discrete-time models are often directly applicable to time-series

data and, in some cases, may more accurately represent contacts which are restricted

to a specific time or time period. In addition, discrete-time models often exhibit a

wider array of behavior than their continuous analogues. Discrete spatial models or

patch models are one of the simplest types of spatial settings to examine the effects

of spatial heterogeneity on disease persistence. In this investigation, competitive

exclusion and coexistence of multiple pathogen strains within the host population are

studied in discrete-time epidemic, patch models. Our two-patch models are similar

to the ones formulated by Castillo-Cha\'e/, and \ 'akubu [19, 20]. However, in their

models, there is only on(- pathogen strain.

In the next section, the singh* patch epidemic model is formulated and analyzed.

Then, the model is extended to two i)atches. Analytical results for these models are

stated and verifiinl. Numerical results illustrating competitive exclusion and coexis­

tence and future res(>arcli directions an- diseussed in the last two sections.

2,2 Single Patch

2.2.1 Model

The host population is subdivided into several disease states: individuals that are

susceptible 5 and individuals infected by one of n different strains, /fc, fc = 1, 2 , . . . , n.

Assume in a fixed interval of time. At, susceptible individuals may be infected with

strain k with probability \k{t) or individuals infected with strain k may recover from

infection with probability 7fcAt. The model is known as an SIS epidemic model

because there is no immunity to infection; individuals recover and can immediately

become reinfected. A compartmental diagram of the n—strain SIS epidemic model is

given in Figure 2.1.

r- - >

I N /

II

c o

N

1

l2

e^

s;

• • • •

N I

In

Figure 2.1: SIS epidemic model with n pathogen strains. The solid arrows represent the infection of susceptibles and the dashed arrows represent recovery of infected individuals.

The SIS epidemic model tak(>s the form of a, system of did'erenee equations:

S{t + At) = 5 ( 0 ( l - 5 ^ A , ( / ) ) + ^ / , ( 0 7 . A / , \ k=\ / k=l

hit + At) = h{t){l^j,Al) + \,{t)S{t), - = l , 2 , . . . , n , (2.1)

where .s (0) > 0, 7,(0) > 0 for k = 1,2,... ,7i, ELi h{0) > 0, and 5(0) +

Ylk—i '.(O) — ^'^- I i straightforward to see that the total population size is constant,

n

S{t) + J^h{t) = K. k=l

In this basic single-patch model (2.1), there are no births, no deaths, and no

disease-related deaths. These restrictions can be relaxed in some cases. Each strain

is transmitted horizontally by direct contact between an infected individual and a

susceptible one. Infection with one strain confers complete cross protection, that

is. infection with one strain provides protection from infection by another strain.

Therefore, there is no coinfection nor superinfection, that is, no individual can be

infected simultaneously with two or more strains and infection by one strain cannot

be superseded by infection with a second strain. These simplifying assumptions are

made so that when the model is extended to two patches, only the effects of dispersal

on the coexistence or exclusion of multiple pathogens in two patches are examined.

The function Afc(t) in model (2.1) is referred to as the force of infection. It is

assumed that Afe(t) = Xk{h{t)/K) = Xk{ik{t)), where ik = Ik/I< is the proportion

of the host population infected with virus k. The following assumptions are made

regarding A and the parameters jk and f3k iov k = 1,2... ,n.

(i) 0 < Afc(zfc) < pkikAt.

(ii) Xkiik) e C2[0,1], dXk{ik)/dik > 0 and (fXk{ik)ldii < 0 for u G [0,1].

(iii) Afe(O) = 0 and dXk{ik)/dik\i,=o = Pk^t > 0.

(iv) 0 < jkAt < 1 and 0 < f PkAt < 1. k=l

Model (2.1) with assumptions (i) (iv) is a generalization of a single strain SIS epi­

demic model studied by Allen and Burgin [6]. Exam])les of A ; satisfying the above

assumptions inchule

AA(a) = Pkik^t and A,(v:,) = 1 - exp(-/5,/ ,A/.) .

When the force of infection satisfies Xkin) = A^^A/., then Xk{'ik)S/At = PkhS/K is

known as the standard incidence rate of infection [.34]. The standard incidence rate

is frequently the form used in many applications [34].

Associated with each strain A: is its basic reproduction number,

Ik

The basic reproduction number has been defined as the number of secondary infec­

tions caused by one infected individual in an entirely susceptible population [9, 34].

The dynamics of the SIS epidemic model depend on the magnitude of the basic re­

production numbers.

The disease-free equilibrium of model (2.1) is denoted EQ and satisfies EQ =

(5 . II. . In) = (A', 0 , . . . ,0) . There exists endemic equilibria with a single infection,

where 5* > 0, Ij > 0, and Ylik^i k^j - ^ ~ * ' denoted as Ej, e.g., i?i = (5, Ji, 0 , . . . , 0),

where 5 > 0, Ji > 0, and 5 + / i = K. Expressed in terms of proportions, s = S/K

and ij = Ij/K, the endemic equilibrium Ej satisfies

_ - , . , - • , ijjjAt

s + ij = 1 and Xj{ij) = \ -• •

1 — ij

The assumptions (i)-(iv) imply there is a unique positive solution, 0 < ij < 1, to the

latter equation, provided Tlj > 1. Equihbria with two or more strains in the host

population require

i a ^ = A^_, j^k. (2,2) AJ( ;•) Afc(zfc)'

If the force of infection satisfies A; - PiiiAt, I = j , k, then (2.2) implies 7 ^ = TZk- In

general, condition (2.2) will not be satisfied and therefore, an equilibrium with two

or more strains is unlikely. 6

2.2.2 .\iialytical Results

The assumptions (i)-(iv) on the force of inleelion and the initial conditions imply

that .solutions are positivt- and cwist for / > 0. It is clear tluit if 4 ( 0 ) = 0, then

h{t) = 0 for all time t and if 4 ( 0 ) > 0, then 4.(t) > 0 For all time. We state these

results in the folk)wing lemma. Denote a solution of (2.1) as {S{t),h{t),... ,In{t)).

Let X = (.ri, ,r,., r„+i) and R'[+^ = {x € R"+i : x, > 0, / = 1, 2 , . . . , n + 1}. The

solution space of model (2,1) is a subset of R " + \

L e m m a 2 . 2 . 1 . The set.s {} = {x e R'^+i : Z['=! ^^ = ^ } «^^ ^k = {x E n :

•'"A- = 0} for k = 2, ,n + l arc invariant subsets for the solution space of system

(2.1). In additixm. solutions of (2.1) with initial conditions 5(0) > 0 and 4 (0 ) > 0,

A- = 1, 2, , n. are positive for all time t G [At, 2At,...].

Several additional analytical results for model (2.1) are verified. It is shown that if

7 /c < 1 for A- = 1, 2 , . . . , n, then none of the strains persist. However, some susceptible

individuals always persist. Finally, the strain with the largest basic reproduction

number is the dominant strain, provided it is greater than one. This latter result

demonstrates the competitive exclusion principle for the discrete-time model.

T h e o r e m 2 . 2 . 1 . Assume in model (2.1) that TZk < 1 for some k, then limt^oo hit) =

0. IfHk < 1 for all k = 1,2,... ,n, then

lim is{t),Tlk{t)] ={K,0). (2.3)

Proof Apply assumption (i) and the fact that S = K - J2%i h - ^"^ ^ h- Then,

Xk{ik)S < PkikSAt < i^khil - Ik/K)At.

Using this inequality in (2.1) and dividing both sides by K, yields

ik{t + At) < ik{t){l - ik^t + l5kAt[l - ik{t)]) = g{ik{t)).

where g{x) = x{a - hx) for x = ik{t), a = 1 + [Pk - lk]^t and b = ^k^t. Note that

0 < a < 1 and g'{x) > 0 for x e [0,a/(26)). The solution ik{t) can be compared to

the solution of the did'erenee e(|uat,i()n,

u{t + Al)=g{u{t)).

Let (/(O) = ?^(0) < 1. Since 0 < g{.r) < x for ,/; G {0,a/b) and a/b > 1, it follows that

l im,^ , u{t) = 0. Also, ,,{At) < giikW) = g{uH))) = u{At). But u{At) < a/{2b)

and g is increasing on the interval [0,a/{2b)) so that

ik{2At) < giuiAt)) < g{u{At)) = u{2At).

By induction it follows that

ik{t + At) < u{t + At)

for t = luAi. m = 0,1,2 Hence, limt^oo4(^) = 0 from which the conclusion

of the theorem follows: limt^oo4(^) = 0. In addition, if limt_^oo4(^) = 0 for A; =

1,2, , n. then linit^-^, S{t) = K. D

In the proof of Theorem 2.2.1, it can be seen that the number of infected individu­

als decreases monotonically to zero when Hk <1. It is of interest to note that the Ja-

cobian matrix for the system ( 4 , 4 , • • • iln) evaluated at the disease-free equiUbrium

EQ is a diagonal matrix, J — diag{\-\-\fi\ — '^\\At, l-|-[/32 —72]At,... , l + [/5„ —7„]At).

Hence, £"0 is locally asymptotically stable if TZk < 1 for all A; = 1, 2 , . . . , n.

Theorem 2.2.1 applies to more general models with disease-related deaths. If

c\kAt denotes the probability of death due to strain k and 0 < (7^ + ak)At < 1, then

Theorem 2.2.1 with basic reproduction number satisfying

Pk Ilk = ^ 7 — < 1

Ik + Ctk

applies to the following SIS epidemic model with disease-related deaths,

/ n \ n

S{t + At) - S{t)ll-J2Xk{t)]+J2lk{thkAt \ k=l / i=l

Ik{t + At) = 4 ( t ) ( l -7 fcAt -a fcAt ) - f Afc(t)S(t), A; = l , 2 , . . . , n (2.4)

and to the SIR epidemic model with disease-rc-latcd <leat,lis,

S{t + At) = S(t)ll-J2h{t) \ k=l /

Ik{t + At) = 4 ( / ) ( l - l A . A / - n , A t ) + A,(t)5(0, A: = 1,2,...,,;, (2.5) II

R{t + At) = R{t) + Y,lklk{n^t. k = l

In the SIR model (2.5), state 7?(/) is the number of immune individuals at time t;

individuals develop immunity to strain A: after recovery. For models (2.4) and (2.5),

^^(0 + T.'Li hit) < K and S{t) + E L i h{t) + R{t) < K, respectively.

Lemma 2.2.2. Soltitions to model (2.1) satisfy l imsupj^^ ^(t) > 0.

Proof. Suppose limj^oo •5(0 = 0. Then it follows from the equation for S in (2.1)

that n

lim V4( t )7fcAt = 0. t->-oo ^—'

fc=l

Since 7/i.Af > 0 from assumption (iv), it follows that limt_>.oo4(0 = 0 for A; = 1.2. . , n, a contradiction. D

Theorem 2.2.2. Assume in model (2.1) that the force of infection satisfies Xk{ik) =

Pkik-^t, k = 1,2.. . ,n.

(i) If Km > 1 for some m G {1, 2 , . . . , n), then

n

lim sup V^ 4 ( 0 > 0. *^°° k=i

(ii) IfIZi>l> ma^Xk^i{Tlk}, then

, ^ ^ ( . W . A W , i : W 0 ) ^ ( ^ . ^ ^ ^ . 0 ) . (2,6)

(iii) If IZi > maxfc^i{l,7^fc} o-nd Y\m.t^ooS{t) exists, then solutions to model (2.1)

satisfy (2.6).

Proof For t\w proof of part (i), suppose lim(, oo E L i fk{t) = 0. Then, lim(_,oo 4(i) =

0 tor A' = 1, 2, , /, and lini/ .x' S{t) = A'. For st.rain in, cluxjse a monotone increas­

ing sul.seciuence {tj}^^^ C {lAl}^^ such that I„{lj + At) < I,„\lj). Choose e > 0

sufficiently small sueli that /3,„[l -e] > 7,,, and j sufficiently large such that for t, > r,

S{tj)/K > 1 - e. Then,

'"^/Vlf^^ > 1 + {Pm[l - e] - o,„)A^ > 1.

But then. I,n{tj + At) > /,„(/j) for tj > r, contradicting the choice of the subsequence.

For the proof of part (ii). Theorem 4.3.1 is applied. When Ilk < 1, 4( t ) decreases

monotonically to zero, 4 ( t + At) < Ik{t) for A; = 2 , . . . ,n. Given any e > 0, t can be

chosen sufficiently large, t > T, such that

0 < A' - Ii{t) - Ke < Sit) < K ^ hit).

Then, applying model (2.1) and dividing by K,

iiit)il - -:iAf + /3iAt[l - e - i,it)]) <iiit + At) < zi(t)(l - j^At + /5iAt[l - nit)]).

Let the left and right sides of the above inequality be denoted as gix) and gix), where

gix) = x(a — be — bx) and 'gix) = x(a — bx) (ov x = iiit), a = 1 + [/?i — 71]A/, and

b = f^iAt. The difference equation

uit + At)=giuit)) (2.7)

is a discrete logistic equation with 1 < a < 2, so that for u(0) G iO, a/b), lim uit) =

, where (a — l)/b is the unique positive fixed point of (2.7) [25]. Also, note that b

uimAt) is a monotonic sequence for m = 1, 2, The monotonicity follows because

the values of u are in the region where g is increasing.

Now, e can be chosen sufficiently small such that the difference equation vit+At) =

givit)) is also a discrete logistic equation, 1 < a-be <2. This equation has the same

properties as (2.7), that is, lim vit) = r > 0 and vim At) is a monotonic

sequence, m = 1, 2 , . . . .In addition, given zi(0), e can be chosen sufficiently small such

10

that for ('(0) = «(0) = z(0), the se(|uenees {w(„/.A/)} and {u(77iAt)} for //,, = 1,2,. . . ,

are both either increasing or decn^asing and u(A/) < (a — be)/{2b). This latter

condition ensun-s that, both (/(.r) and 7/(./:) are increasing for :/: G (0,u(A/.)]. Let

/\(0) = e(0) = »(0). Then,

r(A/) = £(r(0)) = gi>m < /(A/) < ^(^(O)) = (77,(0)) = n(At).

It follows that

K2A/) = giiiAi)) < gitiAt)) < i{2At) < giziAt)) = ^(u(Ai)) = u(2At).

B>- induction, it follows that r{in.At) < iiiniAt) < uimAt). Taking the limit,

< liminfii(t) < limsupii(i) < —r—.

b t^oo j^oo b

Because e is arbitrary,

llm^l(t) = —— = 1 - — .

t->oo 0 /Cl

The limit in (2.6) follows.

For the proof of part (iii), define u,fc(t) as follows: 1

Then, Ukit) = Sit)/K, so that

5 lim Ukit) = —, A; = 1,2,... ,n, t^oo A

where S is the limit of S{f). By Lemma 2.2,2 and part (i) of this theorem, 0 < 5 < A'

The above limit implies

l i m M i ± M = /3,A4-7.At + l = c,, k = l,2,...,n. t^oo 4 ( 0 -'

Let e > 0. Choose r sufficiently large such that for t > r,

ick - e)Ikit) < hit + At) < ick + e)4(0- (2-8)

11

Since lim,^oo E L i hit) = K - S > 0, there exists at least one A: siicJi that

lim sup 4 ( 0 = 7/ . > 0. (2.9) t—>CX3

Then,

i''k~()Vk <Vk < in+e)ilk- (2.10)

If r , < 1, then e can be eln)sen su(4i that r ,, + c < 1, which contradicts the second

inequalit>- in (2.10). If c^, > 1, then e can be chosen such that c - e > 1, which

contradicts the first inetiuality in (2.10). Hence, if (2.9) holds, then Ck = 1 which

means 5 = K/'Rk < K and TZk> 1.

lik ^ 1. then 7 l > Uk implies Ci > 1. The inequalities in (2.8) hold with k = 1

and e > 0 chosen such that ci - e > 1. The first inequality in (2.8) can be applied

repeatedly to obtain

4 ( r + 777A/) 4 ( r + (77^-l)A0 4 ( r + A0 > . _ w 4 ( r + ( 7 7 i - l ) A 0 4 ( r + ( 7 n - 2 ) A 0 " ' 4 ( r ) - ^^^ '''

or

W r + m A ^ > (c, - e)™, Ilir)

It follows that limm-,>oo 7i(T-|-7nAt) = oo which is a contradiction to the boundedness

of solutions. Hence, A: = 1 and Ci = 1.

Next, consider j ^ 1. Since IZj < IZi, it follows that Cj < 1. Apply (2.8) with

A; = j and choose e > 0 such that Cj + e < 1. By repeatedly applying the second

inequality in (2.8), we obtain

4 ( r + 777A0 , .m / , ( r ) - ^ ' ^ ^ ' ^ •

Hence, limj^oo hit) = 0 for j 7 1. D

The result of Theorem 2.2.2 (i) applies to the SIS and SIR models (2.4) and (2.5)

with disease-related deaths.

Theorem 2.2.2 is a competitive exclusion principle for 7i strains. The strain with

the largest reproduction number is the dominant competitor (Theorem 2.2.2 (ii) and

12

(iii)). The result in Theorem 2.2.2 (iii) is not as strong as the analogous continuous-

time model since the existence of the limit of 5(0 is recinired. It has been shown for

some discrete-time epidemic models that periodic and chaotic behavior are possible

[5]. However, this type of Ix^havior re(|uired that ii,,At > 1. Theorem 2.2.2 is based

on assumptions (i) (iv) which may rule out i)eriodie and chaotic behavior. In the

next section, the single patch epidemic model is extended to two patches.

2.3 Two Patches

2.3.1 Model

Assume a population inhabits two different regions or patches and a proportion of

the population moA-es randomly between the two patches. Let Sj denote the number

of susceptible individuals in patch j and Ijk denote the number of individuals infected

with strain A' in patch j . The host susceptibility to each strain may differ between

the two patches. When there are double subscripts on a parameter or variable, jk,

the first parameter j always refers to the patch and the second parameter k always

refers to the pathogen strain. The disease parameters depend on the particular patch

j and strain A', 7 / , and Pjk. Let Njit) denote the size of the subpopulation in patch

n

N,it) = S,it) + J2hkit), k=\

and let A^(0 denote the total population size,

2 / n

Nit) = M(0 + N2it) = Y,[ 5,(0 + 5]/,.(o j=l \ k=l

It is assumed that the total population size is constant, A^(0 = A'. The two-patch

epidemic model is formulated as a two-stage model, infection followed by dispersal. A

compartmental diagram of dispersal between the two patches is graphed in Figure 2.2.

