c 2012, glenn e. lahodny jr. - tdl

Persistence or Extinction of Disease in Stochastic Epidemic Models andDynamically Consistent Discrete Lotka-Volterra Competition Models

by

Glenn E. Lahodny Jr., M.S.

A Dissertation

In

Mathematics and Statistics

Submitted to the Graduate Facultyof Texas Tech University in

Partial Fulfillment ofthe Requirements for the Degree of

Doctor of Philosophy in Applied Mathematics

Approved

Linda Allen(Co-chair)

Lih-Ing Roeger(Co-chair)

Ed Allen

Peggy Gordon MillerDean of the Graduate School

August, 2012

Texas Tech University, Glenn E. Lahodny Jr., August 2012

ACKNOWLEDGEMENTS

First, I would like to acknowledge the faculty in the department of Mathematics

and Statistics at Texas Tech University. It was a privilege and a pleasure to learn

from and work with such wonderful professors. I will never forget the experiences I

had in this department and at Texas Tech University. The staff in the Mathematics

and Statistics department also deserve special thanks for their hard work, kindness,

and willingness to assist in any matter.

I would also like to thank my advisors, Drs. Linda Allen and Lih-Ing Roeger, for

taking the time to mentor me. Thank you both for your hard work, suggestions,

kindness, and always taking the time to help. Thanks to Dr. Edward Allen for all of

his helpful comments, teaching me about stochastic differential equations, taking

time to write recommendation letters, and serve as a committee member.

Thanks to my parents, Glenn and Janice Lahodny, for the love, encouragement,

and support you have given me. I am very fortunate to have parents who are so

supportive and caring despite the circumstances, and I love you both very much.

To my fiancee, Megan Trenck, thank you for being the incredible person that you

are. You are a kind, beautiful, and intelligent women and you have been the most

positive influence in my life since we’ve met. Thank you for all of your love and

support over the years. I love you very much and look forward to a long, happy life

together.

Last, I would like to thank Mrs. Ramona Pelkey for pushing me to be a better

student, work hard, and always do my best. Without your influence, I am certain

that I would not have the education and opportunities that I do today.

ii


TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Dynamically Consistent Discrete Lotka-Volterra Competition Models . . 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Discrete Lotka-Volterra Competition Models . . . . . . . . . . 9

2.2.1 Existence and Local Stability of Equilibria . . . . . . . . . 9

2.2.2 Monotonicity, Positive Invariance, and Global Stability . . 11

2.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. Extinction or Persistence of Disease in Stochastic Epidemic Models . . . 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Single Infectious Group . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Multiple Infectious Groups . . . . . . . . . . . . . . . . . . . . 29

3.4.1 SEIR Model . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.2 Vector-Host Model . . . . . . . . . . . . . . . . . . . . . . 33

3.4.3 Stage-Structured Model . . . . . . . . . . . . . . . . . . . 37

3.4.4 Treatment Model . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4. Stochastic Multi-Patch Epidemic Models . . . . . . . . . . . . . . . . . 51

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Multi-Patch SIS, SIR, and SIRS Models without Demographics 52

4.2.1 Deterministic Models . . . . . . . . . . . . . . . . . . . . . 53

4.2.2 Markov Chain Models . . . . . . . . . . . . . . . . . . . . 59

4.2.3 Stochastic Differential Equation Models . . . . . . . . . . . 62

4.2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . 67

4.3 Multi-Patch SIS, SIR, and SIRS Models with Demographics . . 76

4.3.1 Deterministic Models . . . . . . . . . . . . . . . . . . . . . 76

iii


4.3.2 Markov Chain Models . . . . . . . . . . . . . . . . . . . . 83

4.3.3 Stochastic Differential Equation Models . . . . . . . . . . . 84

4.3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . 85

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

iv


ABSTRACT

Two distinct topics are considered in this dissertation. The first topic concerns

nonstandard finite-difference (NSFD) schemes for the Lotka-Volterra competition

model, and the second topic concerns the persistence or extinction of disease in

stochastic epidemic models.

A problem of interest in numerical analysis is to derive a discrete-time model

which can be used to approximate the solution of an ordinary differential equation

(ODE) or system of ODEs. These discrete models can be constructed by applying

finite difference schemes to a given ODE. The goal is to derive a discrete model

which preserves the properties of the corresponding continuous model. In Chapter

2, the Lotka-Volterra competition model is introduced and some well-known

properties of the model are stated. A general class of discrete-time competition

models constructed from a NSFD scheme is considered. Sufficient conditions are

derived such that the elements of this class of difference equations are dynamically

consistent with the Lotka-Volterra competition model. The discrete models are

shown to preserve the positivity of solutions, existence and stability conditions of

the equilibrium points, boundedness of solutions, and monotonicity of the

Lotka-Volterra system.

The second topic concerns thresholds for epidemic models. In deterministic

epidemic theory, the basic reproduction number, R0, and type reproduction

numbers, Ti, are well-known thresholds used to determine whether a disease will

persist or become extinct. For stochastic epidemic models, there are similar

thresholds which are used to estimate the probability of disease persistence or

extinction. Typically, the deterministic and stochastic thresholds are discussed

separately. In Chapter 3, some deterministic (ODE) epidemic models from the

literature are considered. The basic reproduction number and type reproduction

numbers are calculated for each of these models and a corresponding

continuous-time Markov chain (CTMC) model is derived. For each of the CTMC

models, a stochastic threshold is computed as well as the probability of disease

persistence or extinction. In addition, a new relationship is illustrated between the

deterministic and stochastic thresholds.

v


Factors such as spatial heterogeneity, connectivity, and dispersal of individuals

through worldwide travel can significantly affect disease dynamics. The effects of

these factors have been studied for several infectious diseases including influenza,

severe acute respiratory syndrome (SARS), and tuberculosis. In these studies, the

population is split into several groups or patches and dispersal is allowed between

these patches. In Chapter 4, deterministic and stochastic multi-patch epidemic

models with and without demographics are derived and analyzed. As in Chapter 3,

the basic reproduction number is calculated for the deterministic models. Two types

of stochastic multi-patch models are explored: CTMC models and stochastic

differential equation (SDE) models. For the CTMC models, the stochastic threshold

is computed as well as the probability of disease persistence or extinction.