During each time interval At, susceptible individuals become infected or recover,

then a proportion of the population in each patch is assumed to move to the adjacent

13

Patch 1

Si

111

Il2

<

^ V

^ V, "

pi

P2

pi l

p21

pi2

p22

>» T?

^ >»

^ ^

Patch 2

S2

l21

I22

Figure 2.2: Dispersal of susceptibles and infectives between two patches. The solid arrows represent the dispersal probabilities out of patch 1 and the dashed arrows represent the dispersal probabilities out of patch 2.

patch. The model is a system of difference equations:

5,(t-fA0 = il-pj)fjit)^pifiit),

Ijkit + AO = (1 - Pjk)fjkit) + Pikfikit), (2.11)

where

4(0 = 5,(0 I - X ; A J , ( 0 +J]4fcW7,fcAt k^l k=l

fjkit) = Ijkit)il-Jjk^t) + X,kit)Sjit) (2.12)

for j , / = 1, 2, j 7 /, and A; = 1,2,... , ri, with initial conditions, 5j(0) > 0, Ijki^) > 0,

and 2

E i=i L

5.(o) + E^^-^(o) k=i

14

K.

It is assumed that the probabilit,\' of contracting an infection and rec(n'ery from infec­

tion may differ between the two regions, that, is, the parameters Xjk and ^jk depend

on patc4i j . This difference accounts for the xariat.ion in transmission and recovery

which may depend on the en\ironment.

Model (2.11) is a simple metapopulation model [33], where subpopulations of the

lu)sts represent the pat.(4ies. Such typ(\s of models are applicable to wildlife diseases

[39]. For example, haiita.\-iruses are easily spread by dispersal of rodents between

different geographic regions (se(> e.g., [69]).

Model (2.11) is similar in form to a two-stage, two-patch SIS epidemic model

formulated by Castillo-Chavez and Yakubu [19, 20]. However, only one strain was

included in their model. In [19], the effect of susceptible or infective dispersal on per­

sistence of the disease was studied. A similar type of discrete-time, two-stage growth

and dispersal model (without infection) was formulated by Doebeli and Ruxton [23].

The assumptions (i)-(iv) on the force of infection need to be extended for model

(2.11). Assume the force of infection Xjkit) = Xjkiljkit)/Njit)) = Xjkiijkit)), where

ijkit) is the proportion of individuals in patch j infected with strain A;,

''^'~ N,it)'

j = 1,2 and k = 1,2,... ,n. The force of infection and the parameters satisfy

assumptions similar to those stated in (i)-(iv). For j = 1,2 and A; = 1, 2 , . . . , 77, it is

assumed that

(i)' 0 < Xjkiijk) < /3jkijk^t

iii)' Xjkiijk) G C^[0,1], dXjkiijk)/dijk > 0 and d^Xjkiijk)/di% < 0 for ijk G [0,1].

(iii)' XjkiO) = 0 and dXjkiijk)/dijk\ijk^Q = ^jfeAt > 0,

(iv)' 0 < 7,fcAt < 1 and 0 < E E ^]k^t < 1. j=ik=i

The parameter pj is the probability a susceptible individual moves from patch j

to / and Pjk is the probability an individual infected with virus k moves from patch

j to /, j / /. One more assumption is added regarding these movement probabilities:

15

(v) 0 < pj < 1 and 0 < p^^ < 1, y = 1, 2 and A; = 1, 2 , . . . , n.

It is straightforward to check that the total population size is constant,

N{t + Ai) = Nif) = K.

Hence, solutions to (2.11) satisfy ^ ; _ | [Sjit) + E l ' .i hdt)] = A- The basic repro­

duction number for strain A- in patch j e(iuals

•p __ Pjk l^jk - •

^Jk

The basic reproduction numbers are important determinants of persistence for a par­

ticular strain A'.

Some of the results for model (2.1) can be generalized to model (2.11). However,

unlike the single-patch model, there are cases of coexistence for multiple pathogen

strains when infected individuals disperse between the two patches. These results are

illustrated in the numerical examples. First, the extension of the analytical results

on competitive exclusion to two patches are stated and verified.

2.3.2 Analytical Results

It can be shown that solutions of (2.11) are nonnegative, bounded, and exist for

all time t > 0. The following lemma states these results more precisely. Denote a

solution of (2.11) as

(5i(0, 52(0, / l l (0 , ^2l(0. • • • : hnit), hnit)).

Solutions lie in a subset of R ""*" . Note that if strain A; is initially zero, 4fe(0)+4/!c(0) =

0, then it remains zero for all time.

Lemma 2.3.1. The sets f] = {x G R+"+^ : Yli.T ^i = ' '} andfli = {x G 1 : X21+1 + X21+2 = f

for 1 = 1,... ,n are invariant subsets for the solution space of system (2.11). In ad­

dition, solutions of (2.11) with initial conditions Sjit) > 0 and IjkiO) > 0, j = 1,2,

A; = 1, 2 , . . . ,n, are positive for all time t G {0, At, 2At, . . . }.

16

Theorem 2.3.1. Assume in model (2.11) that TZik < 1 an.d n.,, < 1 for some k,

then lim,_.,.,.[4^(t) + 4fc(t)] = 0. / / R,^ < 1 for j = 1,2 and all k = 1,2,... , n, then

lhnU(/) + 5,(0,X]X^4.(0)=(A',0).

Proof. Let (/ , = max {1 + H^^k - 7iit)At, 1 + ( 2fc - 72fc)At}. Since the basic repro­

duction numbers for strain A' in patches 1 and 2 are less than one, 0 < a < 1.

Then,

hkit + At) + E,it + At) = / i ,(t) + f2kit).

By assumption (i)', it follows that

hkit + At) + hkit + At) < 4/c(t)(l + [/3ik - 7ifc]A0 + 4^(0(1 + [P2k - 72fc]A0

< Ok[hkit)+l2kit)].

The conclusions follow. D

Theorem 2.3.1 applies to SIS and SIR two-patch models with disease-related

deaths, where Qj^At is the probability of death due to strain k in patch j , 0 <

["fjk + o:jk]At < 1. These models are extensions of the SIS and SIR models given by

equations (2.4) and (2.5), where the basic reproduction numbers are

^ _ f3jk l<-]k — ; •

Ijk + CXjk

It is of interest to note that the Jacobian matrix for model (2.11) with states

(5i, 4 i , - 21, • • • , hn, hn) evaluated at the disease-free equilibrium has the form,

/ i _ . . _ . „ 4, A. ... A \

0

J= 0 0 Do ••• 0

0 0 Dr.

17

where subniatric'cs

D,= (1 - pyk)il + [l^Xk - l^k\Al) p;kil + [/32fc - 72fe]At)

Pikil + [Hik - 7ifc]A/) (1 - P2fc)(l + [/32fc - 72;.]At)^

A' = 1,2, .. , /;. Matrix J is a block diagonal matrix. Hence, local stability of EQ is

determined by 1 ~ p\ —pi and the eigen\alues of Dk, A: = 1,2, ,n. For example, if

there is only dispersal of susceptibles, pi > 0, p> > 0, and jjjk = 0, then tlu' disease-free

equilibrium is locally asymi)tt)tically stable if all of the basic reproduction numbers

•Rik< 1.

Lemma 2.3.2. Solutions to system (2.11) satisfy

limsup[5i(0 + 52(t)] > 0 . t—^oo

In addition, if pi > 0 and p2 > 0, then

(2.13)

limsup5i(t) > 0 and lim sup 52 (t) > 0.

Proof. Suppose limt^oo[5'i(t) + 52(t)] = 0. Then it follows from the equations for Si

and 52 in (2.11) and (2.12) that

lim (1 - PJ) 5^ 4fc(t)7;fcAt + PiJ2 Iikit)lik^t = 0 fc=i fc=i

for ], 1 = l,2,j ^ I. Since 0 < p, < 1 and jjkAt > 0 from assumptions (iv)' and (v),

it follows that

lim 4fe(t) = 0

for j = 1, 2 and A; = 1, 2 , . . . , n, a contradiction.

Next, suppose pi > 0 and P2 > 0. Solutions to (2.11) satisfy

5i(t+At) >P2S2it) 1 - J2 f^2kAt k=l

> 0 and 52(t+A0 > P i 5 i ( 0 1-J] /JuAt k=l

>0.

The inequalities above imply limt^^o 5'i(t) = 0 if and only ff limt„,oo 52(t) = 0. But

this contradicts (2.13). ^

18

Theorem 2.3.2. Assume m model (2.11) that thr force of infection satisfies X-jki'jk)

dj^.ijkAt for J = 1, 2, A: = 1, 2,. , 7;. an.d lim,,^^ N^it) = A ,- exists for :j = 1, 2.

(i) If 'Ri,„ > 1 and R^^,,, > 1 for some ni G {1, 2 , , . . , //,}, then

2 71

limsupj]^4^(f) >0. /^<x>

i=\ k=\

(li) If'Rji > 1 > max;,^i{7^,4 for j = 1,2, then

l i insup^4i(0 > 0. i—>oo

i = i

(m) Ifpi > 0, p2 >0,Pjk = Oforj = l,2,k = l,2,... ,n, IZji > max^^i {1, IZjk} , j =

1.2. and lim,^-^. 5j(t) = Sj exists for j = 1,2 then solutions to model (2.11)

satisfy

lim ISjit),hiit),Y^h,it)] = (5„A^,-5„0). k=2

Proof For the proof of part (i), suppose limt^^ Yl"^-^! E L i hkit) = 0. Then, limt^oo['S'i(t)+

52(t)] = A' and limt^oo 5,(t) = limj^oo A^j(t), j = 1,2, so that linij^oo 5j(t)/Aj(t) =

1, j = 1, 2. Since limt^oo ^7=1 hmit) = 0, choose a monotone increasing subsequence

{ti}^, C {mAt}-^i such that E''j=ihmiti + At) < Y.]=ihmiti)- Then, apply (2.11),

4 m ( t / + A 0 + 4 m ( t i + A 0 = flmitl) + f2mitl)

= hmitl)+l2mitl)+hmitl)At

S2itl)

^'""Niiti) ^'"^

+ l2miti)At /5: 2m A^2(t;)

- 7 2 r i

Choose sufficiently large / and sufficiently small e > 0 so that ti > r implies Sjiti)/Njiti) >

1 - e and /3jm(l - 0 > 7jm, for j = 1, 2. Then, for t; > r,

4 m ( t ( + A t ) + 4 m ( t ; + A t ) > 4 m ( t ; ) + 4 m ( t i ) ,

contradicting the choice of the subsequence.

19

For the proof of part (ii), apply Theorem 2.3.1, which sliows that 4^( t + At) +

likit + At) for A- = 2 ,7; api)roa.ches zero monotonically. Sui)i)ose, in addition,

lim,^,,,^, V-_. J 4 i ( t ) = 0. Then, the method in the proof of part (i) of this theorem can

be applied tt) obtain a contradiction.

For the proof of part (iii), define Uji,.{t) for A; = 1, 2 , . . . , 77 as follows:

1 'OA-(0 - !\,kAt

Then Uj,it) = 5,( / ) /Aj( t) so that

hkit + At) + jjkAt - 1 hkit)

SJ lim 7;j^,(t) = - ^ = i 'j , A; = 1,2 t^^ - w jY^ n.

From Lemma 2.3.2 and Theorem 2.3.2 (i) {p^k = 0), it follows that 0 < s, < 1 for

J = 1,2. so that limt-^ooELi-^jfclO = A' - Sj > 0, for j = 1,2. The above limit

implies hkit + At)

i ™ T u\— = f^jkAtSj - jjkAt + 1 = Cjk-t^oc hi^(t)

The remainder of the proof uses the same arguments as in Theorem 2.2.2 (iii). D

The result of Theorem 2.3.2 (i) apphes to SIS and SIR two-patch epidemic models

with disease-related deaths. Theorem 2.3.2 (iii) is a competitive exclusion result

for the case when only susceptibles disperse. It will be shown through examination

of the equilibria and numerical examples that when infectives disperse, competitive

exclusion may not occur; coexistence is possible. To illustrate the differences between

dispersal of infectives and susceptibles, the equilibria for the two-patch model (2.11)

are calculated for two cases. In the first case, it is assumed there is only dispersal

of infected individuals, model (2.11) when pi = 0 = p2 and Pjk > 0, j = 1,2 and

A; = 1, 2 , . . . , 77. Such a case might be reasonable in a disease such as rabies, where

rabid or infectious individuals exhibit erratic movements and are more likely to leave

their home territory than susceptible individuals. The second case assumes there is

only movement of susceptibles, model (2.11) when Pjk = Q, j = 1,2, k = 1,2,... ,n,

Pi > 0, and p2 > 0. For diseases where the illness results in morbidity, the second

case may be more biologically reasonable.

20

2.3.3 Infecti\e Dispersal

Assume pi = 0 = p^. Pjk > 0, for j = 1, 2 and A: = 1, 2 , . . . , 77, and

'V = ^ . . - ^ A / , = /Vv,.A/,.

This t\-pe of dispersal is referred t.o as "disease-enhanced dispersion" by Castillo-

Chavez and \akubu [19]. In their single-strain model, the population grows geo-

metricall,\- and for some parameter regions, then- exists a locally stable disease-free

equilibrium and two endemic equilibria. One of the endemic equilibria is locally

asymptotically stable (bistability). Thus, dispersal can enhance persistence of the

disease.

In our model, there exists a disease-free equilibrium,

^ o : 5i -f 52 = A'l -I- A'2 = K and h^ = 0,

for j = 1, 2 and A' = 1, 2 , . . . , n (superscript / means infected individuals are dispers­

ing). There exist endemic equilibria where individuals can only be infected with one

strain, p / . ^ _ rfc(7^2, - 1) ^ _ (Teife - 1)

^k D{ '

rkin2k - i)inik -1) J in2k - i)inik -1) hk = Yhr ' ' 2k Di ' ' " Di

and hi = 0, li-k where r^ = P2fc/pifc and Di = irkIlikiTl2k - 1) + ^2^(7^1^ - 1)) /K.

The equilibrium Ej. is feasible only if Hik > 1 and 7l2k > 1. Existence of an equi­

librium with a single strain requires that the basic reproduction numbers in both

patches be greater than one.

Equilibria where the population is infected with two or more strains are also

possible. However, explicit expressions for these equilibria are difficult to calculate.

In a single patch model, equilibria with two or more strains are not stable when the

conditions of Theorem 2.2.2 are satisfied. With two patches, however, the numerical

examples illustrate that two or more strains can persist in the host population in both

patches when infected individuals move between the patches. Persistence can occur,

21

it, for example, strain A- dominates in patch 1 and strain 1,1 ^ k, dominates in patch

2, Tv^, > ma.Xj^^. {1,7^1 J and Tvj/ > max^//{1,';^,._, J . This is demonstrated in the

numerical (wamples; a stable equilibrium E[.2 exists.

2.3.4 Susceptible Dispersal

.Assume p,/,, = 0 for j = 1, 2, A' = 1, 2 , . . . , 7/, pi > 0, P2 > 0, and

^3k = / ^ . A - t ^ A / = i\,,i,,At.

This type of dispersal is referred to as -diseasi^-suppressed dispersion" by Castillo-

Cha\-ez and \ 'akubu [19]. For one-way, disease-suppressed dispersion from patch 1 to

2. they showed that there are parameter regions for their model capable of supporting

two endemic equilibria and a locally asymptotically stable disease-free equilibrium.

One endemic equilibrium is locally asymptotically stable. Hence, in their model,

dispersal supports persistence.

For our model, the disease-free equilibrium satisfies

5 o _ V2K ^ _ Pi A' - _ -C'o • Ji — , , ^2 — ; , -i]k — U

Pi + P2 Pi + P2

for j = 1,2 and k = 1,2,... ,n (superscript 5 means susceptible individuals are

dispersing). There exists endemic equilibria with only one strain present. Strain k

may be present in patch 1,

Elk • Si - —g-, hk - f^s , S2 - jTg-, hk - 0, (2.14) ^Ik ^Ik ^Ik

and Iji = 0 for j = 1,2 and / 7 k, where r = P2/P1 and Df^ = (rT^u--|-1)/A',

EquiUbrium Ef/^ is feasible ii IZik > 1- Strain k may be present in patch 2,

D2k '• S2 = jr^-, hk = — j ^ — , ^i = —^, hk = 0, (2.15) ^2k ^2k ^2k

and Iji = 0 for j = 1,2 and / ^ A;, where D|). = ir + IZ2k)/K- Equilibrium A'f is

feasible ifIZ2k > 1- Strain k may be present in patch 1 and strain I in patch 2,

pS . q _ ^ f _ ^i^lk - 1) ^ 1 7 _ I l 2 l - l . ^ l k , 2 l • ^l - 755 ' -'Ife - f^ ' - 2 - 7^5 , hk - -J^ , ( --LOj

-^lfc,2( ^lk,2l ^lk,2l ^lk,2l

22

and 4 , = 0 for q^ k and 4 , = 0 for y / /, where Df _2; = (' ifc + ^2/)/A. The

equdibrium E^^.,^ is feasible" if 7 ,,,, > 1 and R-ii > 1.

Equilibria with two or more strains in one pal(4i re(|uire that

A , A, SJ

' J _ " J

Rjk Rji

k / /, which implies 7 ^ , = 7^,, > 1. Thus, with only susceptibles dispersing,

it is unlikely that there will be (X)existence of two or more strains (as shown in

Theorem 2.3.2 (iii)).

2.4 Numerical Examples

Some numerical examples are presented which complement the analytical results.

In the numerical examples, it is assumed that the force of infection is of standard

type,

Afc = Pkh-At and Xjk = PjkijkAt.

It is assumed that the total population size in the one-patch model is A' = 100 and the

total population size in the two-patch model is K = 200. In addition, it is assumed

that there are two pathogen strains, n = 2. The following local stability results were

verified by calculating the eigenvalues of the Jacobian matrix at each of the feasible

eqmlibria. Although the stability results are only verified to be local, the numerical

simulations indicate that they may be global.

First, consider the single patch model (2.1), where T i = 2 and 7 -2 = 1-5 iPjAt =

0.1, j = 1,2, 7iAt = 0.05 and 72At = 1/15). Strain 1, with the largest basic

reproduction number, is the dominant competitor. Solutions converge to the stable

endemic equilibrium Ei :

l im(5(0,4(0,^2(0) = (50,50,0). (2.17) t—KX

Solutions to model (2.1) with four different sets of initial conditions are graphed in

the 4 — 4 phase plane in Figure 2.3 (a). All solutions converge to (71,72) = (0-5,0),

the equilibrium given by (2.17).

23

0,4 0.6 i „( t)

0,4 0,6 S,{t)

0,4 0,6 i„{t)

Figure 2.3: Solutions to (2.1) and (2.11) are graphed in the phase plane for four different sets of initial conditions. In (a), solutions to model (2.1) with parameters piAt = 0.1 = P2At, 7iAt = 0.05, and 72At = 1/15, K = 100, IZi = 2 and 7 2 = 1.5. In Figures (b), (c) and (d) solutions to the epidemic model (2.11) are graphed in the hi-iu phase plane with parameters (b) PjkAt = 0.1, jjjAt = 0.05 and 7(fcAt = 1/15, j,k,l = l,2,l^k, K = 200, pi = 0.05 = P2, Pjk = 0, i, Ac = 1,2, 7^ll = 2 = 1122, and Tlu = 1.5 = 7^2l, (c) PjkAt = 0.1, jjjAt = 0.05, 7;feAt = 1/15, j,k,l = 1,2, I ^ k, K = 200, Pjk = 0.05, j . A; = 1, 2, pi = 0 = P2, Tin = 2 = 7l22, and Tl^ = 1.5 = Tlzi, (d) pjkAt = 0.1, 7jjAt = 0.05, ^ikAt = 1/9, j,k,l = 1,2, I ^ k, K = 200, pj = 0, Pjk = 0.05, j , A; = 1, 2, 7^u = 2 = 1122, and Tlu = 0.9 = 7^2l.

Next, consider the two-patch model (2.11). Suppose

Tin = 2 = 7 22 and Tin = 1.5 = K 21 (2.18)

i(3jkAt = 0.1, ^jjAt = 0.05, 7,fcAt = 1/15, j,k,l = 1,2, I ^ k). In patch 1, strain

1 has the largest basic reproduction number and in patch 2, strain 2 has the largest

24

basic reproduction number:

7 ,1 > max {l,7li2} Jind R-,-, > max {l,7^2i} •

We consider two cases, susceptible dispersal and infective dispei'sal. If only sns-

ceptible individuals disperse, pi = 0.05 = P2 and pj^ = 0, j,k = 1,2, the basic

reproduction numbers detinniine the outcome. There are nine feasible equilibria

i^o ^ E\k^ E-ik^ E\'k.-2h ^'^^ = 1,2)- flowever, of th(>se nine equilibria, only equilib­

rium Afi._,._, is locally asymptt)tically stable. Figure 2.3 (b) shows that solutions

( ' i i ( t ) . / i2(0) -^ (0.5,0). In addition, (72i(0,'«22(t)) ^ (0,0.5); solutions converge

to the equilibrium solution 7? ^ 22 given by

(5i, 52, 4 i , 721, 4 2 , 4 2 ) = (50, 50, 50,0,0, 50).

Competitive exclusion occurs in each patch; only the dominant strain persists. How­

ever, there is persistence of both strains in the two patches. Now, suppose only

infected individuals disperse, pjk = 0.05, j,k = 1,2, and pi = 0 = p2. Assume the

basic reproduction numbers are the same as in (2.18). In this case, there are four

feasible equilibria iEQ,E{,E.2, and i?(2)- 0^ these four equilibria, only equilibrium

EI2 is locally asymptotically stable, where both strains persist in both patches.

(5i , 52, 4 i , /21, 42 , 42) ~ (57.7, 57.7, 22.7,19.6,19.6, 22.7).

Solutions in the 4 i — 12 phase-plane converge to

( ^ l l ( 0 , n 2 ( 0 ) ^ (0.227,0.196).

See Figure 2.3 (c). Coexistence of all strains occur in both patches when infected

individuals disperse.

We consider another example which is related to the last one. The parameter

values are assumed to be the same as in Figure 2.3 (c), except 7 a A t = 1/9, I ^ k, so

that

7^n = 2 = 7^22, and 7^l2 = 0.9 = 7^2l.

25

For these parameter \-aIues, there exist only two feasible e<iiiilibria {El^ and El^). Of

these two ec[uilil)ria, only E{.-, is locally asymptotically stable,

(5i , 52, 4 i , 4 i , 42 , 4-2) ~ (T2.7, 72.7,17.2,10.1,10.1,17.2).

Solutions are shown to c-onverg(> in the in - 712 phase-plane to

( M I ( / ) , / I 2 ( 0 ) - > (0.172,0.101)

in Figure 2.3 (d). There is still persistc-nce of both strains in both patches with

infective dispersal.

The examples in Figure 2.3 (b)~(d) illustrate that infective dispersal (disease-

enhanced dispersion) is capable of increasing disease persistence, provided that the

disease persists in at least one patch. Susceptible dispersal (disease-suppressed dis­

persion) does not increase disease persistence. However, we show that these results

depend ^•ery much on the magnitude of the basic reproduction numbers and the dis­

persal probabilities.

To illustrate the importance of the magnitude of the dispersal probabilities, we

consider again the examples described in Figures 2.3 (b) and (c). The values of

the parameters are the same as in these figures with the exception of the dispersal

probabilities. The basic reproduction numbers satisfy (2.18). Strain 1 dominates in

patch 1 and strain 2 dominates in patch 2. We consider patch-dependent dispersal,

where p n = pi2 and P21 = P22. In Figure 2.4, the stable equilibria are plotted in

Pik — P2k parameter space; either .strain 1 persists in both patches, strain 2 persists

in both patches, or both strains persist in both patches, coexistence. It can be seen

that if dispersal out of patch 1 is greater than dispersal out of patch 2, pik » P2t,

then strain 2 dominates in both patches, but if the reverse occurs, then strain 1

dominates in both patches; competitive exclusion occurs. The region of coexistence

appears small compared to the regions where there is competitive exclusion. But

if the dispersal probabilities are relatively small, of the same magnitude or smaller

magnitude than the transmission and recovery parameters, PjkAt and 7j/oAt, then

the coexistence region is comparatively large.

26

08

Strain 1

Strain 2

0,4 0,6 0,8

0,1

0 08

cPoe

0.04

0,02

0,04 0,06 Pi rPi2

Figure 2.4: The stable equilibria are identified in pik — P2k parameter space. There is patch-dependent infective dispersal, pn = pi2 andp2i = P22. The remaining parameter values are the same as those in Figure 3.1 (b) and (c). The basic reproduction numbers are T^n = 2 = Tvoo and IZu = 1.5 = 7?.2i. In the figure on the left, pik x p2k G [0.1] X [0,1] and in the figure on the right, the parameter region is magnified so that the coexistence region is visible, pu x p2k G [0,0.1] x [0,0.1]. Regions where strain 1 persists in both patches (Strain 1), strain 2 persists in both patches (Strain 2), or coexistence, persistence of both strains in both patches (Coexist), are identified. The coexistence region bounded by sohd curves represents a stable coexistence equilibrium. However, the coexistence region bounded by a dashed curve represents existence but not stability of the equilibrium. In this region, there are locally stable equilibria with strain 1 or strain 2 persisting.

We compare infective dispersal with infective and susceptible dispersal. Suppose

the parameter values are the same as in the last figure. Figure 2.4, except that both

susceptibles and infectives disperse:

Pi=Pik and P2=P2k, A; = 1,2.

Figure 2.5 shows the stable equilibria in pi - P2 parameter space. Either strain 1

persists in both patches, strain 2 persists in both patches, or both strains persist

in both patches, coexistence. It is interesting to note that the coexistence region in

Figure 2.5 is smaller than in Figure 2.4. Infective dispersal enhances persistence more

than both susceptible and infective dispersal.

27

0.08

0,06

0,04

0,02

Coexisty

Strain 1 y/

•

strain 2

0.02 0.04 0,06 0.08 0.1 Pl

Figure 2.5: The stable equilibria are identified in pi - p2 parameter space. There is patch-dependent susceptible and infective dispersal, pi = pn = pi2 andp2 = P21 = P22. The remaining parameter values are the same as those in Figure 2.4. In the figure on the left, Pl X p2 G [0,1] x [0,1] and in the figure on the right, the parameter region is magnified so that the coexistence region is visible, pi x p2 G [0,0.1] x [0,0.1]. Regions where strain 1 persists in both patches (Strain 1), strain 2 persists in both patches (Strain 2), or coexistence, persistence of both strains in both patches (Coexist), are identified.

In the next examples, we do not assume that in each patch there is a dominant

strain. We decrease the magnitude of the basic reproduction numbers in patch 2 so

that

7 2l =0 .5 = 7 22. (2.19)

The basic reproduction numbers in patch 1 are the same as in the previous examples:

7^ll = 2 and 7li2 = 1.5. (2.20)

In the absence of dispersal or with susceptible dispersal, Pj = 0.05 and Pjk = 0, strain

1 persists in patch 1 and neither strain persists in patch 2. However, when infectives

disperse, Pjk = 0.05 and Pj = 0, we obtain the opposite of the expected outcome;

persistence is suppressed. In both patches, there is disease extinction. Solutions

converge to the disease-free equilibrium,

(5i, 52, 111, /21,42, /22) = (0, 200,0,0,0,0)

28

(see Figure 2.6 (a)). The population size in patch 1 approaches zero, limt^ooM(0 =

0 and all individuals eventually reside in the patch with the least disease impact,

patch 2. This latter result depends very much on the assumption that there is only

dispersal of infectives. If both susceptibles and infectives disperse, pj = 0.05 = Pjk,

disease extinction does not occur; strain 1 persists in both patches (see Figure 2.6

(b)). Disease persistence is enhanced. In Figure 2.6 (b), the locally stable endemic

equilibrium is

(5i, 52,4i, /21, 42, /22) « (69.5,80.2,30.5,19.8,0,0).

200

S150 •5

">100»

50

/

•ff^

Patch 1 Patch 2

(a)

200

,>150

">100

50-

- - . , , . _ _

Patch 1 Patch 2

(b)

..^:^>^ 200 400 600

Time 800 1000 200 400 600

Time 800 1000

Figure 2.6: Solutions to model (2.11) with parameter values AiAt = 0.1, ^yiAt = 0.075 and ^2iAt = 0.025 = ^22At, 7jfcAt = 0.05, f, k = l,2,K = 200, and pjk = 0.05, j , k = 1,2. The basic reproduction numbers are TZn = 2, 72-12 = 1.5, and 72-21 — 0.5 = 7 -22. The initial conditions satisfy 4A;(0) = L i, A; = 1, 2 and 5^(0) = 98, j = 1,2. In (a), Pl = 0 = P2, only infectives disperse, limt_>oo A^i(0 = 0, and hmt^oo A 2(t) = 200. In (b), Pj = 0.05, j = 1, 2, infectives and susceptibles disperse, solutions converge to a stable endemic equilibrium.

In the last examples, the impact of the magnitude of the dispersal probabilities

on disease persistence is illustrated. Both susceptibles and infectives disperse and

the dispersal probabilities pj and Pjk are assumed to be patch-dependent, that is,

Pl = Pik and P2 = P2fc for strains A; = 1,2. Suppose the parameters are the same

as in Figure 2.6 (b), except the dispersal probabilities out of patch 1 are reduced to

Pl = 0.01 = pik. The basic reproduction numbers satisfy (2.19) and (2.20). Strain

29

1 still persists in both patches, (41,-^21) = (76.65,9.55). This is illustrated in the

phase-plane diagram in Figure 2.7 (a).

Figure 2.7: Solutions to model (2.11) graphed in the 4 i - - 21 phase plane for four different sets of initial conditions. Parameter values are the same as Figure 2.6, /3iiAt = 0.1, ppAt = 0.075 and /52iAt = 0.025 = /322At, jjkAt = 0.05, j , Ac = 1,2, and K = 200. Basic reproduction numbers are Tin = 2, IZ^ = 1.5 and IZ2i = 0.5 = 7 22. There is patch-dependent dispersal with p2fc = 0.05 = P2, k = 1,2. In (a), the dispersal out of patch 1 is less than the dispersal into patch 1: pi = 0.01 = pi^, A; = 1,2. An eridemic equilibrium is approached with strain 1 persisting in both patches: 1; ( 4 i , 4 i ) = (76.65,9.55). Strain 2 does not persist in either patch. In (b), the dispersal out of patch 1 is greater than the dispersal into patch 1: pi = 0.2 = Pik, A; = 1, 2. The disease does not persist; solutions converge to the zero equilibrium.

Next, suppose all of the parameters are the same as in Figure 2.6 (b), except the

dispersal probabilities out of patch 1 are increased to 0.2, pi = 0.2 = pik- In this case,

both patches become disease-free (Figure 2.7 (b)). There is greater dispersal out of

patch 1 into patch 2, where the reproduction numbers are less than one. In patch 2,

individuals are recovering faster than becoming infected and the greater proportion

of individuals reside in patch 2.

The results of the last two examples can be seen in pi - P2 parameter space,

Pl = pifc and P2 = P2fe, A; = 1,2. Regions in parameter space are identified where

there is a stable equilibrium with strain 1 persisting in both patches or a stable

disease-free equilibrium (Figure 2.8). When there is greater dispersal out of patch 1,

Pl ^ P2, then there is disease extinction.

30

1

0.8

0.6

M

0.4

0.2

n

Strain 1

•

•

^ ^

•

-

^^^^^

No Strains

0.2 0.4 0.6 0.8

Figure 2.8: The stable equilibria are identified in pi - p2 parameter space. The dispersal probabilities are patch-dependent, and satisfy pi = pu = pi2 and p2 = P21 = P22. The remaining parameter values are the same as those in Figures 2.6 and 2.7. Regions where strain 1 persists in both patches (Strain 1) or neither strain persists, where the disease-free equilibrium is stable (No Strains), are identified.

2.5 Summary

Competitive exclusion is studied in discrete-time SIS epidemic models. The ana­

lytical results for the SIS epidemic model show that the competitive exclusion princi­

ple generally holds for a single patch and for a two-patch model when only susceptibles

disperse. Such results have been demonstrated for a single patch in continuous-time

epidemic models (see e.g., [15]), but have not been demonstrated for the discrete-time

epidemic model. The strain with the largest basic reproduction number is the one

that persists, provided it is greater than one. Some of the simplifying assumptions

that lead to competitive exclusion in these models are that there are no births and

deaths, no disease-related deaths, no coinfection, and no superinfection. Some of the

disease extinction results can be extended to include disease-related deaths, but the

competitive exclusion results are unlikely to hold with coinfection, superinfection or

density-dependent mortality [1, 10, 11, 37, 49, 54, 55, 58, 62, 66]. For example, in

a discrete-time, two-patch epidemic model, Castillo-Chavez and Yakubu [19] showed

31

that if there is get)metric growth and susce|)t.il)le dispersal, then disease persistence

can be enhanced. These other assiinii)tions need further consideration in future re-

sea rt-h.

The numerical examples illustrate that, the magnitude of both the basic reproduc­

tion numbers and the dispersal probabilities are important in deciding whether there

is comi)etiti\e exclusion or ct)existence. If a single strain A: dominates in both patches,

then the numerical simulations indicate that it is only possible for strain k to persist

in both patches. On the other hand, if strain 1 dominates in one patch and strain

2 dominates in another patch, then it is possible for both strains to persist in both

patches when infectives disperse between the two patches. However, the magnitude

of the dispersal probabilities pj and Pjk and their relative magnitude with respect to

the transmission and reco\'ery parameters Pjk At and 'jjkAt play an important role

in persistence and extinction. For example, patch-dependent dispersal can change

the outcome predicted by the basic reproduction numbers. It can be seen from the

numerical examples, presented in Figures 2.4, 2.5, and 2.8, for a fixed set of basic

reproduction numbers, that there are regions in the dispersal parameter space where

there is competitive exclusion (persistence of strain 1 or strain 2 but not both) or

coexistence.

In future research, the relationship between the dispersal probabilities and basic

reproduction numbers needs further investigation. In addition, it will be of interest to

develop and analyze stochastic versions of these deterministic models. For example,

the stochastic models can be used to estimate the probability of disease extinction.

32

CHAPTER III

DISCRETE-TIME MARKOV CHAIN SIS EPIDEMIC MODELS WITH

MULTIPLE PATHOGENS

3.1 Introduction

Epidemic models with multiple pathogen strains have been applied to many in­

fectious diseases including HIV/AIDS, gonorrhea, dengue fever, myxoma virus, and

influenza (e.g., [8, 10, 16, 18, 17, 27, 53]). Questions of interest are whether one or

more than one strain will persist over time and the virulence of the persistent strains.

The virulence of a particular strain is generally defined as the disease-induced host

mortality. A classical example demonstrating the evolution of virulence is myxoma

virus in rabbits. Data on the proportion of the various grades of myxoma virus in wild

populations of rabbits showed that the grade of virulence changed from a highly viru­

lent grade to an intermediate level over time [9, 28]. For simple mathematical models

without the complicating effects of coinfection (individuals infected with two or more

strains), superinfection (individuals infected with one strain can be infected by a more

virulent strain) or density-dependent mortality, it is the maximal basic reproduction

number that determines which strain is persistent (e.g., [15, 47, 49, 53, 71]). The

basic reproduction number is the ratio of the transmission rate (3 to the recovery rate

7 plus the disease-related death rate a,

J + a

One can see that if a, /3, and 7 are independent parameters, then 7^ is a maximum

when /5 is a maximum or when a or 7 are a minimum. However, the parameters P

and 7 may be linked to a: 7 = 7(0;) and /3 = ^(a) . Then, there may not a unique

maximum value of IZ and the maxima may occur at different values of a. It is often

assumed that 7(0;) is a decreasing function of a and ^(a) is an increasing function of

a with a concave shape [8, 9, 53], but the relationship between these parameters is

not known.