Numerical examples illustrate the differences between the deterministic and

stochastic patch models.

vi


LIST OF FIGURES

2.1 Four approximate solutions to the Lotka-Volterra competition model

in the phase plane. For figure (a), the step size is h = 0.5 and for figure

(b), the step size is h = 0.1. . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Four approximate solutions to the Lotka-Volterra competition model

as a function of time. Parameter values and initial conditions are the

same as in Figure 2.1 with a step size of h = 0.1. . . . . . . . . . . . . 21

3.1 The ODE solution and one sample path of the CTMC SEIR model.

Parameter values are Λ = 1, d = 0.005, β = 0.25, ν = 0.1, γ = 0.05,

and α = 2d = 0.01 with initial conditions S(0) = 200, E(0) = 0,

I(0) = 2, and R(0) = 0. The disease persists with probability 1−P0 =

0.926. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The ODE solution and one sample path of the CTMC vector-host

model. Parameter values are Λ = 0.5, d = 0.005, γ = 0.1, Γ = 500,

µ = 0.5, and βm = βh = 0.2 with initial conditions S(0) = 100,

I(0) = 2, M(0) = 1000, and V (0) = 0. The disease persists with

probability 1− P0 = 0.970. . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 The ODE solution and one sample path of the CTMC staged-progression

model. Parameter values are Λ = 1, d = 0.005, β1 = 0.4, β2 = 0.1,

ν1 = 0.2, ν2 = 0.05, d1 = 0.02, and d2 = 0.01 with initial conditions

S(0) = 200, I1(0) = 1, I2(0) = 1, and R(0) = 0. The disease persists

with probability 1− P0 = 0.917. . . . . . . . . . . . . . . . . . . . . . 42

3.4 The ODE solution and one sample path of the CTMC treatment model.

Parameter values are Λ = 1, d = 0.005, β1 = 0.1, β2 = 0.04, βT =

0.25β2, r1 = r2 = 0.05, p = 0.5, q = 0.1, ν1 = 0.5, ν2 = 0.1, and

d1 = d2 = 1.5d ν1 = 0.2, ν2 = 0.05, d1 = 0.02, and d2 = 0.01. The

reproduction numbers are R1 = 2.58 and R2 = 5.08. The disease

persists with probability 1− P0 = 0.936. . . . . . . . . . . . . . . . . 49

vii


4.1 The ODE solution and one sample path for the CTMC two-patch SIS

model. Parameter values are β1 = 0.5, γ1 = 0.1, β2 = 0.2, γ2 = 0.4,

ds12 = ds21 = 0.1, and di12 = di21 = 0.1 with initial conditions S1(0) =

199, I1(0) = 1, S2(0) = 200, and I2(0) = 0. An outbreak occurs with

probability 1− P0 = 0.6590. The locally stable endemic equilibrium is

(S1, I1, S2, I2) ≈ (68, 132, 161, 39). . . . . . . . . . . . . . . . . . . . . 69

4.2 The ODE solution and one sample path for the CTMC three-patch SIS


β3 = 0.1, γ3 = 0.4, dskj = dikj = 0.1 for k 6= 3 and j 6= 3, and ds13 =

ds31 = di13 = di31 = 0 with initial conditions S1(0) = 149, I1(0) = 1,

S2(0) = 150, I2(0) = 0, S3(0) = 150, and I3(0) = 0. An outbreak

occurs with probability 1 − P0 = 0.6491. The locally stable endemic

equilibrium is (S1, I1, S2, I2, S3, I3) ≈ (53, 97, 126, 24, 144, 6). . . . . . . 71



α1 = α2 = 0.5, ds12 = ds21 = 0.1, and di12 = di21 = 0.1 with initial

conditions S1(0) = 195, I1(0) = 5, S2(0) = 185, and I2(0) = 15. . . . . 72

4.4 The ODE and SDE solutions for the two-patch SIS model. Parameter

values are β1 = 0.8, γ1 = 0.1, β2 = 0.3, γ2 = 0.4, α1 = α2 = 0.5, ds12 =

ds21 = 0.1, and di12 = di21 = 0.1 with initial conditions S1(0) = 195,

I1(0) = 5, S2(0) = 185, and I2(0) = 15. . . . . . . . . . . . . . . . . . 72

4.5 Solution of the ODE and one sample path for the CTMC two-patch

SIRS model. Parameter values are β1 = 0.8, γ1 = 0.1, β2 = 0.2,

γ2 = 0.4, ν1 = ν2 = 0.1, α1 = α2 = 0.4, and dskj = dikj = drkj = 0.1 for

k, j = 1, 2 with initial conditions S1(0) = 195, I1(0) = 5 R1(0) = 0,

S2(0) = 185, I2(0) = 15, and R2(0) = 0. . . . . . . . . . . . . . . . . . 73

4.6 The ODE and SDE solutions for the two-patch SIRS model. Parameter

values are β1 = 0.8, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, ν1 = ν2 = 0.1,