33

We formulate a st(K4iastic discrete-time SIS epidemic model based on the model

m Chapter II. This d(>t(>rniinistic mod(4 was also analyzed by Allen, Kirupaharan,

and Wilson [7]. Susci-ptibU^ individuals become infected, then recover and become

susceptible again, 5 ^ 7 ^ 5. Individuals may be infected with one of n differ­

ent strains through direct contact l)etw(>en an infected individual and a susceptible

one. Infection with one strain confers complete cross protection. There is no coinfec­

tion nor superinfec-tion. The stocliastic single patt4i model is extended to a stochas-

tu- two-patch model, where dispersal of infectives and susceptibles occurs between

the two patches. Some properties unique to the dynamics of stochastic models are

investigated-probability of disease extinction and the quasistationary distribution.

.\s shown in Chapter II, in the underlying deterministic models, a competitive

exclusion principle applies to the single patch model. The pathogen strain with the

maximal basic reproduction number is the one that persists and outcompetes all other

strains provided its magnitude is greater than one. When infected individuals dis­

perse between two patches, coexistence may occur. However, in the stochastic model,

eventually disease extinction occurs. The probability of disease extinction approaches

one in the limit as t —> oo. Because the convergence to one may be very slow, the

probability of disease extinction may remain relatively constant for a long period of

time. An estimate for this probability of disease extinction for finite time is obtained.

In addition, the frequency distribution of the population conditioned on nonextinc­

tion, reaches an approximate stationary distribution during this time, known as the

quasistationary distribution. This quasistationary distribution is compared to the

solution of the deterministic model. Through extensive numerical simulations and es­

timates based on the theory of branching processes, the quasistationary distribution

and the extinction dynamics of the stochastic model are studied. It is shown that

competitive exclusion and coexistence in the stochastic two-patch models are much

more complicated than the deterministic models and the outcome depends critically

on the initial number infected by each strain, the dispersal probabilities, and the basic

reproduction numbers for each strain.

34

The stochastic SIS epidemic model for a single patch is formulated in section 3.2

and is then extimded tt) two i)atches in section 3..'5. The probability of disease ex­

tinction and the tiiiasistatit)nar>' distribution are tliscnssed. In section 3.4, numerical

results illustrating disease extinction and the ([uasistationary distribution over time

are presented and discussetl. In the last section, the results are summarized and

generalizations of the basit' model are discussed.

3.2 Single Patc4i

3.2.1 Stochastic Model

The deterministic model is extended to a stochastic model, a discrete-time Markov

chain model. Let S[t) and 4t(t) denote discrete random variables for susceptible

individuals and individuals infected with strain k at time t. Assume

n

Sit).Ikit)e {0,1.2 A}, ^ = 1,2,... ,n, and Sit)+ J2Mt)<K, (3.1) k=l

where t G {0, At, 2At , . . . }. The initial conditions satisfy <S(0) + X]"=i - i(O) = A',

where 5(0) > 0 and Yl'^^i hi^) > 0- During the time period At, there is either a new

infection caused by strain k, a recovery from the infection, or a disease-related death.

Let Sit) = a and 4fc(t) = a for A; = 1, 2 , . . . ,n. Then, the transition probabilities

satisfy the following:

Prob{iS,lk)it + At) = ia-l,ak + l)\iS,Xk)it) = ia,ak)] (3.2)

= —r^^TT—-oAt, a + Z^fc=l (^k

Pro6{(<5, Jfc)(t + AO = ( a + l , a f c - l ) I(<S,Jfc)(0 = (a, Gfc)} = IkakAt, (3.3)

Prob{iS,Xk)it + At) = ia,ak-l)\iS,Xk)it) = ia,ak)) = (^k^k^t, (3.4)

where Pk and 7^ are positive constants and a^ is nonnegative for A; = 1, 2 , . . . , 77. The

transition probabilities (3.2), (3.3), and (3.4) represent probabilities of an infection,

a recovery, and a disease-related death, respectively. In addition, the probability of

35

no transition or no change in state in t,li(> time period At is

To ensure that (3.5) is a probability, it must be the case that

5Z ;7T^ " + [7 + ^kW I At < 1 (3.6)

for all values of a and a^ satisfying (3.1). Sufficient conditions for (3.6) to hold are

(A - a) a— -h 7 -t- a h At < 1, (3.7)

where a G {0,1,2 K}, p = supj/,i;..}, 7 = supfc{7fc}, and a = s up j a^} . There­

fore, inequality (3.7) holds when A[7 + a]At < 1 if 7 o = P/[j + a] < 1 and when

R'lJ + 7 + afAt < 4:3 if KQ > 1. Inequality (3.7) holds in all of the numerical

examples.

3.2.2 Probability of Extinction

This multivariate, finite Markov chain model with state variables

iSit),Iiit),...,Init))

has a unique absorbing state given by (A', 0, 0 , . . . ,0). All other states are transient.

If Jfe(t) = 0 for all k and some time t, then the probability of no change given by (3.5)

is equal to one. In addition, if 4c(t) = 0 for some k and time t, then Ikit + At) = 0,

i.e., if the number of individuals infected with strain k reaches zero, it remains at

zero. This can be seen from the transition probabilities (3.2), (3.3), and (3.4).

Denote the probability of extinction of strain k at time t as

pg(0 = Pro6{J,(0 = 0}

and the probability of extinction of all strains as

Poit) = Prob lY^Ikit) = 0 I fc=i

36

Then, it follows from finite Marko\- chain theory [42] t4iat.

lim p§(t) = 1 = limpo(/).

This extint'tion belnwior is unique to the sto(4iastic epichunic model; disease extinc­

tion does not occur in tlu- deterministic nio(l(>l unless all of the basic reproduction

numbers are less than one. In the stochastic model, eventual termination of the epi-

deniii- or absorption t)ccurs with probability one, regardless of the value of the basic

reproduction numbers. However, it ma>' take a long time until these probabilities

reach one, if, for example, the population size is sufficiently large. A' ^ 1, or the

initial number of infectives is large, Xfc(O) ^ 1. We shall consider the case where

A' : ^ 1 but X/c(0) = 1 for some A;. It will be seen that the probabilities Po(t) and p^it)

increase rapidly initially, but then reach plateaus which rise very slowly. We will use

the theory of branching process to estimate the level of these plateaus. The plateau

reached by po(t) can be used as an estimate of the probability that the epidemic ends

rapidly.

An estimate for the probability that the epidemic ends rapidly can be obtained

from the basic reproduction number (see e.g., [6, 41, 13]). If, there are initially a^

individuals infected with strain k, Xfc(O) = a^ and Xj(0) = 0, j ^ k, then po(t) = p'^it).

An estimate for the level of the plateau reached by p^it) is based on the theory of

linear branching process and is given by

f 1, ff 7 fc < 1

where TZk- = - % - . The value of p^(t) is constant for a long time interval before

eventual absorption occurs.

If more than one strain is introduced, strains k and 777, Xfe(O) = a/c > 0, X^(0) =

Om > 0, and Xj(0) = 0 for j / A;,7n, then the probability that the epidemic ends

rapidly, the level of the plateau for po(t) can be estimated from the theory of multitype

branching processes [35, 43, 56]. An estimate for the probability the epidemic ends

37

rapidh- is

po(0«pS(/)p;;'(0^(^y'(^) , (3.9)

provided 7 , > 1 and 'R,„ > 1 (the fractions in (3.9) ar(> replaced by the number

one if the basic- reproduction nuinb(-rs are l(>ss than or equal to cuie). This estimate

assumes the events are independent. Due to competition between the strains, the

events are not independent, but dependent. The numerical examples will illustrate

how competition affects the estimates given above.

3.2.3 Quasistationary Distribution

Given the infection process has not ended or is conditional on nonextinction of the

infection process, a quasistationary distribution may exist. Let p'^it) = Prob{Xkit) =

i). I = 0,1, 2,. , K. Define gf (t) = Prob{Xkit) = i\Xkit) > 0}. Then,

pno i - p § ( 0

lHt) = T^m7j„ 1 = 1,2,... ,K,

are the probabilities associated with the distribution for the number of infectives of

strain A; conditioned on nonextinction, i.e., provided Po(t) ^ 1. For some parame­

ter values, this probability distribution may approach an approximately stationary

distribution, known as the quasistationary distribution (see e.g., [6, 59, 60, 61]). An

approximation to the distribution conditioned on nonextinction for strain k is graphed

in the numerical examples. It is this distribution that can often be related to the de­

terministic model. When a stable endemic equilibrium exists in the deterministic

model, the mean of the conditional distribution may have a value close to this equi­

librium. For some simple epidemic models with one pathogen strain, estimates for

the quasistationary distribution can be obtained (see e.g., [6, 59, 60, 61]). For our

model, an approximate quasistationary distribution is illustrated by the numerical

simulations.

In the next section the single patch model is extended to two patches.

38

ess

3.3 Two Pat(4ies

3.3.1 Stochastic Model

The sti)chastic model for two pat.(4ies is a two-st,a,ge process, an infection pro(

followed by a dispersal prot'(>ss. The infection process is confined to the individual

patches and is analogous to the infection process for the single patch model with the

exception that the population size in each patch is not fixed. Let Sj and Xjk denote

discrete random variables for the number of susceptible individuals in patch j and

the number of individuals infected with strain A: in patch j , respectively. Let J\fj and

.V denote discrete random \'aria])les for population size in patch j , j = 1,2 and the

total population size, A/"i(t) + Miit) = A/'(t). In addition.

n

Sj.Xj.e {0,1,2. .,Afj}, k = l,2,...,n, 5, + J ^ X,, = AT,, k=l

for j = 1,2. The total population size satisfies

Me{0,l,2,...,K).

During the time period At, in patch j , one of 377 events may occur. Either

a susceptible individual becomes infected with strain k, or an individual infected

with strain k recovers from infection or dies from the infection. Since there are 77

strains, there are 377 possible events. Also, there is a possibility that during the time

period At, no event occurs. After infection, each individual may disperse to another

patch with a given probability. Assume at time t, the number of susceptible and

infected individuals in patch j equal Sjit) = Uj and Xjkit) = ajk, for j = 1,2 and

A- = 1, 2,. . ,n. The transition probabilities for the infection process in patch j for

strain k satisfy the following:

Prob{iSj,Xjk)it + At) = (a, - l,ajk + 1) \iSj,Xjk)it) = (a,,a,fc)} = Jf^fi^At

Prob{iSj,Xjk)it -h AO = (a ' + 1, a-fc - 1) \iSj,Xjk)it) = (a^, ajk)} = jjkttjkAt

Prob{iSj,Xjk)it - AO = iaj, ajk - 1) \iSj,Xjk)it) = (a^, ajk)] = OjkttjkAt

39

h)r 7 - 1 , 2 and A: = 1, 2 , . . . , n, where Afjit) = a + f ajk. The probabihty of no • f . . . k=l

mtection or rectn'er>- m the time period At in ])atch j is

„ / \

E A ; = l

"jk + Ujk + <yjk\<^jk At (3.10)

\ A=:l /

lor all Oj + 22k=i "JA- ^ {0,1, 2 , A 'j}. As in the single patch model, it is assumed

that probability (3.10) is nonnegative. For standard incidence, sufficient conditions for

(3.10) to be nonnegative are AT/jAt < KSjAt < 1 ilTZj < 1 ovMjPjAt < KpjAt < 1,

if Rj > 1, where 7 ^ = pj/6j, Pj = sup,{/?,fc} and 5j = supj^.^} < 1. These

conditions are satisfied in the numerical examples.

Alter the first stage, the infection process, there is the dispersal process. Assume

the susceptible and infected random variables satisfy S'j and XJ . after the infection

process. In the dispersal process, every individual is treated separately. For example,

if 5j = a J and X'j^ = Ojk, then each of the aj susceptible individuals in patch j ,

moves from patch j to patch /, j ^ I, with probability pj and remains in patch j

with probability 1 —Pj.- Each of the ajk individuals infected with strain A- in patch

i moves from patch j to patch /, j ^ I, with probability pjk and remains in patch j

with probability 1 — Pjk-

3.3.2 Probability of Extinction

The multivariate Markov chain model for two patches has state variables

(<Si(0,<S2(0,Xii(0,X2i(0,...,Xi„(0,X2„(0).

The only absorbing states are those with all infectives equal to zero, (<Si, <S2,0,0,... , 0,0),

w 2 here Si + §2 = K. All other states are transient. In addition, if Yl,]=i'^jki'^) = 0,

then Ej=i^i/c(t) = 0 for t > r. Denote the probability of extinction for strain k

in patch j as pl (t) = Prob{Xjkit) = 0}, the probability of extinction of strain A: as

Po(t) = Prob{Y^j^^Xjkit) = 0}, and the probability of extinction of all strains as

40

Po(t) _ P7-o6{5]; _, ^'Uihkit) = 0}. Then, eventual extinction of the infection is

^•ertam.

lim/r;,'(0 = l = l impg(t)= lmipo(0.

Prior to absorption, if a small number of individuals are initially infected and the

population size is sufficiently large, the probability an epidemic ends quickly can be

estimated based on the assumption of independence. Supp(js(>, in addition, that the

dispersal probabilities are of tlu> same magnitude, so that there is no differential effect

resulting in greater movement to or away from a particular patch. Then, assuming

independence of e^•ents, the probability strain A; dies out is approximately

where a^k = Xj^iO) and TZjk > h j = 1,2. In addition, po(0 ^ n^=iPo(0- More

realistic approximations are obtained for p'^it) and po(t) in the numerical examples

and compared to these estimates. Competition and dispersal make the events in this

stochastic process dependent.

3.3.3 Quasistationary Distribution

Let p ' -(0 = Prob{EUhkit) = ^} and pf(0 = Prob{j:UT.Lihkit) = 0-

Probability distributions conditioned on nonextinction can be defined for each strain:

gf (t) = Prob {Xjkit) = I \Xjkit) > 0 } = ^'^^^ l - 7 > ^ o ' ( 0

and

iit) = Prob\Y,Xjkit)=i E i , . W > o ' - *'•<" ^=1 , l - P o W

It will be shown in the numerical examples that the distribution conditioned on

nonextinction often approaches a stationary distribution for a range of t values, which

is known as the quasistationary distribution. The quasistationary distribution is

compared to the dynamics of the deterministic model.

41

3.4 Numerical Examples

It is assumetl that tlu- force of infection is of standard incidence type. In the single

patch model,

\kit)=PkAt'^, A

where «/., is the number of indi^•iduals infected with strain A;. In the two-patch model,

Xj,{t) = Pj,At- "'" "j + Efc = l "-jk '

where Oj is the number of individuals susceptible in patch j and (ijk is the number

of individuals infected with strain A' in patch j . In addition, it is assumed that there

are two strains in each patch, 77 = 2 and there are no disease related deaths, ajk = 0.

This latter assumption allows comparison with the deterministic models in Chapter

2. For each set of parameter values, 10,000 sample paths are generated. Probability

frequency histograms for the number of infectives at fixed times are constructed based

on the 10,000 sample paths. In addition, an approximation to the frequency distri­

bution of the population conditioned on nonextinction is graphed over time. These

graphs are plotted as a function of T, where T is the number of time steps At, e.g.,

T = 1000 means t = lOOOAt.

Frequency histograms are graphed rather than probability histograms (see Figure

3.1). In Figure 3.1, these histograms approximate the probability function,

p'^{T) = Prob{XkiT) =i}, i = 0,l,2,...,K,

k = 1,2. If the units on the vertical axis are divided by 10,000, then they repre­

sent probabilities rather than frequencies. For the two-patch models, the frequency

histograms approximate the probability histograms corresponding to

p>,'iT) = Prob{XjkiT) = 1}, I = 0,1,2,..., K

j,k = 1,2. In addition to the frequency histograms, an approximation to the fre­

quency distribution conditioned on nonextinction is given by the dashed curve. Statis­

tics from the frequency histograms (mean and variance) are used to fit a normal fre­

quency distribution, where the the range is adjusted to fit the frequency histogram.

42

llie normal approximations to the distributions conditioned on nonextinction, vf(t)

and q] (t), are given by the dashed curves in Figure 3.1. As time progresses, for exam­

ple, at T = 5000, the normal approximation provides a good fit to the quasistationary

distribution.

In Figure 3.1, the stochastic model for a single patch is numerically simulated.

The basic reproduction numbers for each strain satisfy

TZi = 2 and 112 = 1.5.

Initially, there are two infected individuals, one infected with strain 1 and the other

infected with strain 2. The frequency histograms in Figure 3.1 are bimodal. The

peak at zero represents the probability of extinction for a particular strain, Po(X) =

Pro6{4(X) = 0}, A' = 1, 2. The total frequency at zero cannot be seen in Figure 3.1;

this value is larger than 50,000 in some cases. To better view the distribution for

values of 4(X) > 1 in Figure 3.1, the vertical axis is shortened (maximum frequency

is 1000 or 1200). The frequency for values at zero is graphed separately in Figure 3.4.

43

1200

1000

O 600

I 400

S

Figure 3.1: Frequency histograms at T = 1000,3000 and 5000 for the single patch stochastic model and the approximate frequency distributions conditioned on nonex­tinction for the number of infectives over the interval T G [100, 5000] (based on the normal distribution, dashed curve) are given in the bottom two figures. Parameter values satisfy ^lAt = 0.01, ^2At = 0.01, 71 At = 0.005, 72At = 1/150, Xi(0) = 1 = X2(0), and K = 100. The basic reproduction numbers are Hj = 2 and 7 2 = 1-5. The width of the intervals of the histograms are 3 units: [—1.5,1.5), [1.5,4.5), —

The solution to the deterministic model with the same parameter values as in

Figure 3.1 is graphed in Figure 3.2. Note that the distribution conditioned on nonex-

44

tinction (bottom two graphs in Figure .3.1) follows the deterministic solution for strain

1 but not for strain 2. In the stocliastic model, strain 1 may die out, then strain 2

may dtuninate. This behavior is not possible in the deterministic model since strain 1

is the dominant strain. Two sample paths, graphed in Figure 3.3, illustrate two of the

nuun- possible dynamics for the stocliastic model. In Figure 3.3 (a), strain 1 persists

and in Figure 3.3 (b) strain 2 persists. Compare the sample path in Figure 3.3 (a) to

tlu- tleteniiinistic solution in Figure 3.2.