α1 = α2 = 0.4, and dskj = dikj = drkj = 0.1 for k, j = 1, 2 with initial

conditions S1(0) = 195, I1(0) = 5 R1(0) = 0, S2(0) = 185, I2(0) = 15,

and R2(0) = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

viii


4.7 Solutions of the ODE and SDE nine-patch SIS models. Parameter

values are βj = 0.3, γj = 0.2 for j = 2, 4, 6, 8, βj = 0.2, γj = 0.4, for

j = 1, 3, 7, 9, β5 = 0.6, and γ5 = 0.3 with initial conditions S5(0) = 95,

I5(0) = 5, and Sj(0) = 100, Ij(0) = 0 for j 6= 5. . . . . . . . . . . . . . 75

4.8 Solutions of the ODE and SDE nine-patch SIS models. Parameter

values are βj = 0.1, γj = 0.4 for j = 1, 2, 3, 4, 7, βj = 0.1, γj = 0.2

for j = 5, 6, 8, and β9 = 0.5 and γ9 = 0.1 with initial conditions

S9(0) = 99, I9(0) = 1, and Sj(0) = 100, Ij(0) = 0 for j 6= 9. . . . . . . 76


model with demographics. Parameter values are Λ1 = Λ2 = 2, µ1 =

µ2 = 0.01, β1 = 0.5, γ1 = 0.2, β2 = 0.2, γ2 = 0.5, and dskj = dikj = 0.1

for all k, j = 1, 2 with initial conditions S1(0) = 199, I1(0) = 1, S2(0) =

200, and I2(0) = 0. An outbreak occurs with probability 1−P0 = 0.4265. 87

4.10 Solution of the ODE and SDE two-patch SIS models with demograph-

ics. Parameter values are Λ1 = Λ2 = 2, µ1 = µ2 = 0.01, β1 = 0.5,

γ1 = 0.2, β2 = 0.2, γ2 = 0.5, and dskj = dikj = 0.1 for k, j = 1, 2 with

initial conditions S1(0) = 195, I1(0) = 5, S2(0) = 195, and I2(0) = 5. . 87

ix


CHAPTER 1

INTRODUCTION

Most ordinary differential equations (ODEs) or systems of ODEs cannot be solved

analytically. Therefore, numerical methods such as Euler’s method or Runge-Kutta

methods are used to solve the equation or system. These numerical methods can be

derived by applying finite difference schemes to the ODE or system of ODEs

resulting in a corresponding difference equation or system of difference equations.

Ideally, the resulting difference equation should have the same properties as the

differential equation. A differential equation and its corresponding difference

equation are said to be dynamically consistent if they have the same properties [52].

Properties of interest may include the existence of periodic solutions, stability of

equilibrium points, or bifurcations.

Liu and Elaydi [52] applied a nonstandard finite-difference (NSFD) scheme to the

Lotka-Volterra competition system and derived a dynamically consistent system of

difference equations which preserves many properties of the ODE system. Using a

NSFD scheme similar to that of Liu and Elaydi, Roeger and Lahodny [63] also

derived a dynamically consistent discrete competition model. In Chapter 2, the

Lotka-Volterra competition system is introduced and some well-known properties of

the system are stated. A general class of difference equations is considered.

Conditions are stated to guarantee that the members of this class are dynamically

consistent with the Lotka-Volterra competition model.

A problem of major interest in mathematical epidemiology is to determine

whether an outbreak of an infectious disease will result in an epidemic. For

deterministic epidemic models, a well-known threshold is the basic reproduction

number, R0. This threshold is defined as the expected number of secondary

infections resulting from the introduction of one infectious individual into a

completely susceptible population [3]. In general, if R0 ≤ 1, the disease will become

extinct and if R0 > 1, the disease persists. Of course, the basic reproduction

number is an expectation. Thus, even if R0 > 1, there is a nonzero probability that

the disease will become extinct. For instance, if an outbreak begins with only one

infectious individual, there is a chance that individual will recover or die before they

1


can infect a susceptible individual. The benefit of using stochastic epidemic models

is that the probability of disease extinction or persistence can be calculated. For

stochastic epidemic models, there is a threshold similar to the basic reproduction

number which is used to predict whether a disease will persist.

In Chapter 3, several epidemic models from the literature are introduced. For

each of these models, the basic reproduction number is calculated and a

corresponding continuous-time Markov chain (CTMC) model is described. Using

the theory of multi-type branching processes, a stochastic threshold is derived to

predict the probability of disease extinction and a new relationship between the

deterministic and stochastic thresholds is illustrated. For specific models, analytical

expressions for the probability of disease extinction are derived. The content in

Chapter 3, co-authored with Linda J.S. Allen, has been published in The Journal of

Biological Dynamics [8].

Infectious diseases in humans or domestic and wild animals such as influenza,

tuberculosis, SARS, and foot-and-mouth disease can be easily transmitted from

region to region. Factors such as spatial connectivity and the dispersal of

individuals through travel can significantly affect the likelihood that a disease will

persist in a given region. For instance, a disease may not be prevalent in rural areas,

but if individuals from an urban center travel to these more remote regions, then

the disease may be more likely to persist. Alternatively, the reverse may be the case

where the disease is present in rural areas and is spread to urban areas [68]. Thus,

dispersal and environmental heterogeneity should be accounted for in epidemic

models.

The role of dispersal on disease dynamics has been examined for general epidemic

models [5, 24, 41, 72] and models focused on particular diseases such as tuberculosis

[25], malaria [33], and influenza [42, 66]. Many of these models involve dividing a

region into multiple patches and allowing dispersal of susceptible or infectious

individuals between these patches [5, 41, 42, 72].

The role of movement also has important implications for disease control. For

instance, it may be possible to stop the spread of a disease by conducting border

checks and restricting the movement of infectious individuals or restricting the

movement to and from high-risk patches where a disease is more prevalent. Ruan,

2


Wang, and Levin [65] showed that the spread of SARS can be contained by

implementing border screening for infectious individuals. On the other hand,

screening at the borders is only effective in identifying individuals exhibiting

symptoms. Gao and Ruan [33] state that the dispersal of exposed or latent

individuals can contribute to the spread of disease and that ineffective border

screening may adversely affect disease transmission.

In Chapter 4, stochastic multi-patch epidemic models that are analytically

tractable are developed. These models are used to examine how dispersal between

patches and the location of an outbreak affect the probability of disease persistence

or extinction. These results have important implications for disease control.

3


CHAPTER 2

DYNAMICALLY CONSISTENT DISCRETE LOTKA-VOLTERRA

COMPETITION MODELS

In this chapter, the Lotka-Volterra competition model is introduced and a general

class of discrete-time competition models derived from a NSFD scheme is

considered. Sufficient conditions are stated so that the elements of this class of

discrete models are dynamically consistent with the Lotka-Volterra competition

model. The discrete models are shown to preserve the positivity of solutions,

existence and stability conditions of equilibrium points, boundedness of solutions,

and monotonicity of the continuous Lotka-Volterra competition system.

In Section 2.1, the Lotka-Volterra competition model is introduced and some

well-known properties of the model are stated. In addition, dynamically consistent

discrete-time competition models constructed by Liu and Elaydi [52] and Roeger

and Lahodny [63] are discussed. In Section 2.2, a general class of discrete-time

competition models is considered and sufficient conditions are derived to ensure that

the elements of this class of difference equations preserves the properties of the

Lotka-Volterra competition system. In Section 2.3, explicit discrete models are

given and numerical results illustrate dynamical consistency. Finally, in Section 2.4,

the results are summarized and possible future work is presented.

2.1 Introduction

The Lotka-Volterra competition model has the form:

x = x(r1 − a11x− a12y),

y = y(r2 − a21x− a22y),(2.1)

where x and y are the population densities of two competing species, r1 and r2 are

the intrinsic growth rates, and a12 and a21 are the interspecific coefficients. The

“dot” notation denotes differentiation with respect to t, x = dx/dt. Assume that

ri > 0 and aij > 0 for i, j = 1, 2. The properties of system (2.1) are well-known [3]:

1. The positive cone, R2+, is positively invariant. If the initial conditions (x0, y0)

are positive, then the solutions remain positive for all forward time.

4


2. There are at most four nonnegative equilibria: the extinction equilibrium

E0 = (0, 0), the single-species equilibria E1 = (r1/a11, 0) and E2 = (0, r2/a22),

and the coexistence equilibrium

E3 =

(r1a22 − r2a12

a11a22 − a12a21

,r2a11 − r1a21

a11a22 − a12a21

).

The coexistence equilibrium lies in R2+ provided that

a11

a21

<r1

r2

<a12

a22

ora11

a21

>r1

r2

>a12

a22

.

3. The stability properties of the equilibria are:

(a) E0 is always unstable

(b) E1 is locally asymptotically stable ifa11

a21

<r1

r2

(c) E2 is locally asymptotically stable ifr1

r2

<a12

a22

(d) E3 is locally asymptotically stable if and only if (iff)

a11

a21

>r1

r2

>a12

a22

.

Moreover, if E3 is stable, it is globally asymptotically stable.

4. The system is competitively monotone.

5. The solutions are eventually bounded in the set S = [0, r1/a11]× [0, r2/a22].

The set S is positively invariant.

The term “dynamically consistent” was first used by Liu and Elaydi [52]. If a

discrete system has the same properties as the corresponding continuous system,

then the discrete and continuous systems are said to be dynamically consistent.

Properties of interest may include positivity of solutions, existence and stability of

equilibrium points, periodicity of solutions, and bifurcations. A formal definition is

given by Mickens [54].

5


Definition 2.1. (Mickens, [54]) Consider the differential equation

dx

dt= f(x, t, λ) (2.2)

and a discrete model of it given by

xk+1 = F (xk, tk, h, λ). (2.3)

We assume that f is such that the proper existence-uniqueness theorem holds; λ

represents the parameters defining the system; the time step-size is h = ∆t with

tk = hk, k an integer; and xk is an approximation to x(tk). Let the differential

equation and/or its solutions have property P . The discrete model, given by

equation (2.3), is dyamically consistent with equation (2.2), with respect to property

P , if it has and/or its solutions also have property P .

The following example illustrates the application of finite-difference schemes to a

one-dimensional ODE in order to derive a dynamically consistent difference

equation.

Consider the decay equation

x = −ax, (2.4)

where a > 0. The solution of this equation is well-known, x(t) = x0e−at. A forward

Euler scheme for (2.4) isxn+1 − xn

h= −axn,

where h = ∆t, tn = hn, and xn is an approximation of x(tn). It follows that

xn+1 = xn(1− ah),

and so

xn = x0(1− ah)n.

Note that the solution of the differential equation, x(t) = x0e−at, and the solution of

the difference equation, xn = x0(1− ah)n, do not necessarily have the same

properties. For instance, if h = 2/a, the solution xn = x0(1− ah)n oscillates between

positive and negative x0 while x(t) = x0e−at decreases monotonically to zero. Only

6


if h is sufficiently small, 0 < h < 1/a, will the solution xn = x0(1− ah)n decrease

monotonically to zero. Alternatively, a backward Euler scheme for (2.4) is

xn+1 − xnh

= −axn+1.

Solving for xn+1,

xn+1 =xn

1 + ah.

It follows that

xn =x0

(1 + ah)n. (2.5)

Since ah > 0, the solution xn = x0/(1 + ah)n decreases monotonically to zero

regardless of the choice for h. Thus, (2.5) preserves positivity and monotonicity of

the soluion to (2.4). For additional examples in one dimension, see [11, 31, 62, 64].

Liu and Elaydi [52] constructed the following NSFD scheme for the

Lotka-Volterra system (2.1):

X − xφ

= r1x− a11xX − a12yX,

Y − yφ

= r2y − a21xY − a22yY,(2.6)

where X = xn+1, x = xn, and φ = h+O(h2).

The NSFD scheme of Liu and Elaydi leads to the system of difference equations

X = x

(1 + φr1

1 + φ(a11x+ a12y)

),

Y = y

(1 + φr2

1 + φ(a21x+ a22y)

).

(2.7)

Cushing et al. [28] showed that the system of difference equations (2.7) preserves

many properties of the differential equation system (2.1) including positivity of

solutions, existence and stability conditions of equilibrium solutions, monotonicity,

and boundedness of solutions.