2000 3000 Time Steps, A t

4000 5000

Figure 3.2: The solution to the deterministic model with the same parameter values and initial conditions as in Figure 3.1.

45

1000 2000 3000 Time Steps, A t

4000 5000 1000 2000 3000 4000 5000 Time Steps, A t

Figure 3.3: Two sample paths for the stochastic model for the parameter values and mitial conditions given in Figure 3.1. In (a) strain 1 persists and in (b) strain 2 persists.

The parameter values in Figure 3.1 correspond to a stable endemic equilibrium

in the deterministic model, (5,4,72) = (50,50,0) (see Figure 3.2). The mean and

standard deviation (/i and a) of the quasistationary distribution (conditioned on

nonextinction) for Xi and X2 computed from the numerical simulations at T = 5000

satisfy

/ii(5000) ^ 48.8, ai(5000) ^ 7.3

and

/i2(5000) ^ 30.3, ^2(5000) ^ 9.0.

The mean of strain 1 is close to the stable endemic equilibrium X2 = 50 and the mean

of strain 2 is close to the endemic equilibrium where strain 1 is absent Xi = 33.3.

The probability strains 1 or 2 are eliminated, pl and pl, are approximately con­

stant for a large range of times. An estimate for these constants, assuming inde­

pendence, is given by equation (3.8). For strain 1 and strain 2, the estimates are

pJ(T) ^ l/Ui = 0.5 and p^(r) « 1/112 = 2/3, respectively (see Figure 3.4). The

probability both strains are eliminated by time T (assuming independence) is given

by the estimate po(X) PU (7^i7^2)~^ = 1/3. The 10,000 samples paths give a more

accurate approximation to the true values of these probabilities,

pj(5000) « 0.52, Po(5000) ^ 0.83, and po(5000) fa 0.36.

46

See Figure 3.4. Bast'd on these i)rol)a,bilit.i(-s, the proportion of saini)le paths where

there is coexistc-nce can be calculated. If pr(t) = Prob{Xiit) > 0 and X2(t) > 0},

then

Pr(5000) ft; 0.01; (3.11)

coexistence is an unlikely outcome. The approximations obtained from the simula­

tions for PI and po agree with the predicted estimates, but the value for strain two

is larger than the estimate but the rate of increase of Po(t) slows down considerably

when it reaches 2/3. Since the estimate was based on the events being independent,

we should not expect the estimate to be accurate. Competition between the two

strains makes the events dependent and results in the weaker strain being eliminated

at a faster rate. Note that the vertical axis in Figure 3.4 gives the frequency rather

than the probability.

10000

o o o

8000

•^ 6000

o o c cr CD

4000

2000

Strain 1 Strain 2 Both Strains

2000 3000 4000 5000

Figure 3.4: The frequency distributions for elimination of strain 1, strain 2 or both strains for the parameter values and initial conditions given in Figure 3.1.

In the next example, the dynamics of a single strain in two patches are simulated.

Individuals infected with strain 1 are introduced into patch 2 and the disease is

transmitted to patch 1 by dispersal of infected individuals, X2i(0) = 20 and Xii(O) = 0.

47

In this exami)le, the basic reproduction numbers for patch 1 and 2 satisfy

7 11 =0.5 and 'R-,i = 20.

The freciuenc>- histograms of the stochastic model at T = 1000, 3000, and 5000 for

4 i and 4 i are graphed in Figure 3.5. The normal approximation (dashed curves)

is a good approximation to the friHiueucy histograms. Eventually, the disease dies

out in both i)ati-hes. The ilisease-free equilibrium is locally asyniptot.ically sta­

ble in the deterministic model and solutions approach the disease-free equilibrium,

(5i, 52. 4 i . 4 i ) = (200,0,0,0), see Figure 3.6. The means and standard deviations

for the simulations at T = 1000, 3000, and 5000 are given in Table 3.1 and agree

closely with the solution to the deterministic model. Figure 3.6. Close agreement be­

tween the stochastic and deterministic models occurs because both the initial number

of infectives and the population size are large.

48

Figure 3.5: Frequency histograms of the stochastic model with one strain when in­fectives disperse. The histograms are graphed for T = 1000, 3000 and 5000. The dashed curves are the normal approximation to the frequency distribution conditioned on nonextinction; the normal approximation is graphed in the bottom two figures for T G [100,5000]. The parameter values and initial conditions satisfy PnAt = 0.001, p2iAt = 0.005, 7iiAt = 0.002, 72iAt = 0.00025, pu = 0.005 = P2i, Pi = 0 = P2 Xii(O) = 0, X2i(0) = 20, 5i(0) = 100, 52(0) = 80, and K = 200. The basic reproduc­tion numbers are Kn = 0.5 and 7 2i = 20.

49

40

35

•I 30 n

§25 Q.

120

£l5 O

CD 1 1 1

1 1

/ / / /

- - - ' 1 1

— '21

^ ^ v \

/ ^ \

^ N.

^ ^ :>.

^^"^^

0 1000 2000 3000 4000 5000 Time Steps, A t

Figure 3.6: The solution to the deterministic model corresponding to the parameter values and initial conditions of Figure 3.5.

Table 3.1: The mean values and standard deviations for the stochastic simulations of Xii and X21 at T = 1000, 3000 and 5000 are compared to the values of the determin­istic solution for the parameter values given in Figure 3.5.

Time

T = 1000

T = 3000

T = 5000

Stochastic (/i ± a)

Xu

28.6 ±5.5

13.1 ±4.2

5.0 ±2.9

X21

36.3 ±5.5

14.6 ±4.2

5.5 ±2.9

Deterministic

4 i

29.3

13.0

5.0

4 i

37.0

14.6

5.6

In the next set of figures either susceptibles or infectives disperse. The basic

reproduction numbers satisfy

1111 = 2 = 1122 and Tin = 1-5 = TZii- (3.12)

In the deterministic model, when susceptibles disperse, the competitive exclusion

principle holds and in patch one, strain 1 dominates, but in patch two, strain 2 dom­

inates. The dynamics of the stochastic model are more complicated. In Figure 3.7,

there are graphs of the frequency histograms at T = 1000, 3000 and 5000 and the

approximation (based on the normal distribution) to the distribution conditioned on

nonextinction for Xu and X21.

50

Figure 3.7: Frequency histograms of the stochastic model with two strains in two patches when susceptible disperse. Frequency histograms are given at T = 1000, 3000 and 5000. The normal approximation for the distribution conditioned on nonextinc­tion is the dashed curve which is graphed in the bottom two figures for T G [100,5000]. The parameter values satisfy PjkAt = 0.005, -fjjAt = 0.0025, jikAt = 1/300, j,k = 1,2, I ^ k, pj = 0.01, Pjk = 0, XjkiO) = 1, and 5^(0) = 98 for j,k = 1,2. The basic reproduction numbers satisfy T -n = 2 = TZ22 and 7li2 = 1.5 = 7l2i.

Note that the mean values of the quasistationary distribution (conditioned on

nonextinction) for Xu and X22 are close to 50, the stable solution in the deterministic

51

)se lo nuidel, (Figure 3.8 (a)). See Table 3.2, The mean values for X12 and X i are clos

-0, much greatc-r than tlu' deliMiuinistic solution. However, the dynamics can be very

complicated in the stothastic model. For example, if the dominant strain 1 in patch 1

IS eliminated, the weaker strain 2 coidd persist in patch 1. In addition, the dominant

strain 2 in jiatcli 2 could he eliminated and the weaker strain 1 could persist.

Table 3.2: The mean \aliu-s and standard deviations for the distributions conditioned on nonextinction for the stoc4iastic simulations of Xjk at T = 5000, /ijA:(5000) ± o-jfc(5000), for the parameter values and initial conditions given in Figure 3.7.

Strain 1

Strain 2

Patch 1

54.6 ±12.7

25.7 ±14.1

Patch 2

25.4 ±13.7

54.3 ±12.9

r40

'30

= 10-

(a) , - - ' ' " '

/ /

1 / 1

1 1

1

/ / / / /

''-—" "

- - - '11

— '21

25

= 20

c10

(b) ^ '

/ /^ / / / / // 1/ 1/

1/ 1/

// 1/ 1/

1/ u

---'„ — '21

0 1000 2000 3000 4000 5000 Time Steps, A t

0 1000 2000 3000 4000 5000 Time Steps, A t

Figure 3.8: The solutions to the deterministic model corresponding to the parameter values and initial conditions of (a) Figure 3.7 and (b) Figure 3.10.

The frequency distributions for elimination of a particular strain for the parameter

values given in Figure 3.7 are graphed in Figure 3.9. Note that

pj(5000) ^ 0.41, p^(5000) « 0.42, and po(5000) « 0.13.

- 1 The first two values do not agree with the approximations given by p\it) = (7^ll7^2l)

1/3 and pHt) = (7^227^l2)"^ = 1/3. However, it can be seen in Figure 3.9 that there

52

IS a change in the rate of increa.s(> when these probabUities reach 1/3 (frequency

3,333). The rate- of increase is rapid until 1/3, then this rate slows down consid­

erably. The probabilitA- both strains are eliminated agrees more closely with the

estimate, po(t) = ilZu'RiiR 12^22)'^ = 1/9. The estimate for po shows it is smaller

than for a single patch, either iTinRii)'^ for patch 1 or (7^l27^22)"^ for patch 2. The

probability of coexistence, pdt) = Pro6{^^^^iX,i(t) > 0 and E?=i^ i2( t ) > 0} for

T = 5000 satisfies

Pc(5000) ^ 0.30.

This estimate can be compared to the single patch model. Figure 3.4 and equation

(3.11), at time T = 2500, pc(2500) ^ 0.04. Coexistence has a greater probability for

the two patch model. Dispersal causes the strains to behave similarly with respect

to extinction (7^ii7?-2i = 7 i2'7^22) and competition causes the extinction probabUities

for each of the strains to increase over time.

5000

4000 o o o

-53000

o

§2000 :D cr 0)

1000

strain 1 Strain 2 BotIn Strains

/ ' *

i f . 1

1 1

i 1 t

i / 1 /

1000 2000 3000 4000 5000

Figure 3.9: The frequency distributions for elimination of strain 1, strain 2, or both strains in both patches for the parameter values and initial conditions in Figures 3.7.

In Figure 3.10, the parameter values are the same as for Figure 3.7, except only

infectives disperse, Pj = 0 and Pjk > 0 for j , k = 1,2. In the deterministic model, since

53

the basic reproduction numbers satisfy (3.12), there is coexistence of both strains. The

deterministic model has a stable eciuilibrium at

iS^.S-,. 4 i , 4 i , 42, 4-) ~ (58,2, 58.2, 21.3, 20.5, 20.5, 21.3) (3.13)

See Figure 3.8 (b). The stochastic model in Figui(> 3.10 does not appear to follow the

deterministic model. The mean values and standard deviations for the distribution

conditioned on nonextinction for Figure 3.10 are given in Table 3.3. The mean values

in Table 3.3 are larger than the deterministic endemic equilibrium values in (3.13)

but approximately equal in value.

54

I , atT=1000

Figure 3.10: Frequency histograms of the stochastic model with two strains when infectives disperse. The histograms are graphed for T = 1000, 3000 and 5000. The normal approximation for the distribution conditioned on nonextinction is the dashed curve and is graphed in the bottom two figures. The parameter values satisfy PjkAt = 0.005, -fjjAt = 0.0025, 'jjkAt = 1/300, j ^ k, pj = 0, Pjk = 0.01, XjkiO) = 1, and SjiO) = 98 for j,k = 1,2. The basic reproduction numbers satisfy TZn = 2 = 7Z22 and 7 12 = 1.5 = 7?-2i.

In the stochastic model of Figure 3.10, rapid extinction of a particular strain may

55

Table ,3.3: The mean values and standard de\iations for the distributions conditicmed on nonextinction for the stochastic simulations of 4/,, at T = 5000, /4fc(5000) ± (3-j/..(5000), for the parameter values and initial conditions given in Figure 3.10.

Strain 1

Strain 2

Patch 1

29.2 ± 13.7

29.1 ± 13.7

Patch 2

28.8 ±13.8

29.4 ±13.4

occur causing sample paths to approach different equilibria. Three sample paths,

graphed in Figure 3.11, illustrate diflFerent types of behavior in the stochastic model,

either all strains persist, only strain 1 persists or only strain 2 persists. There are

also sample paths where all strains are eliminated rapidly. Compare Figure 3.11 (a)

with Figure 3.8 (b).

1000 2000 3000 4000 5000 Time Steps, A t

60

atio

ns

,£40 S • 30

Sso CO

10

1

21

'12

- - . ' 2 2

j(jTVTir

1

s . 1 y i

MmVrillVl

(b)

1* Ll><

im

1000 2000 3000 4000 5000 Time Steps, i t

550

£40

S • 30

js^^L^

m 1000 2000 3000 4000 5000

Time Steps, i t

Figure 3.11: Three different sample paths from the stochastic model where infectives disperse, see Figure 3.10. In (a), all strains persist, the sample path oscillates near the deterministic stable equilibrium defined in (3.13). In (b), strain 1 persists, the sample path oscillates near the equilibrium 4 i = 40 = 4 i and 42 = 0 = 42- Jn (c), strain 2 persists, the sample path oscillates near the equilibrium 4 i = 0 = 4 i and 42 = 40 = 42.

Frequency distributions for elimination of the disease are graphed in Figure 3.12 for

the parameter values in Figure 3.10. Approximations for the probability of extinction

based on the numerical simulations at T = 5000 are

pj(5000) ^ 0.41, Po(5000) ^ 0.40, po(5000) = 0.13.

56

Based on these probabilities, the probability of cot-xist.ence of both strains S IS

P,.(5000) ?«0.32.

These \-alues agree with those for Figmv 3.9, where susceptibles disperse. Dispersal

and competition cause the intectives to behave similarly and disease extinction for a

single strain to occur more rapidly than t'xpected (estimated values for pJ and pl are

1/3).

5000

4000 o o o o ^3000 "3 o

§2000 cr

1000

strain 1 - - Strain 2

Both Strains

// //'

f

. ;

•

' ^ ^ —

1000 2000 3000 4000 5000 T

Figure 3.12: The frequency distributions for elimination of strain 1, strain 2, or both strains in both patches for the parameter values and initial conditions given in Figure 3.10.

3.5 Summary

In the stochastic SIS epidemic models, the disease is eventually eliminated from

the population. This is due to the fact that there is a single absorbing state, where all

infectives equal zero, so that \im.t^ooPoit) = 1. Since complete elimination may take

a long time, a better estimate of disease elimination is the value of po(t) after a small

number of infectives have been introduced, the value of po(t) satisfies 0 <C po(t) < 1 for

a large range of times. Estimates of po(t) depend on the initial number of infectives

and the basic reproduction numbers. Generally, in two patches, the value of po(t)

57

is smaller and the probability of coexistence pdt) is larger than in a single patch

environment. Spatial heterogeneit.\- promotes coexistence of more than one strain.

The dynamics of the stiK4iastic models are more complicated than the determin­

istic models. If more than one strain is introduced in the single patch model, the

competiti\e exclusion principle applies to the deterministic model, but it may not

apply to the stochastic model. If the dominant strain dies out rapidly (which de­

pends on the initial number of inlecti\-es and the basic reproduction number), then

the weaker strain may i)ersist (Figmx> 3.3).

These results have implications for some emerging diseases in wildlife, e.g., han­

tavirus and arenavirus infection in rodents. To determine whether a viral strain could

persist in the host population in a particular environment, attention should be paid

to the dispersal capability of susceptibles and infectives and the magnitude of the

basic reproduction number of the viral strain (or magnitude of the parameters which

define the basic reproduction number).

58

CHAPTER IV

STOCHASTIC DIFFERENTIAL EQU.4TI0N SIS AND SIR EPIDEMIC

MODELS WITH MULTIPLE PATHOGENS

4.1 Introduction

There have been numerous theoretical studies concerned with the evolution, per­

sistence, and extinction of multiple pathogen strains based on deterministic epidemic

models [1, 2, 15, 16, IS, 22, 27, 37, 47, 54, 55, 58, 62, 66], These types of models

have been applied to diseases such as infiuenza, dengue fever, malaria, and HIV-AIDS

[10, 16, IS, 26, 27, 32], In these models, it is often the basic reproduction numbers

of each strain that pla>- an important role in determining which strains persist and

which strains do not. The evolution and persistence of multiple pathogens on the

disease dynamics have not been investigated in stochastic epidemic models. The pur­

pose of this investigation is to formulate new stochastic epidemic models based on SIS

and SIR deterministic epidemic models with multiple pathogen strains and to com­

pare the disease dynamics predicted by the deterministic models to the corresponding

stochastic models.

The deterministic epidemic models differ from the model of Chapter II, These de­

terministic models are ordinary differential equations, not difference equations, and

births and deaths are included. However, we do not consider two patches. In partic­

ular, it is shown that with density-dependent mortality there can be coexistence in

these models. This was not the case for the model in Chapter II, unless there was

dispersal of infectives between two patches. Because we do not include dispersal, it

is the inclusion of the density-dependent mortaUty that promotes coexistence.

The new stochastic models are expressed as stochastic differential equations (SDEs).