7


Following the discretization example (2.6), Roeger and Lahodny [63] constructed

the following NSFD scheme for system (2.1):

X − xφ1

= r1[(1 + θ1)x− θ1X]− a11x[(1 + θ11)X − θ11x]

−a12y[(1 + θ12)X − θ12x],Y − yφ2

= r2[(1 + θ2)y − θ2Y ]− a21x[(1 + θ21)Y − θ21y]

−a22y[(1 + θ22)Y − θ22y],

(2.8)

where φi > 0, θi ≥ 0, and θij ≥ 0 for i, j = 1, 2. In the special case that φ1 = φ2,

θi = 0, and θij = 0 for all i, j = 1, 2, this system is reduced to (2.6).

The NSFD scheme of Roeger and Lahodny leads to the system of difference

equations

X = x · 1 + φ1r1 + φ1(r1θ1 + a11θ11x+ a12θ12y)

1 + φ1(r1θ1 + a11θ11x+ a12θ12y) + φ1(a11x+ a12y),

Y = y · 1 + φ2r2 + φ2(r2θ2 + a21θ21x+ a22θ22y)

1 + φ2(r2θ2 + a21θ21x+ a22θ22y) + φ2(a21x+ a22y),

(2.9)

where X = xn+1, x = xn, φi > 0, θi ≥ 0, and θij ≥ 0 for i, j = 1, 2.

If θ11 ≥ θ12 and θ22 ≥ θ21, then the system of difference equations (2.9) preserves

the positivity of solutions, existence and stability conditions for each equilibrium

solution, and monotonicity [63]. Note that (2.9) can be written as

X = x · F1(x, y) + φ1r1

F1(x, y) + φ1(a11x+ a12y),

Y = y · F2(x, y) + φ2r2

F2(x, y) + φ2(a21x+ a22y),

(2.10)

where

F1(x, y) = 1 + φ1(r1θ1 + a11θ11x+ a12θ12y),

F2(x, y) = 1 + φ2(r2θ2 + a21θ21x+ a22θ22y).

Similarly, (2.7) can be written as (2.10), where F1(x, y) = F2(x, y) = 1. The system

of difference equations (2.9) is a complicated system with many parameters. The

8


goal is to find other, more simplistic, functions F1 and F2 such that a system of the

form (2.10) is dynamically consistent with the Lotka-Volterra competition system

(2.1). In the next section, a generalized discrete competition model of the form

(2.10) is explored.

2.2 Discrete Lotka-Volterra Competition Models

Consider a discrete Lotka-Volterra competition model of the form:

X = x · F1(x, y) + φ1r1

F1(x, y) + φ1(a11x+ a12y),

Y = y · F2(x, y) + φ2r2

F2(x, y) + φ2(a21x+ a22y),

(2.11)

where X = xn+1, x = xn, and φi > 0 for i, j = 1, 2.

To ensure that solutions of (2.11) remain in the positive cone, R2+, assume that

the functions F1 and F2 are positive-valued for all (x, y) ∈ R2+.

2.2.1 Existence and Local Stability of Equilibria

A discrete competition system of the form (2.11) is shown to be dynamically

consistent with the Lotka-Volterra competition system (2.1) with respect to the

existence and stability conditions for each equilibrium regardless of the choice of the

functions F1 and F2.

Note that (2.11) can be written as

X = x+φ1x(r1 − a11x− a12y)

F1(x, y) + φ1(a11x+ a12y),

Y = y +φ2y(r2 − a21x− a22y)

F2(x, y) + φ2(a21x+ a22y).

The following lemma states that the discrete system (2.11) has the same

equilibrium solutions as the Lotka-Volterra competition system (2.1).

Lemma 2.1. The discrete competition system (2.11) is dynamically consistent with

the continuous system (2.1) with respect to the equilibrium points.

9


Proof. To find the equilibria of (2.11), set X = x and Y = y. It follows that

x(r1 − a11x− a12y) = 0,

y(r2 − a21x− a22y) = 0.

The solutions of this system are the equilibrium solutions of the Lotka-Volterra

competition model (2.1).

The next lemma determines the stability of the equilibrium points for the discrete

model (2.11).

Lemma 2.2. The discrete competition system (2.11) is dynamically consistent with

the continuous system (2.1) with respect to the local stability of the equilibrium

points E0, E1, and E2.

Proof. The Jacobian matrix for (2.11) evaluated at E0 is

J(E0) =

1 +r1

F1(0, 0)0

0 1 +r2

F2(0, 0)

.Since the eigenvalues are both greater than one, E0 is unstable. At E1,

J(E1) =

1− r1

F1(E1) + r1

− a12(r1/a11)

F1(E1) + r1

0 1 +r2 − a21(r1/a11)

F2(E1) + a21(r1/a11)

.The eigenvalues of this matrix are the diagonal entries. Thus, the equilibrium E1 is

locally asymptotically stable provided that r2 − a21(r1/a11) < 0. That is,

a11

a21

<r1

r2

.

Similarly, the equilibrium E2 is locally asymptotically stable if

r1

r2

<a12

a22

.

10


The following theorem states that the stability condition for the coexistence

equilibrium E3 of the discrete system is the same as for the continuous system.

Systems (2.1) and (2.11) are dynamically consistent with respect to the local

stability of all equilibrium points.

Theorem 2.1. The coexistence equilibrium E3 of (2.11) is locally asymptotically

stable if and only if a11a22 − a12a21 > 0.

Proof. The Jury conditions [3] state that an equilibrium point (x, y) of a

two-dimensional system is locally asymptotically stable if

|trace(J(x, y))| < 1 + det(J(x, y)) < 2.

At the coexistence equilibrium E3 = (x, y),

J(E3) =

1− a11x

F1 + a11x+ a12y− a12x

F1 + a11x+ a12y

− a21y

F2 + a21x+ a22y1− a22y

F2 + a21x+ a22y

.Clearly trace(J) > 0. So |trace(J)| < 1 + det(J) iff 1 + det(J)− trace(J) > 0. Now

1 + det(J)− trace(J) =(a11a22 − a12a21)xy

(F1 + a11x+ a12y)(F2 + a21x+ a22y).

which is positive iff a11a22 − a12a21 > 0. Moreover, 1 + det(J) < 2 iff 1− det(J) > 0.

Now

1−det(J) =a11x

F1 + a11x+ a12y+

a22y

F2 + a21x+ a22y− (a11a22 − a12a21)xy

(F1 + a11x+ a12y)(F2 + a21x+ a22y).

It is straightforward to show that this expression is positive. By the Jury conditions,

the equilibrium E3 is locally asymptotically stable iff a11a22 − a12a21 > 0.

2.2.2 Monotonicity, Positive Invariance, and Global Stability

Conditions are derived so that the discrete system (2.11) is a competitive

monotone system. Let X = X(x, y) and Y = Y (x, y).

Lemma 2.3. Suppose that

11


(i)∂F1

∂xand f1(x, y) = x(r1 − a11x− a12y) have opposite signs for all (x, y) ∈ R2

+,

(ii)∂F2

∂xand f2(x, y) = y(r2 − a21x− a22y) have the same sign for all (x, y) ∈ R2

+.

If 0 ≤ x1 ≤ x2 and y ≥ 0 is fixed, then for system (2.11)

X(x1, y) ≤ X(x2, y) and Y (x1, y) ≥ Y (x2, y).

Proof. Fix y ≥ 0 and for i = 1, 2 set

D1i = F1(xi, y) + φ1(a11xi + a12y),

D2i = F2(xi, y) + φ2(a21xi + a22y).

Then

X(x2, y)−X(x1, y) = (x2− x1) +φ1x2(r1 − a11x2 − a12y)

D12

− φ1x1(r1 − a11x1 − a12y)

D11

.

This expression is nonnegative iff

D11D12(x2 − x1) + φ1D11x2(r1 − a11x2 − a12y)− φ1D12x1(r1 − a11x1 − a12y) ≥ 0.

Applying a computer algebra system such as Maple, the above inequality holds iff

φ21r1a12y(x2 − x1) + φ1r1(F1(x1, y)x2 − F1(x2, y)x1) + φ1a11x1x2(F1(x2, y)− F1(x1, y))

+φ1a12y(F1(x2, y)x2 − F1(x1, y)x1) + F1(x1, y)F1(x2, y)(x2 − x1) ≥ 0.

Since x2 ≥ x1,

φ1r1(F1(x1, y)x2 − F1(x2, y)x1) + φ1a11x1x2(F1(x2, y)− F1(x1, y))

+φ1a12y(F1(x2, y)x2 − F1(x1, y)x1)

≥ φ1r1(F1(x1, y)x1 − F1(x2, y)x1) + φ1a11x21(F1(x2, y)− F1(x1, y))

+φ1a12y(F1(x2, y)x1 − F1(x1, y)x1)

≥ φ1(F1(x2, y)− F1(x1, y))x1(a11x1 + a12y − r1)

= −φ1(F1(x2, y)− F1(x1, y))f1(x1, y) ≥ 0.

12


Similarly,

Y (x1, y)− Y (x2, y) =φ2y(r2 − a21x1 − a22y)

D21

− φ2y(r2 − a21x2 − a22y)

D22

.


φ2D22y(r2 − a21x1 − a22y)− φ2D21y(r2 − a21x2 − a22y) ≥ 0.


φ22r2a21y(x2 − x1) + φ2r2y(F2(x2, y)− F2(x1, y)) + φ2a21y(x2F2(x1, y)− x1F2(x2, y))

+φ2a22y2(F2(x1, y)− F2(x2, y)) ≥ 0.

Since x2 ≥ x1,

φ2r2y(F2(x2, y)− F2(x1, y)) + φ2a21y(x2F2(x1, y)− x1F2(x2, y))

+φ2a22y2(F2(x1, y)− F2(x2, y))

≥ φ2(F2(x1, y)− F2(x2, y))(a21x1 + a22y − r2)

= −φ2(F2(x1, y)− F2(x2, y))f2(x1, y) ≥ 0.

By a symmetric argument, the following lemma is not difficult to prove.

Lemma 2.4. Suppose that

(i)∂F1

∂yand f1(x, y) = x(r1 − a11x− a12y) have the same sign for all (x, y) ∈ R2

+,

(ii)∂F2

∂yand f2(x, y) = y(r2 − a21x− a22y) have opposite signs for all (x, y) ∈ R2

+.

If 0 ≤ y1 ≤ y2 and x ≥ 0 is fixed, then for system (2.11)

X(x, y1) ≥ X(x, y2) and Y (x, y1) ≤ Y (x, y2).

13


Proof. Fix x ≥ 0 and for i = 1, 2 set

E1i = F1(x, yi) + φ1(a11x+ a12yi),

E2i = F2(x, yi) + φ2(a21x+ a22yi).

Then

Y (x, y2)− Y (x, y1) = (y2 − y1) +φ2y2(r2 − a21x− a22y2)

E22

− φ2y1(r2 − a21x− a22y1)

E21

.


E21E22(y2 − y1) + φ2E21y2(r2 − a21x− a22y2)− φ2E22y1(r2 − a21x− a22y1) ≥ 0.


φ22r2a21x(y2 − y1) + φ2r2(F2(x, y1)y2 − F2(x, y2)y1) + φ2a22y1y2(F2(x, y2)− F2(x, y1))

+φ2a21x(F2(x, y2)y2 − F2(x, y1)y1) + F2(x, y1)F2(x, y2)(y2 − y1) ≥ 0.

Since y2 ≥ y1,

φ2r2(F2(x, y1)y2 − F2(x, y2)y1) + φ2a22y1y2(F2(x, y2)− F2(x, y1))

+φ2a21x(F2(x, y2)y2 − F2(x, y1)y1)

≥ φ2r2y1(F2(x, y1)− F2(x, y2)) + φ2a22y21(F2(x, y2)− F2(x, y1))

+φ2a21xy1(F2(x, y2)− F2(x, y1))

= −φ2(F2(x, y2)− F2(x, y2))f2(x, y1) ≥ 0.