The derivation of the SDEs is based on a method developed by Allen [3]. We gen­

eralize the method of Allen [3] to n state variables where the population dynamics

can include births, deaths, and migrations (or transitions). We apply this method to

the SIS and SIR deterministic epidemic models with multiple strains. This method

59

ot deri\ation assumes that the variability inli(>reiit in the system is due only to de­

mographic variabilitA' and not environnu'utal \ariability. The SDEs take the form of

Ito SDEs, The SDEs are limiting systems (At -> 0) of discretr>-time Markov c4iain

models. In [3, 4], SDEs for simple SIS and SIR epidemic models with constant pop­

ulation size were derived. No conii)arisons were ma.de b(>tween the deterministic and

stochastic models.

Many stochastic epidemic models have bc-en formulated to study questions relevant

to epidemiology (see e,g,, [6, 12, 13, 21, 24, 30, 36, 40, 41, 57, 59, 60, 61] and references

therein), but these models have often taken the form of discrete or continuous-time

Markn- chain models rather than SDEs, The SDEs formulated here are new models.

The stochastic- epidemic models in [30] have been used for estimation and inference for

measles outbreaks where the stochastic processes explain extinction and recurrence of

epidemics observed in measles, A stochastic SIS model with one infected individual

among a population partitioned into M sites and N individuals in each was modeled

by a continuous-time .Markov chain in [12] and the limit behavior of the system as M

and N go to infinit}' was analyzed. In [36], a continuous-time Markov chain method

was used to calculate the reproduction number HQ in complicated stochastic epidemic

models. In [6] and [24], the epidemic models studied were discrete-time Markov chain

models. In [59, 60, 61], the epidemic models were continuous-time Markov chain

models. In none of these stochastic models was the effect of multiple pathogens

studied, SDEs have an advantage over some of the other stochastic formulations in

that they can be easily derived directly from the deterministic system of ordinary

differential equations and have a relatively simple form. In addition, the numerical

computations for multiple pathogens are simpler than in either discrete or continuous-

time Markov chain models. However, it is often the case that the results from SDEs

are similar to those corresponding to discrete and continuous-time Markov chain

models.

We compare some of the outcomes predicted by the deterministic models to the

SDEs via extensive numerical simulations. When the initial size of the infected popu-

60

lation is small, it can be exi)ect(Hl that the deterministic and stochastic models should

have diffcn-ing results. For exaiupl(>, dis(>as(> extinction can occur in the stochastic

model (the- dis(>as(--free state is often a.n absorbing st;ite). We estimate the \noh-

abilit>- of disease extinction in the stochastic models and show how it depends on

the initial infected population sizi- and th(> basic reprodiution number of the partic­

ular strain. These estimations are check(>d via numerical simulations of the SDEs.

Estimates of the probabilit\- distribution conditioned on nonextinction are obtained

and the results are compared with the deterministic models. We find that in all

of our numerical examples where coexistence of multiple strains is predicted by the

deterministic models that the SDEs predict the opposite, either complete disease ex­

tinction or persistence of only one strain. This finding has important implications for

the evolution of particular disease strains.

In the next section, SDEs are derived from a system of n ordinary differential

equations representing n interacting populations or states. In section 4.3, the deter­

ministic SIS and SIR epidemic models with multiple pathogen strains are described

and their dynamics are summarized. Then the method described in section 4.2 is

used to formulate a system of SDEs for these SIS and SIR models. The probability

distribution for the SDEs and the probability of disease extinction are discussed in

section 4.4. In the last two sections, numerical examples of the SDE epidemic models

are presented and the results are compared to the deterministic epidemic models.

4.2 Derivation of Stochastic Differential Equations

Stochastic differential equations of the Ito type are derived from a system of

n ordinary differential equations. Consider the following system of n differential

equations, where Xi represents the number of individuals in state or population i:

dxi = X i U i - 4 - 2 ^ mji\+ 2 ^ rriijXj i = l,2,... ,n. (4.1)

The parameters 6j and 4 for z = 1, 2 , , , , , n, are the per capita birth and death rates,

respectively, for population i and the ruji (j / i) are the per capita rates of migration

61

frcmi population ; to population j ((-.g,, a sustvptible becomes infected or a transfer

ot iiulniduals between stat(\s). Each paranu'ter may depend on time t as well as on

X,. that is, 6, = b,{xi ,x,„t), d, = d.ixi,....,xn,t), ami m,,, = ^...(.xi, ,.„,a;„, t).

Ito SDEs are derivcnl from model (4.1) assuming that changes in ./:., over short

time steps are normally- distributed and that the random variability is only due to

births, deaths, and migrations, i,e,, demographic variability. We do not consider

environmental \ariabilit\-. We follow the derivation given in Allen [3] and extend it to

/) state variables. Stochastic differential equations for SIS and SIR epidemic models

and Lotka-\'olterra predator-prey and competition models have been derived by this

method, but these models were restricted to two state variables [3, 4].

Let Ai. A'o,.. . A'„ denote the random variables for the number of individuals in

state 1,2. , , 7), Assume that the time interval At is sufficiently small such that the

probabilities of a birth or a death in population i are given by

Prob{AAi(t) = l|A,(t)} = 6,A;(t)At + o(At),

Prob {AA-,(t) = -l\Xiit)} = d^Xiit)At + o(At),

respectively, where AA'j(t) = A'j(t + At) — A'j(t), i = 1,2,,,, ,n. These types of

assumptions generally give rise to a continuous-time Markov chain model when At ->

0 (e.g., [4, 42, 51, 67, 70]). The probability that an individual of population i is

transformed into an individual of population j (migration or transition) is

Prob{AA',(t) = - 1 , AA,(t) = l|(A,(t),Xj(t))} = m,,AMt)At + o(At)

for i,j = l,2,...,n,ij^ j . Then the probability that there are no changes in any of

the n populations in the small time interval At is given by

Prob{AXi(t) = 0, AX2(t) = 0 , . . . , AA4(t) = 0|(A-i(t), A2(t), . . . , A-„(t))}

= l-J2U + d^+ Yl ^iO ^ ^ ( ) ^ + «( ) =^-PAt + o(At). 1=1 \ j=l,j¥'i /

62

Let

/ A.Xiit) \

AA(0

V ^-v„(t) I be the \-ecti,n' for the incremental (4ianges in the p(i])ulation sizes during the time

interval At. Let o(At) represent an /;-vect.or where ea,(4i c(unij(jnent is o(A/). It

is nec'essary to find the mean and covariaiice matrix for the change AA'(t). The

probability of multiple births, deaths, or transformations in time At have probabilities

which are o(At). There are n(// + 1) + 1 different possibilities for AA'(t) in time At,

neglecting the multiple changes which are o(At).

The expectation of AA'(t) is

£(A,V(f)) = Yl 1 = 1 L

bie, + d,i-ei) + ^ rujiiej - e,) j = l , i ^ i

Xiit)At+ [ l-PAt]+o(At).

\v Neglecting terms of o(At) leads to

/

EiAXit))

bi - di - Y. "^ji

62 - 4 - S i^j2

bn-dn- E ^jn j=l,j^n \

A'l

/"2

\\^nj

\ Xi + E rnijAj

A2 + E rn2jXj

A'n + E "^njXj

At

At = /iAt, (4.2)

where e is the standard unit vector in R" with the i^^ component equal to 1. For

simplicity, we have omitted the t in the terms on the right side of the above expression,

x, = x,it).

63

The ctn-ariance matrix foi- A.X'it) is

1 •(AA(t)) = EiAX{A.\y) - EiAX)iEiAX))^.

The product E{AX){E{A.\))^ ^ l-H^'^iAt)'' is of order (At)2. Therefore, the covari-

ance matrix ]', neglecting terms of o(A/), is

r (AA ' ( / ) )«P (AA(AA) ' ' ) .

Expressed in t(M-ms of the birth, death, and migration rates, the covariance matrix

has the form:

r(AA'(t)) =. Yl 1=1

n

- E

?^\T

! = 1

ke,e^' + c/ ,(-e,)(-eJ^ + ^ rUj^iej - ei)iej - e

n

(6, + 4 ) e . e f+ Y. rnj^iej^ - ej^ - e^e^ + eie[)

X,At

X,At.

If we denote

r(AA(t))

' (Jii . . . air ^

yJni . . . <7nnJ

At = CAt,

where

a„ = 6, + 4 ± Y '^ji ^^^ + Y "^iJ- 'i' ^ = 1,2,...

cTzj = Oji = -rUjiX^ - niijXj i ^ j , i, j = 1,2,. .. ,n. (4.3)

it is straightforward to see that the matrix C is symmetric and diagonally dominant.

Therefore, the eigenvalues of C are real and positive [64]. Matrix C is positive definite,

and therefore, has a unique positive definite square root B [64]:

B = VC.

Because of the Cental Limit Theorem, when A'(t) is suflficiently large and At is

sufficiently small, the probability distribution of the change AA'(t) can be assumed

64

to be approximately normally distributed, NipAt,CAl). The normal approximation

IS often a good approximation ewn when A.Y(/) is not so large [3].

If I] is a normally distributcxl vcrtor, r; = (r/^,... , 7;,,)^ - A(0, / ) , then the vector

pAt + B\/At 7] has an approximate normal distribution,

pAt + BV^7] ~ NipAt.CAt).

The distribution of A'(t + A/) is approximately normally distributed:

A(t + At) = A'(t) ± AA(t) ^ A(t) + pAt + B^/AIT].

This latter equation is an Euler approximation of an Ito stochastic differential equa­

tion for A(t) with time step At [44, 45]. If At -^ 0, then T v/At -^ dW„ i =

1,2... . n, where \\\ is the Wiener process [31, 44], If the coefficients p and B satisfy

certain smoothness assumptions and growth conditions on an interval [0,T], in the

time and state variables [31, 44, 45], then A'(t) converges in the mean square sense

to the solution of the Ito SDE:

dA" ^dW _ = , + S _ , A(0) = ,V„ (4,4)

where B = v/C and i r ( t ) is an n-dimensional Wiener process.

4.3 SIS and SIR Epidemic Models

First, we introduce the SIS and SIR deterministic epidemic models with multiple

pathogen strains and summarize their dynamics. Then we use the method in section

4.2 to write the corresponding SDEs for these epidemic models.

4.3.1 Ordinary Differential Equations

We summarize the dynamics for the deterministic epidemic models with n pathogen

strains that were formulated and analyzed in [2]. In these models, it is assumed that

the host population can be infected with n different pathogen strains. Infection with

one strain confers complete cross protection; there is no coinfection and no superin­

fection.

65

The first model is of 575 tyix-; the population consists of susceptibles, 5, and

mdividuals inlected with strains 1 through /;, 4 , j = 1,2,. ,/;,. There is no immu­

nity to infection which means after reco\'(-ry an individual can immediately become

susceptibh- again. The SIS epidiMiiic niodt-l is giv(>n by the system of//.+ 1 differential

ec[uations:

f - s[^-<^i^)-Y^^±hib^.k)

^ = 4 (-ct(AO - 7. - " . ± ) , A: = l , 2 , . . . , n , (4.5)

where 5(0) > 0. 4(0) > 0, A- = 1, 2, .. ,n, and

^ = Nib-diN))-Yo^kIk. (4.6)

fc=i

In model (4.5), b is the birth rate and diN) is the density-dependent death rate.

The parameter Pk is the transmission rate, and jk is the recovery rate of an individual

infected with strain k. The disease-related death rate is ak. We assume all of the

parameters, b, 7/e, ak, and P^, for A; = 1, 2 , , , , ,n, are positive. The density-dependent

death rate d(A') satisfies the following conditions:

(i) deC^[0,oo)

(ii) 0 < d(0) < b

(iii) diN) is increasing for N G [0, 00)

(iv) There exists a constant K > 0 such that diK) = b.

Conditions (i)-(iv) lead to logistic growth for the total population size, where K is

the carrying capacity. In the absence of infection, limj^oo A''(t) = A".

A second epidemic model we consider is an SIR model. In this model, there are

immune or recovered individuals, R. After recovery individuals develop immunity to

the infection. The model is given by the system of n + 2 equations:

66

f = sU-iiw-±'M,)^Y.^j^,,^ k=l / k=l

' ^ = 4 ( ^ - c i ( A f ) - 7 . - n , + /3,^^,A: = l , 2 , . . . , n (4.7)

— = p(-4.v)) + 5]4o. A;=i

where 5(0) > 0,4(0) > 0, A' = 1,2,.. , n, and 4(0) > 0, The total population size

satisfies (4,6),

In [2], there are two parameters that determine competitive exclusion:

^k = . \ (4,8) b + jk + dk

and

Bk = (4 9) ct(O) ± 7. + cv4 ^ •''

for A- = 1, 2,, , n. If for each k = 2,... ,n, one of the following two sets of conditions

holds:

III > Ilk and 7fc + a;fc > 7 i + Q;I, (4.10)

Bi > Bk and Pl > Pk, (4.11)

and

(v) b > d(0) +m<iXk{ak},

then it was proved for models (4.5) and (4.7) that a competitive exclusion result

holds.

Theorem 4.3.1 (see [2]). If conditions (i)-(v) hold, either condition (4-10) or (4-11)

hold, and TZi > 1, then in models (4-5) and (4-^),

l i m i n f 4 ( t ) > 0 and lim 4( t ) = 0, A; = 2, 3 , . . . , n.

67

Theorem 4.3.1 states that strain 1, the one with the largest basic reproduction

number, Tvi > 1, is the dominant strain. A similar competitive exclusion result was

proved for the continuous time SIR ei)id(Miiic model with ma.ss action incidence rate

in .Ackleh and Allen [1]. Numerical examples in [1, 2] showed that the conditions in

Theorem 4.3.1 are sufficient but not lUHcssary for coni])etitive exc lusion. In addition,

there are c-ases when coexistcnct- of more than one strain occurs if the conditions in

Theorem 4.3.1 are not satisfied.

4.3.2 Stochastic Differential Equations

The SDEs for the SIS and SIR epidemic models (4.5) and (4.7) are derived using

the method in section 4.2. Let A'(t) denote the random variable for susceptibles at

time t. Let ^^.(t), denote the random variables for individuals infected with strain

A', A' = 1, 2 n. and let Z(t) denote the random variable for recovered individuals.

The sum Mit) = A'(t) + Efc=i^'c(^) + ^it) is the random variable for the total

population size. All of these random variables are continuous-valued. In addition, we

assume the values of the random variables are bounded, Mit) G [0, 2A ].

We relate system (4.1) and the system of SDEs (4.4) to the SIS epidemic model " bYk

(4.3). The birth rate 6i into the susceptible class is 5i = 6 + E ~T^, the death rate k=l A

from the susceptible class di = diM), and the birth rate into the infective class k is

bk+i = 0 because no individual is born infected. The death rate from the infective

class A; is 4 + i = diM) ± ak, the transition from a susceptible individual to an

infected individual is m^+i^i = -\^, and the recovery rate of an infected individual

represents a transition from an infected to a susceptible individual, mi,fc+i = 7fc for

A; = 1, 2 , . . . ,n. The other transition rates, m^j = 0 for i, j = 2, 3 , . . . ,n, i^ j . Then

the Ito SDEs for the SIS epidemic model with n pathogen strains can be derived by

substituting the values given above for births, deaths, and transition into equations

68

(4.2), (4,3), and (4.4):

^ = '''{^~di^n-Yl'4]+Y''kib + 7k) k=\ / k=\

n+1

+E^ </ir,(/)

• 1 ' ' '"

^ = -ykidiM) + j , + n,) + ^ ^ (4.12)

"+1 r/ir,(t)

^k+lj

n ^ _ L 1 T. 1 1

diy,(t)

k=l 1=1 j = l

E D ' " ' A ' /

Bk+ij—j^ A: = 1,2, . . . , n ; 1 r " n+1 n+1

where Hj(t), j = 1,2 ,n ± 1, are n + 1 independent Wiener processes. The

in ± 1) X {n + 1) matrix B = (%) = x/C, where

_ / c^ii S i 2

yE2i E22

The element

( n \ n

6 + 4i\4 + J]/3,r,/i\/ +^r,(6±7fc), /c=l / k=l

submatrix E12 is a 1 x n matrix, E12 = ia^ a^ ... ai,„+i], where cri,fc+i =

—PkX}'f:/M — jk^'k, k = 1,2,3,.. ,n, and E21 = Efg. Submatrix E22 is an n x n

diagonal matrix with diagonal elements

afc+i,fc+i - Yk idiM) + ak + jk + PkX/M)

k = 1,2,3,... ,n.

For the special case of an ShhS epidemic model with two infected classes, the

SDE's have the form

69

= - v ( 6 - c i ( A 4 - ^ / 3 , ^ ) + ^ r , , ( 6 ± 7 . ) k=l / k=l

dt

EB..-J = l

dt

lYi PihX dt

dYo

dM

~dt

-4 (4,U) + 71 + o,) + ^^^4^ + ^P2 , ,

= - r 2 ( d ( M ) + 7 2 + a.,)±

M

P2Y2X

M

3

+ E^3,, i= i

dWjjt)

dt

dU'j(t)

dt

3 3

Mib-diM))-Yyk<^k + YT.BK fc=i j=i j = i

dWjit) dt

\

(4,13)

where B = \/C and

a n ai2 cri3

^ = 0-21 (722 0

y £731 0 ass )

Next we find the Ito SDE for the SIR epidemic model (4,7), It is similar to the

SIS epidemic model except that recovery of infectives is a transition from infectives

to immune individuals, that is, m„_|_2,fc+i = 7fc for A; = 1,2,,.. , n. The Ito SDE for

the SIR epidemic model with n infected subpopulations follows from (4.2), (4.3), and

70

(4.4):

dX ~dT X [b^d[M)-YPkUjM\ +Ybyk + bz

k=l k=l n+2

+ E« j=i

ij-dWjitl

dt

dY,

dt -YddiM) + ^, + a,)+P,Y,X/M

nvjit)

dt k = 1,2,... ,n

n+2

~ = -ZdiM)+YjkYk + YBn+2,j dZ

dM

"df

k=i j = i n

dWjit) dt

(4.14)

M(b-diM)) + Yc^kYk k=l

n+2 n+2

+EE^^^ 1=1 j=i

dWjjt) dt

where irj(t), j = 1.2 n + 2, are n±2 independent Wiener processes. The matrix

B = iB,j), B = VC. where

^^11 Ei2 0 \

^ = E21 E22 E23

y 0 Es2 Cr„+2,n+2y

The element

/ n \ n

an = A 6 + diM) + Y ^ ^ '^1^^ + E ^ '' + • k=l k=l

Submatrix E12 is a 1 x n matrix, E12 = fai2 ais , . . ai,„+ij, where ai,fe+i

-pkXYk/M k = 1,2,3,... ,n and E21 = £[2- The element

n

^n+2,n+2 = C?(M)Z± ^ ^ 7 ^ 1 ' ; , . k=l

where Submatrix E32 is a 1 x n matrix, E32 = ( a„+2,2 crn+2,3 •.• o"n+2,n+ij'

an+2,k+i = -Yklk k = 1,2,3,... ,n and E23 = £^2- Submatrix E22 is an n x n

71

diagonal matrix with diagonal (>lenieuts

afc+1, ,+1 = Yk id{M) + a;, + 7fc + pk.\/M) , A: = 1, 2, 3,..., /t.