Similarly,

X(x, y1)−X(x, y2) =φ1x(r1 − a11x− a12y1)

E11

− φ1x(r1 − a11x− a12y2)

E12

.


φ1E12x(r1 − a11x− a12y1)− φ1E11x(r1 − a11x− a12y2).

14



φ21r1a12x(y2 − y1) + φ1r1x(F1(x, y2)− F1(x, y1)) + φ1a12x(y2F1(x, y1)− y1F1(x, y2))

+φ1a11x2(F1(x, y1)− F1(x, y2)) ≥ 0.

Since y2 ≥ y1,

φ1r1x(F1(x, y2)− F1(x, y1)) + φ1a12x(y2F1(x, y1)− y1F1(x, y2))

+φ1a11x2(F1(x, y1)− F1(x, y2))

≥ φ1(F1(x, y1)− F1(x, y2))(a11x1 + a12y − r1)

= −φ1(F1(x, y1)− F1(x, y2))f1(x, y1) ≥ 0.

Theorem 2.2. Suppose that conditions (i) and (ii) in Lemmas 2.3 and 2.4 hold. If

0 < x1 ≤ x2 and 0 < y2 ≤ y1, then for system (2.11)

X(x1, y1) ≤ X(x2, y2) and Y (x1, y1) ≥ Y (x2, y2).

That is, the discrete system (2.11) is a competitive monotone system.

Proof. Applying the previous two lemmas,

X(x1, y1) ≤ X(x2, y1) ≤ X(x2, y2)

and

Y (x1, y1) ≥ Y (x2, y1) ≥ Y (x2, y2).

Note that the conditions in Theorem 2.2 are sufficient but not necessary. Indeed,

the discrete system (2.11) preserves monotonicity if F1 and F2 are positive constants

or if F1 and F2 are positive linear functions of x and y, respectively. That is, if

F1(x, y) = a1x+ b1 and F2(x, y) = a2y + b2, where a1, a2, b1, and b2 are positive

constants.

Theorem 2.3. For the discrete model (2.11), the set S = [0, r1/a11]× [0, r2/a22] is

positively invariant.

15


Proof. Let X = X(x, y) and Y = Y (x, y) in (2.11) and suppose (x, y) ∈ S. That is,

0 ≤ x ≤ r1/a11 and 0 ≤ y ≤ r2/a22. Since F1 and F2 are positive-valued for all

(x, y) ∈ R2+, both X and Y are positive. Moreover,(

X − r1

a11

)−(x− r1

a11

)= X − x

= x

(F1 + φ1r1

F1 + φ1(a11x+ a12y)

)− x

≤ x

(F1 + φ1r1

F1 + φ1a11x

)− x

= x

(1 +

φ1(r1 − a11x)

F1 + φ1a11x

)− x

= x

(φ1(r1 − a11x)

F1 + φ1a11x

)= −φ1a11x

(x− r1/a11

F1 + φ1a11x

).

It follows that

X − r1

a11

=

(x− r1

a11

)(1− φ1a11x

F1(x, y) + φ1a11x

)≤ 0,

and so 0 ≤ X ≤ r1/a11. Similarly, 0 ≤ Y ≤ r2/a22. By induction, the set S is

positively invariant.

Lemma 2.5. Solutions of the discrete model (2.11) are eventually bounded. That is,

lim supn→∞

xn ≤r1

a11

and lim supn→∞

yn ≤r2

a22

.

Proof. As in the proof of Theorem 2.3,(X − r1

a11

)−(x− r1

a11

)≤ φ1x

(r1/a11 − xF1 + φ1a11x

).

16


The right-hand side is negative iff x > r1/a11. It follows that if x > r1/a11, then(X − r1

a11

)<

(x− r1

a11

).

That is, X is closer to r1/a11 than x. Similarly, if y > r2/a22 then Y is closer to

r2/a22 than y. Thus, if (x0, y0) lies outside of the set S, then trajectories approach

the boundary of S. On the other hand, if (x0, y0) ∈ S, then (xn, yn) ∈ S for all

n > 0. In either case, solutions eventually lie in S. That is,

lim supn→∞

xn ≤r1

a11

,

lim supn→∞

yn ≤r2

a22

.

The preceding results have established that if F1 and F2 are positive-valued

functions for all (x, y) ∈ R2+, then a discrete competition system of the form (2.11)

and the Lotka-Volterra competition model (2.1) are dynamically consistent with

respect to the positive invariance of R2+, the existence and local stability conditions

of the equilibrium solutions E0, E1, E2, and E3, positive invariance of the set

S = [0, r1/a11]× [0, r2/a22], and the boundedness of solutions within the set S.

Furthermore, if the functions F1 and F2 satisfy the following conditions:

(i)∂F1

∂xand f1 have opposite signs for all (x, y) ∈ R2

+,

(ii)∂F2

∂xand f2 have the same sign for all (x, y) ∈ R2

+,

(iii)∂F1

∂yand f1 have the same sign for all (x, y) ∈ R2

+,

(iv)∂F2

∂yand f2 have opposite signs for all (x, y) ∈ R2

+,

then the systems (2.11) and (2.1) are both competitive monotone systems.

A theorem by Smith [69] is used to establish the global stability of the coexistence

equilibrium, E3. A 2× 2 matrix J is called K-positive if J11 > 0, J12 < 0, J21 < 0,

and J22 > 0 for all (x, y) ∈ R2+ [63].

17


Theorem 2.4. (Smith, [69]) Suppose a map P : S → S is continuously

differentiable and satisfies the following:

(a) S contains order intervals and is ≤K-convex,

(b) det(DP (x)) > 0 for x ∈ S; the map is orientation preserving,

(c) DP (x) is K-positive in S,

(d) P is injective.