4.4 Probability Distribution of the SDEs

For each of the stochastic ei)idemic models, a multivariate probability density

function (p.d.f.) describes their dynamics tnvr tune For the SIS epidemic model, the

multivariate p.d.f. is

P(-r. yi, 2/2.... , yk. t) = Prob{A'(t) = X,} \it) = yi,Y,{t) = 2/2,... , Y^it) = yk},

where x, yj G [0, 2A'], j = 1.2.... ,k and :i-±Ej=i Vj e [0, 2A"]. For the SIR epidemic

model, the multivariate p.d.L includes the immune state Z'(t):

P[-r. yi,y2. .. .yk,:.t) = Prob{A-(t) = x,Yiit) = yi,Y2it) = ^2,. . • , Vfc(t) = yk, ^(t) = z},

where ,r, yj, : G [0, 2/v'], j = 1, 2 , , , . , A: and x ± z + Ej=i Vj e [0, 2A']. In both of

these models, we have assumed an upper bound for the population size which is twice

the carrying capacity, 2A'. For a specific set of initial conditions, the system of SDEs

has a unique p,d,f. We are interested in the marginal p.d,f, of the infected states

11- and the marginal p.d.L conditioned on nonextinction, which we shall denote as

Pkiy,t) and qkiy-t), respectively,

Pkiy,t) = Vioh{Ykit) = y}

and

qkiy,t) = ?Yoh{Ykit) = y\\\it) > 0}.

It will be seen in the numerical examples that the distribution conditioned on nonex­

tinction may approach a stationary distribution. This distribution is known as the

quasistationary probability distribution. However, because the disease-free equilib­

rium is absorbing and the population size is bounded, solutions will eventually ap­

proach the disease-free equilibrium (and eventually complete population extinction).

72

The length of time until absorption can increase exponentially with the population

size, therehue, it cannot be reali/c-d in the time frame of the simulations [6], Because

the di,s(>ase-free eciuilibrium is absorbing, when the inlected state Ykit) = 0, then

r;.(t + r ) = 0 for r > 0,

4,5 Numerical Examples

For the stochastic SIS and SIR epidemic' models, we use Euler's method [44, 45]

and .MatLab to numerically approximate sample paths for the SDEs, Frc^quency

histograms for the number of infectivc>s at fixed times are constructed based on the

10,000 sample paths. In addition, an approximation to the frequency distribution

conditioned on nonextinction is graphed over time. These graphs are plotted as a

function of time.

Frequency histograms are graphed rather than probability histograms (see Figure

4,2). For example, in Figure 4.2, the histograms are 10,000 x Pkiy,t) and 10,000 x

qkiy,t). If the units on the vertical axis are divided by 10,000, then they represent

probabilities rather than frequencies.

An approximation to the frequency distribution conditioned on nonextinction is

given b}- the dashed curve in addition to the frequency histograms. Statistics from

the frequency histograms (mean and variance) are used to fit a normal frequency

distribution, where the range is adjusted to fit the frequency histogram. The normal

approximations to the distributions conditioned on nonextinction, qkiy,t), are given

by the dashed curves in Figure 4.2, As time increases, for example, at T = 15, the

normal approximation provides a good fit to the quasistationary distribution.

In the first two examples, we consider the case of competitive exclusion in the

deterministic and stochastic SIS and SIR models. We show that Theorem 4.3,1 is

satisfied for these two examples in the deterministic SIS and SIR models. In the

third example. Theorem 4.3.1 is not satisfied, this is a case of coexistence which we

examine in section 4.5.2. In the first two examples, we assume that d(A^) = 1 ± N/K

and 6 = 2 so that conditions (i)-(iv) are satisfied in Theorem 4.3.1, and there are two

73

patlu>geu strains, n = 2.

4,5,1 Competitive Exclusion

In the first example, the SIS epidemic model, we let Pi = 7, P2 = 5,25, 71 = 72 = 1,

oi = a . = 0,5, so that Tli = 2, R-, = 1.5, Bi = 2.8 and B2 = 2.1. Conditions (i)-(v)

and the inetiualitic^s in (4.10) and (4.11) (eitli(>r one is sufficient) are satisfied. It can

be seen in Figure 4.1 that solutions to the deterministic- SIS epidemic model converge

to

lim 4( t ) = 39.35 and hm 4( t ) = 0 , A.' = 2, 3 , . . . ( - ^ x ,n. f—>oo

(4.15

The solution to the deterministic model and the stochastic mean are graphed in

Figure 4.1.

50

•I 40 o

?30

D 2 0

ho

/ / / / /' II

^ .

1,

,.,., 4 Y,

(a)

Time.T 10 15

50

•I 40 o

Cfl

,!2 30 c E S Q 2 0

IIO

if if . if / il-

11/ __

Ii It

/

""*%.

. . .

(b)

>"^~""

'1

I2

4

'•»-«.V.,v......*

Time.T 10 15

Figure 4.1: The solution to the SIS deterministic model is compared to the (a) stochas­tic mean and (b) stochastic mean conditioned on nonextinction. Parameter values satisfy Pl = 1, P2 = 5.25,6 = 2, 71 = 72 = 1, ai = ^2 = 0.5, A(0) = 5(0) = 50, 4(0) = 140) = 1 = 12(0) = 4(0), and K = 100. The basic reproduction numbers are 7 l = 2 and 7 2 = 1.5.

74

Note that the distribution conditioned on noiu'xtjnction (Figure 4.1(b) and the

bottom two graphs in Figure 1.2) closely follows the (leterministic' solution for strain

1 but not for strain 2. In the SDEs, strain 1 may die out, then strain 2 may dominate.

This behavior is not possible in the deterministic model since strain 1 is the dominant

strain and strain 2 alwavs clicks out.

75

Figure 4.2: Frequency histograms at T = 5,10 and 15 for the SIS stochastic model and the approximate frequency distributions conditioned on nonextinction for the number of infectives over the interval T G [0.5,15] (based on the normal distribution, dashed curve) are given in the bottom two figures. Parameter values satisfy Pi = 7, p.^ = 5.25, b = 2, diM) = 1 + M/K, 71 = 72 = 1, oi = a2 = 0.5, At = 0.01, l'i(O) = 1 = ^2(0), A"(0) = 50 and A' = 100. The basic reproduction numbers are IZi = 2 and 7 2 = 1-5. The width of the intervals of the histograms are 3 units: [-1.5,1.5), [1.5,4.5), etc.

In Figure 4.2, the stochastic model for an SIS model is numerically simulated.

76

Initially, there are two infected individuals, one inlected with strain 1 and one infected

with strain 2. The frequenc>' histograms in Figure 4.2 are bimodal. The frequenc'V

value at zero reprcwnts the probability (freciucnc-y) of extinction for a particular

strain, pfc(0,T) = Prob{rfc(r) = 0}, A; = 1,2. The total frequency at zero was not

shown in Figure 4.2: this value is larger than 5,000 in some cases. To better view

the distribution for valuc\s of ^^.(r) > 0 in Figure 4.2, the vertical axis is shortened

(maximum heciuenc>- is 2000). The frequency for values at zero is graphed separately

in Figure 4.4.

Two sample paths, graphed in Figure 4.3, illustrate two of the many possible

dynamics for the stocliastic model. In Figure 4.3 (a) strain 1 persists and in Figure 4.3

(b) strain 2 persists. Compare the sample path in Figure 4.3 (a) to the deterministic

solution in Figure 4.1.

60

50

40

S30 CO

20

10

— Strain 1 • - - Strain 2

4' .

r

ji

ki^ ,1.1 !i0;

:y

TimeT TimeT 10 15

Figure 4.3: Two sample paths for the stochastic SfS model for the parameter values and initial conditions given in Figure 4.2. In (a) strain 1 persists and in (b) strain 2 persists.

The mean and standard deviation (/i and a) of the distribution conditioned on

nonextinction for Yi and Y2 computed from the numerical simulations at T = 15

satisfy

/ii(15) ?«37.0, 4 ( 1 5 ) ^ 8 . 8

77

and

4,(15) Ri 27.1, a2(15) ^ 8.3.

The mean of strain 1 is closc> to tli(> deterministic eciuilibrium given in (4.15), 4 =

39.3o. The mean of strain 2 is close to the endemic- equdibrium where strain 1 is

absent, 4 = 30.06.

The probability strains 1 or 2 are eliminated, pi(0,r) and P2(0,T), are approxi-

matel>- constant for a large range of times. An estimate for these constants, assuming

independence, is gi •en by equation

if ^;c < 1 PkiO,T)^{ a, - (4.16)

i ) , ii'Rk>i.

where o/,, is the initial number of individuals infected with strain k [13, 14, 41]. For

strain 1 and strain 2. the estimates are pi(0, T) ftj l/7^i = 0.5 and p2(0, T) ^ I/XI2 =

2/3. respectively (see Figure 4.4). The probability both strains are eliminated by time

T (assuming independence) is given by the estimate p(0,T) •^ iTZ{R,2)~'^ = 1/3. The

10,000 samples paths give an approximation to the true values of these probabilities,

pi(0,15) ^0 .49, P2(0,15) !=i 0.81, and p(0,15) ^ 0.30.

See Figure 4.4. Based on these probabilities, the proportion of sample paths where

there is coexistence can be calculated. If pc(P) = Prob{YiiT) > 0 and y2(T) > 0},

then

Pc(15)«0 ; (4.17)

coexistence is impossible. The approximations obtained from the simulations for

Pi(0,T) and p(0 , r ) agree with the predicted estimates, but the value for strain 2 is

larger than the estimate. The rate of increase of P2(0,r) slows down considerably

after it reaches approximately 2/3. Since the estimate was based on the events being

independent, we should not expect the estimate to be accurate. Competition between

78

the two strains makes the c-vents dependent and results in the weaker strain (i.e., strain

2) being eliminated at a fasten- rale. Note that the vertical axis in Figure 4.4 gives

the freciuency rather than the probability.

10000

o o o o

8000

•5 6000 "3 O >% c 4000 CD

cr

^ 2000

Strain 1 Strain 2 Both Strains

10 15 t imeT

Figure 4.4: The frequency distributions for elimination of strain 1, strain 2 or both strains in stochastic SIS epidemic model for the parameter values and initial condi­tions given in Figure 4.2.

In the second example, the SIR epidemic model is discussed, we use the same

parameter values as in the previous example: Pi = T, P2 = 5.25, 71 = 7 2 = 1,

ai = Qo = 0.5, b = 2,so that Tei = 2, 7l2 = 1.5, Bi = 2.8 and B2 = 2.1. Conditions

(i)-(v) and the inequalities in (4.10) and (4.11) are satisfied. Therefore, Theorem

4.3.1 is satisfied for the deterministic SIR model. It can be seen in Figure 4.5 that

solutions to the deterministic SIR epidemic model converge to

lim 4( t ) = 28.15 and lim 4( t ) = 0, k = 2,3,...,n. t^OO t->CX)

(4.18)

The solution to the deterministic model and the stochastic mean are graphed in

Figure 4.5.

79

30

o 2 5

o w o20

i i 5 L s Q •g io

' ^ —

^—' 1 1 I

f /•

/ / j 1 It

'" /'

r ^~~--

. . . . . Y,

4 — '1

- - - ' 2

{«)

-

-

30

;25 -

•20

15

eio

-

1

If if If

II tj , if-il:

2f *

1 *"' * If Ii

-.. ^

'

_ , . • „ • • . - - , - , . „ .

'

- - - . . . v „ , . . . . _

. , . , Y,

,,.4

- - ' 2

,...-.,--.-

4 6 Time.T

10 4 6 Time.T

10

Figure 4.5: The solution to the SIR deterministic model is compared to the (a) stochastic mean, (b) stochastic mean conditioned on nonextinction. Parameter values satisfv Pl = 7. ,4 = 5.25, b = 2, diM) = 1 + M/K, 71 = 72 = 1, 0 1 = ^ 2 = 0.5, 5(0) = A-(0) = 50, 4(0) = 51(0) = 1 = r2(0) = 4(0), P(0) = Z(0) = 0, and A =100. The basic reproduction numbers are 7?-i = 2 and 7 2 = 1.5.

As was shown for the SIS epidemic model, the distribution conditioned on nonex­

tinction (Figure 4.5(b) and the bottom two graphs in Figure 4.6) closely follows the

deterministic solution for strain 1 but not for strain 2. In the SDEs, strain 1 may die

out. then strain 2 may dominate. This behavior is not possible in the deterministic

SIR epidemic model.

80

Figure 4.6: Frequency histograms at T = 5,10 and 15 for the SIR stochastic model and the approximate frequency distributions conditioned on nonextinction for the number of infectives over the interval T G [0.5,15] (based on the normal distribution, dashed curve) are given in the bottom two figures. Parameter values satisfy Pi = 7, P2 = 5.25, b = 2, diM) = 1 + M/K, 7 = 72 = 1, ai = a2 = 0.5, At = 0.01, YiiO) = 1 = 12(0), A:(0) = 50, Z(0) = 0 and A: = 100. The basic reproduction numbers are T i = 2 and 7?-2 = 1.5. The width of the intervals of the histograms are 3 units: [-1.5,1.5), [1.5, 4.5), etc.

Now, in Figure 4.6, the stochastic model for an SIR model is numerically sim-

81

ulatcxl. Frequenc-y histograms for the number of infec-tives at fixed times are (-ou­

st ructcxl based on the 10,000 sanipl(> paths. The frequency for values at zero is

graphed separately- in Figure 1.7.

In the SIR model, the mean and standard deviation ip and a) of the distribution

conditioned on nonextinction for Yi and V, computed from the numerical simulations

at T = 13:

/4(15)«26.3, 4 ( 1 5 ) ^ 8 . 0

and

/"'/2(15) ? 18.1, 4(15) «i 7.8.

The mean of strain 1 is close to the deterministic equilibrium given in (4,18), 4 =

28.15. The mean of strain 2 is close to the endemic equilibrium where strain 1 is

absent 4 = 20.52.

From the 10,000 samples paths, we estimate the probability of disease extinction

in the SIR epidemic model:

Pi(0.15)?^0.49, P2(0,15) ^0,86, and p(0,15) f« 0,35,

See Figure 4.7 (Note that pi(0,T) is close to the estimate in (4.16) but not P2(0,T).)

Based on these probabilities, the proportion of sample paths where there is coexistence

shows that pc(15) ~ 0; coexistence is impossible in this case.

82

10000

Q 8000 o o o

•5 6000 • ^

O

c 4000 (D

cr

2000

- - - strain 1 - - Strain 2 - — ^ t ^ S t r a i n s

/ / / /

i ' -/ (

1 /

time T 10 15

Figure 4.7: The frequency distributions for elimination of strain 1, strain 2 or both strains in the stochastic SIR epidemic model for the parameter values and initial conditions given in Figure 4.6.

4.5.2 A Study of Coexistence

An example of coexistence of both strains in the deterministic SIR epidemic model

is studied in the SDE SIR epidemic model. We let the carrying capacity be A' = 100,

birth rate he b = 6. We assume the density-dependent death rate is diN) = l-\-bN/K,

transmission rates for the two strains are Pi = 30, P2 = 15, recovery rates for the two

strains are 7i = 6,72 = 1, and the disease-related death rates for the two strains are

Qfj = 4, Q;2 = 1.4. Then the reproduction numbers Ki = 1.875 > 1.786 = 7?.2 and

;8i = 2.727<4.412 = B2, but Theorem 4.3.1 is not satisfied because Q;i+7i > a2-l-72

and Pl > P2- The following positive equilibrium is locally asymptotically stable for

the deterministic SIR epidemic model,

5 = 42.56, 4 = 12.60, 4 = 11.99 and R = 16.85.

Furthermore, the numerical simulations of the deterministic model suggest that this

equilibrium is globally asymptotically stable. However, we shall find that the SDE

83

uiodel for tlu- abcne c-xam])lc- does not give c-oexistence.

The two coexisting strains in the deterministic model are graphed in Figure 4.8.

It takes a long time for the numerical solution to converge to the equilibrium solution.

Since the initial variation is not visible in Figure 4.8(a), Figure 4.8(b) is graphed up

to time T = 2.

20

15

-10

• r

(a)

•

-

20

15

-10

5

/ \ ('"

/

/ /

1- C

VJ

1 1

•

50 Time.T

100 150 0.5 1 Time.T

1.5

Figure 4.8: Coexistence of strain 1 and strain 2 in the deterministic SIR model for the parameter values and initial conditions. Pi = 30, P2 = 15, 6 = 6, diN) = 1-1- 5N/K, ni = 6,02 = 1, ai = 4,^2 = 1.4, At = 0.01, 4(0) = 1 = 4(0), 5(0) = 50, P(0) = 0 and A" = 100, is graphed in (a) from T = 0 to T = 150 (b) from T = 0 to T = 2

In Figure 4.9, the stochastic model is numerically simulated. Frequency his­

tograms for the number of infectives at fixed times are constructed based on the

10,000 sample paths. The frequency for values at zero is graphed separately in Fig­

ure 4.11.

84

, at lime T=15 I attimeT=15

Figure 4.9: Frequency histograms at T = 5,10 and 15 for the SIR stochastic model and the approximate frequency distributions conditioned on nonextinction for the number of infectives over the interval T G [0.25,15] (based on the normal distribution, dashed curve) are given in the bottom two figures. Parameter values satisfy Pi = 30, /32 = 15, 6 = 6, diM) = 1 + 5M/K, ji = 6,72 = 1, ai = 4,^2 = 1.4, At = 0.01, y^O) = 1 = ^2(0), A:(0) = 50, Z(0) = 0 and A: = 100. The basic reproduction numbers are 7<!-i = 1.875 and 7?-2 = 1.786.

The two coexisting strains in the deterministic model are compared with the

stochastic means from the SDE model in Figure 4.10.

85

40

c35 .2

1 3 0 CO

o •^25

p20

c

CO

/ — ' i

,....4

Y, | \ _.,>-~-.r..Vx.»s'-'v.y»-'-'"v..-'.>^,'.'»'>.»v,,l^'%'''''.^-'

J^-L'-

' • ' • * ^ . . ^ n . . v ' ^ '

F Time.T

10 15

Figure 4.10: The coexistence solution to the SIR deterministic model is compared to the (a) stochastic mean, (b) stochastic mean conditioned on nonextinction. Parameter values and the initial conditions are the same as in Figure 4.9

In the SDE SIR epidemic model, the mean and standard deviation ip and a)

of the distribution conditioned on nonextinction for Yi and Y2, computed from the

numerical simulations at T = 15 satisfy

pi(15) ?« 15.3, 4(15) ?« 8.0

and

/i2(15) ;: 34.2, 4 ( 1 5 ) ^ 7 . 8 .

The mean of strain 1 approximates the endemic equilibrium where strain 2 is absent

4 = 18.65. The value of/ii(15) cannot approximate the coexistence equilibrium value

because as shown next the probability of coexistence at 4 = 15 is zero. The mean of

strain 2 is close to the endemic equilibrium where strain 1 is absent 4 = 35.79. The

10,000 sample paths give pi(0,15) ^ 0.87, P2(0,15) ^ 0.62, and p(0,15) ^ 0.49.

Based on these probabilities, the proportion of sample paths where there is coexistence

satisfies pc(15) ~ 0; coexistence does not occur in this case. Note that p(0,T) and

Pi(0,T) are increasing and p2(0,T) is approximately constant by time T = 15.

86

10000

o o o o

8000

6000

o c 4000

cr

2000

- - - strain 1 -•-• Strain 2

Botti Strains -

, , - - - ' " '

10 15 time T

Figure 4.11: The frequency distributions for elimination of strain 1, strain 2 or both strains in the SDE SIR model for the parameter values and initial conditions given in Figure 4.9.

Although coexistence is possible in the deterministic models at least one strain

dies out in the stochastic models. We have simulated several other coexistence cases

in the deterministic SIR epidemic model and found that the stochastic model predicts

competitive exclusion. Coexistence may be an unlikely occurrence in the stochastic

models. Further analysis is required to come to a conclusion.

4.6 Summary

The stochastic differential equation models for SIS and SIR epidemic models with

multiple pathogens and density-dependent mortality have been formulated. The com­

petitive exclusion and coexistence in continuous-time stochastic SIS and SIR epidemic

models with multiple pathogen strains have been investigated using the SDEs. The

dynamics of the stochastic models are compared to the dynamics of the determin­

istic models. The stochastic mean conditioned on nonextinction closely follows the

deterministic solution for the dominant strain but not for the weaker strain. In the

stochastic models, there is also a high probability of disease extinction as predicted

87

by /)(0,r), esi)ecially for a small initial infected population size.

In the deterministic models with disease-related death and density-dependent mor­

tality, it the parameters satisfy the conditions in Theorem 4,3.1, then competitive

exclusion occurs. The stochastic model corresponding to the same conditions also

predicts competitive exclusion. But, if the parameters do not satisfy the conditions

in Theorem 4.3.1 there is a possibility for coexistence in the deterministic model.

Some of these coexistence cases in the deterministic models have been tested in the

stochastic models, but coexistence has not been obtained in the SDE models. At least

one strain dies out; competitive exclusion is predicted in the SDE's. The possibility

of coexistence in the stochastic models needs further investigation in future research.

CHAPTER V

CONCLUSIONS AND FUTURE DIRECTIONS

Competitive exclusion and coexistence are studied in deterministic and stochastic

epidemic models. In Chapter II, the dynamics of discrcMc-time deterministic SIS

epidemic model with multiple pathogens are investigated analytically and numerically.

Ihe analytical results for the SIS epidemic model show that the competitive exclusion

principle holds for a single patch and for a two-patch model when only susceptibles

disperse. The competitive exclusion may not hold with coinfection, superinfection

or density-dependent mortality. But in two patches, if strain 1 dominates in one

patch and strain 2 dominates in another patch, then coexistence might occur in

both patches when infectives disperse between the two patches with some limitations

on the dispersal probabilities. In this case, the relationship between the dispersal

probabilities and basic reproduction numbers needs to be examined further in future

research.

In Chapter III, the stochastic SIS epidemic model with multiple pathogens are

formulated as discrete-time Markov chain models in a single patch and two patches.

Since the disease-free state is the only absorbing state in the stochastic models, the

disease will eventually be eliminated from the population. However, the elimination

may take a very long time for a large population size. Numerical examples suggest that

spatial heterogeneity promotes coexistence. The dispersal capabdity of susceptibles

and infectives and the magnitude of the basic reproduction numbers of the pathogen

strains have to be studied further in future research to determine the persistence of

the pathogen strains.

In Chapter IV, the competitive exclusion and coexistence in continuous-time

stochastic SIS and SIR epidemic models with multiple pathogens and density-dependent

mortality have been investigated using the stochastic differential equation models.

The epidemiological assumptions of these models are more complicated than the

models in Chapters II and III, but in Chapter IV models we do not include dispersal.

89

\\ hen competitive exclusion occurs in the deterministic models, it is shown that the

SDEs also predic-t the same or complete disease extinction. But when c-oexisteuee

occurs in the deterministic model, the SDEs do not predict coexistence. Since we

have considered only a few examples, this latter result has to be investigated further

in future research.

The study of persistence and evolution of diseases with multiple pathogen strains

will continue to be of interest as new pathogen strains emerge and are identified (e.g.,

influenza, dengue fever, hantavirus). A more thorough understanding of factors that

lead to the coc^xistence of multiple strains, competitive exclusion of all but one strain,

or the exclusion of all strains will help in the development of better treatment, control,

and management strategies of these emerging diseases.

90

BIBLIOGRAPHY

[1] A. Ackleh, L. J. S. .Allen, Competitive exclusion and coexistence for pathogens 111 an epidemic model with variable population size. 4 Math. Biol. (2003), In press.

[-J A. .\c-kleh, L, J. S, Allen, Competitive exclusion in SIS and SIR epidemic- models with total c-ross imniunit\' and density-depiMident host mortality. Submitted for publication.

[6\ E.J. Allen, Stoc-hastic differential equations and persistence time for two inter­acting populations. Dynamics of Continuous, Discrete and Impulsive Systems. 5 (1999) 271-281.

[4J L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biol­ogy. Prentice Hall, Upper Saddle River, N.J., 2003.

[o] L.J.S. Allen, Some discrete-time SI, SIR, and SIS epidemic models. Math. Biosei. 124 (1994), 83-105.

[6] L.J.S, Allen, A.M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time. Math. Biosei. 163 (2000), 1-31.

[7] L. J. S. Allen, X. Kirupaharan, S. M. Wilson. SIS epidemic models with multiple pathogen strains. 4 Difference Eqns. Appl. (2003), In press.

[8] R.M. Anderson, R. M. May, Coevolution of hosts and parasites. Parasitology 85 (1982), 411-426,

[9] R.M, Anderson, R.M, Mav, Infectious Diseases of Humans Dynamics and Con­trol, Oxford Univ. Press, Oxford, 1992.

[10] \ ' Andreasen, J. Lin, S.A. Levin, The dynamics of cocirculating influenza strains conferring partial cross-immunity, 4 Math. Biol. 35 (1997), 825-842.

[11] V. Andreasen, A. Pugliese, Pathogen coexistence induced by density dependent host mortality, 4 Theor. Biol. 177 (1995), 159-165.

[12] F. Arrigoni, A. Pugliese, Limits of a multi-patch SIS epidemic model. Journal of Math. Biology 45 (2002) 419-440

[13] N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and its Appli­cations. Charles Griffin, London, 1975.

[14] N.T.J. Bailey, The Elements of Stochastic Processes with Applications to the Natural Sciences. John Wiley & Sons, New York, 1990.

[15] H. J. Bremermann, H.R. Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol. 27 (1989), 179-190.

[16] C. Castdlo-Chavez, W. Huang, J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases, SI AM J. Appl. Math. 56 (1996), 494-508.

91

[!'] C Castillo-Chavez, W. Huang, J. Li, The effects of females' susceptibility on the coexistence of multiple pathogen strains of sexually transmitted diseases, 4 Math. Biol. 35 (1997), 503-522.

^ ^ * - Castillo-Chavez, W Huang, J, Li, Competitive exclusion and coexistence ot multiple strains in an SIS STD model, SIAM J. Appl. Math. 59 (1999), 1< 90-1811, V '

\'i ^,'^^tillo-Chavez, A.-A, \'akubu. Dispersal, disease and life-history evolution, ^yath. Biosei. 173 (2001), 35-53.

ri- '^^tillo-Chavez, .A..-.\. A'akubu, Intraspeciflc competition, dispersal, and disease dynamics in discrete-time patchy environments, In: Mathematical Ap-pivaehes for Emerging and Reemerging Infectious Diseases An Introduction C. LastiUoChavez, S. Bknver, P. \-an den Driessche, D. Kirschner, A.-A. Yakubu (Eds.), Springer-\'erlag, New York, 2002, pp, 165-181.

[-1] p.J . Daley, J. Gani, Epidemic Modelling: An Introduction. Cambridge Studies m Mathematical Biology, Cambridge Univ. Press, 1999.

[_2J J.H.P. Dawes and J.R. Gog, The onset of oscillatory dynamics in models of multiple disease strains. J. Math. Biol. 45 (2002) 471-510.

[23] M. DoebeU, G.D. Ruxton, Stabilization through spatial oattern formation in metapopulations with long-range dispersal, Proc. R. Soc. Lond. B 265 (1998), 1325-1332.

[24] R. Durrett. S.A. Levin, Stochastic spatial models: a user's guide to ecological applications. Phil. Trans. Roy. Soc. London B. 343 (1994) 119-127.

[25] S. N. Elaydi, An Introduction to Difference Equations. 2nd Ed., Springer-Verlag, New \brk, 1999.

[26] L. Esteva, C. Vargas, Coexistence of different serotypes of dengue virus, 4 Math. Biol. 46 (2003) 31-47

[27] Z. Feng, J.X. Velasco-Hernandez, Competitive exclusion in a vector-host model for the dengue fever, 4 Math. Biol. 35 (1997), 523-544.

[28] F Fenner, Biological control, as exemplified by smallpox eradication and myx­omatosis. Proc. R. Soc. B218 (1983) 259-285.

[29] N.M. Ferguson, R.M. May, R.M. Anderson, Measles: persistence and syn-chronicity in disease dynamics. In: Spatial Ecology: The Role of Space in Pop­ulation Dynamics and Interspecific Interactions. Tilman, D. and P. Kareiva (Eds.), Princeton Univ. Press, Princeton, N.J., 1997, pp. 137-157.

[30] B.F. Finkenstadt, O.N. Bjornstad, B.T. Grenfell, A stochastic model for ex­tinction and recurrence of epidemics: estimation and inference for measles out­breaks. Biostatistics 3,4 (2002) 493-510.

[31] T.C. Gard, Introduction to Stochastic Differential Equations Marcel Dekker, Inc., New York and Basel, 1988.

92

[32] S. Gupta, K. Trenholme, R.M. Anderson, K.P. Day, Antigenic- diversity and the transmission dynamics of Plasmodium falciparum. Science. 263 (1994) 961-963.

[33] I. Hanski, Metapopulation Ecology. Oxford Univ. Press, Oxford, 1999.

[34] H.W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2000), 599-653.

[3o\ T. E. Harris. 1963. The Theory of Branching Processes. Prentice-Hall, Inc., Englewood Cliff's, New Jersey.

[36] CM. Hernandez-Suarez, A Markov Chain approach to calculate RQ in stochastic epidemic models. 4 Theor. Biol. 215 (2002) 83-93.

[3/] M.E. Hochberg, R.D. Holt, The coexistence of competing parasites I, The role of cross-species infection, Amer. Nat. 136 (1990), 517-541.

[38] E.E. Holmes, Basic epidemiological concepts in a spatial context. In: Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Inter­actions. Tilman, D. and P. Kareiva (Eds.), Princeton Univ. Press, Princeton, X.J., 1997, pp, 111-136,

[39] P.J, Hudson, A. Rizzoli, B,T. Grenfell, H, Heesterbeek, A,P Dobson (Eds.), The Ecology of Wildlife Diseases. Oxford Univ. Press, Oxford, 2002.

[40] \ ' IsliAm.G. Medley (Eds.) Models for Infectious Human Diseases: Their Struc­ture and Relation to Data. Cambridge University press, Cambridge, 1996.

[41] J.A. Jacquez, C.P. Simon, The stochastic SI model with recruitment and deaths I. comparison with he closed SIS model. Math. Biosei. 117 (1993) 77-125.

[42] S. Karlin, H. M. Taylor. 1975. A First Course in Stochastic Processes. 2nd Ed. Academic Press, Xew York.

[43] M. Kimmel, D. Axelrod. 2002. Branching Processes in Biology. Springer-Verlag, New York.

[44] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equa­tions. Springer-Verlag, New York, 1992.

[45] P.E. Kloeden, E. Platen, and H.Schurz, Numerical Solution of Stochastic Differ­ential Equations through Computer Experiments. Springer-Verlag, BerUn, 1997.

[46] C.L. Lehman, D. Tilman, Competition in spatial habitats. In: Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. D. Tdman. and P. Kareiva (Eds.), Princeton Univ. Press, Princeton, N.J., 1997, pp. 185-203.

[47] S. A. Levin, Community equihbria and stability, and an extension of the com­petitive exclusion principle, Amer. Nat. 104 (1970), 413-423.

[48] S.A. Levin, Dispersion and population interactions, Amer. Nat. 108 (1974), 207-221.

93

14Jj S. Le\-in, D. Pimentel, Selection of intermediate rates of increase in parasite-host systems, Amer. Nat. 117 (1981), 308-315.

TTT ^P^itch, M. A, Nowak, The evolution of virulence in sexually transmitted Hl\ /AIDS. 4 Theor. Biol. 174 (1995) 427-440.

[51] J.H Matis and T.R. KiHe,Stochastic Population Models. Springer-Verlag, New "lork, 2000. 1 6 6,

I f , ^ ' • ' S])atial chaos and its role in ecology and evolution. Frontiers in i\lathematieal Biology, S. A. Levin (Ed.), Springer-Verlag, Berlin and Heidel­berg, 1994, pp. 326-344.

[53] R.M. May, R.i\L Anderson, Epidemiology and genetics in the coevolution of parasites and hosts, Proc. R. Soc. Lond. B 219 (1983), 281-313.

[o4J R. May. Al..\. Nowak. Superinfec-tion, metapopulation dynamics, and the evo­lution of virulence, J. Theor. Biol. 170 (1995), 95-114.

[ooj R.M. May, M,A, Nowak, Coinfection and the evolution of parasite virulence, Proc. R. Soc. Lond. B 261 (1995), 209-215.

[56] C J. Mode. 1971. Multitype Branching Processes Theory and Applications. American Elsevier Pub. Co., Inc., New York.

[57] D. Mollison, Epidemic Models Their Structure and Relation to Data. Cambridge I niversity Press, Cambridge, 1995.

[58] J. Mosquera, F.R. Adler, Evolution of virulence: a unified framework for coin­fection and superinfection, J. Theor. Biol. 195 (1998), 293-313.

[59] I. Xasell, The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28 (1996) 895-932.

[60] I. Xasell, On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosei. 156 (1999) 21-40.

[61] I. Xasell, Endemicity, persistence, and quasi-stationarity. In: Mathematical Ap­proaches for Emerging and Reemerging Infectious Diseases An Introduction C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu (Eds.), Springer-Verlag, New York, 2002, pp. 199-227.

[62] M.A. Nowak, R.M. May, Superinfection and the evolution of parasite virulence, Proc. R. Soc. Lond. B 255 (1994), 81-89.

[63] A. Okubo, S.A. Levin, Diffusion and Ecological Problems. Second Ed. Springer-Verlag, New York, Berlin, and Heidelberg, 2001.

[64] J.M. Ortega, Matrix Theory Plenum Press, New York and London, 1987.

[65] S.W. Pacala, S.A. Levin, Biologically generated spatial pattern and the coexis­tence of competing species, In: Spatial Ecology: The Role of Space in Popula­tion Dynamics and Interspecific Interactions. D. Tilman and P. Kareiva (Eds.), Princeton Univ. Press, Princeton, N.J., 1997, pp. 204-232.

94

[66] A. Pugliese, On the evolutionary coexistence of parasite strains. Math. Biosei. 177&178 (2002), 355-375.

[67] E. Renshaw Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge, 1993.

[68] P.R. Rohani, R.M. May, M.P. Hassell, Metapopulations and equilibrium stabil­ity: the eff'ects of spatial structure, J. Theor. Biol. 181 (1996), 97-109.

[69] C. Schmaljohn, B. Hjelle, Hantaviruses: A global disease problem, Emerging Infect. Dis. 3(2) (1997), 95-104.

[70] H.M. Taylor and S. Karlin, An Introduction to Stochastic Modeling. 3rd Ed. Academic Press, New York, 1998.

[71] M. van Baalen, M.W. Sabelis, The dynamics of multiple infection and the evo­lution of virulence, Amer. Nat. 146 (1995), 881-910.

[72] P. Yodzis, Competition for Space and The Structure of Ecological Communi­ties, Lecture Notes in Biomathematics. S. Levin (Ed.) Springer-Verlag, Berlin, Heidelberg, New York, 1978.

95