Then P is a competitive map and P nx is eventually componentwise monotone for

all x ∈ S. In this case, if an orbit has compact closure in S, then it converges to a

fixed point of P .

Consider the map P defined by P (x, y) = (X(x, y), Y (x, y)). By Theorem 2.3,

P : S → S is a map from the compact, connected set S into itself.

Lemma 2.6. If conditions (i) and (ii) from Lemmas 2.3 and 2.4 hold, the Jacobian

matrix J is K-positive in R2+.

The preceding result follows directly from Theorem 2.2. The following lemma

shows that the discrete map is orientation preserving. That is, the determinant of

the Jacobian matrix is always positive.

Lemma 2.7. Suppose the two conditions (i) from Lemma 2.3 and Lemma 2.4 hold

and ∂F1/∂y = ∂F2/∂x = 0. Then det(J) > 0 for all (x, y) ∈ R2+.

Proof. For ease of notation, denote the partial derivatives of F1 and F2 with respect

to x and y by F1x, F1y, F2x, and F2y, respectively. The result follows easily using a

computer algebra system such as Maple. The determinant of the Jacobian matrix is:

det(J) =1

[F1 + φ1(a11x+ a12y)]2[F2 + φ2(a21x+ a22y)]2· φ1F

22F1x(−f1)

+φ2F21F2y(−f2) + (F1F2 + φ1a12yF2 + φ2a21xF1)(F1 + φ1r1)(F2 + φ2r2)

+φ1φ2[r1F1F2y(−f2) + r2F2F1x(−f1) + φ1r1a12yF2y(−f2) + φ2r2a21xF1x(−f1)

+xyF1xF2y(−f1)(−f2) + a21xF2F1x(−f1) + a12yF1F2y(−f2) + F1yF2x(−f1)(f2)

+a21yF1y(−f1)(φ2r2 + F2) + a12xF2x(−f2)(φ1r1 + F1)].

18


The only possible non-positive terms are

φ1φ2[F1yF2x(−f1)(f2) + a21yF1y(−f1)(φ2r2 + F2) + a12xF2x(−f2)(φ1r1 + F1)].

Since F1y = F2x = 0, these terms equal zero and det(J) > 0.

In order to apply Smith’s theorem, the map P must be injective. As noted by

Smith, since S = [0, r1/a11]× [0, r2/a22] is compact and connected, it is enough to

show that there exists (x, y) ∈ P (S) ⊂ S such that P−1(x, y) is a single point.

Consider the point (0, 0) ∈ P (S). Since all nonzero points in S are mapped to

nonzero points in S, it follows that P−1(0, 0) = (0, 0), a single point. Applying

Theorem 2.4, we have the following result.

Theorem 2.5. Suppose that the conditions of Lemma 2.7 hold. Then all solutions of

the discrete system (2.11) converge to one of the equilibria. Thus, if the coexistence

equilibrium E3 is locally asymptotically stable, it is also globally stable in R2+.

2.3 Numerical Examples

In this section, discrete competition models of the form (2.11) are compared to

Euler’s method. Applying Euler’s method to the Lotka-Volterra competition system

(2.1) leads to the system of difference equations:

X = x[1 + h(r1 − a11x− a12y)],

Y = y[1 + h(r2 − a21x− a22y)].(2.12)

In order for this system to preserve the positivity of solutions, the step size, h > 0,

must be sufficiently small. That is,

h(a11x+ a12y − r1) ≤ 1,

h(a21x+ a22y − r2) ≤ 1,

for all (x, y) ∈ R2+. However, for any h > 0 there exists (x, y) ∈ R2

+ such that these

inequalities are not satisfied. Thus, Euler’s method does not preserve the positivity

of solutions for (2.1). On the other hand, if the initial conditions are relatively

small, then Euler’s method yields a good approximation.

19


Consider a discrete competition model of the form (2.11). As stated in Section

2.3, it is difficult to find explicit functions F1 and F2 that satisfy the hypotheses of

Theorem 2.2. A trivial example is to choose F1 and F2 to be positive constants. In

the special case where F1 = F2 = 1, (2.11) is the discrete model constructed by the

NSFD scheme of Liu and Elaydi (2.6).

Let us now compare several discrete models of the form (2.11) to Euler’s method.

The solution of (2.1) is approximated using four distinct numerical methods. The

numerical methods used are Euler’s method (2.12), the discrete model of Liu and

Elaydi (2.7), the discrete model of Roeger and Lahodny (2.9), and a discrete model

of the form (2.11) where F1(x, y) = F2(x, y) = 0.2. Consider the parameter values

r1 = 2, r2 = 3, a11 = 0.02, a12 = 0.01, a21 = 0.02, and a22 = 0.03. For the discrete

models of Liu and Elaydi and Roeger and Lahodny, let φ1 = φ2 = φ = h. For these

values, a11a22 − a12a21 > 0, and so the coexistence equilibrium E3 = (75, 50) lies in

R2+ and is globally asymptotically stable. For each of these discrete models with an

initial condition of (x0, y0) = (5, 5), solutions approach the equilibrium E3. The

solutions of the four discrete models are graphed in the phase plane in Figure 2.1

and as a function of time in Figure 2.2. It should be noted that for a fixed time t,

the numerical method does not converge for F1(x, y) = F2(x, y) = 0.2.

Figure 2.1. Four approximate solutions to the Lotka-Volterra competition model inthe phase plane. For figure (a), the step size is h = 0.5 and for figure (b), the stepsize is h = 0.1.

20


Figure 2.2. Four approximate solutions to the Lotka-Volterra competition model asa function of time. Parameter values and initial conditions are the same as in Figure2.1 with a step size of h = 0.1.

2.4 Discussion

A general class of the discrete-time competition models was considered. Provided

that F1 and F2 are positive-valued functions for all (x, y) ∈ R2+, the discrete model

(2.11) preserves the positivity of solutions, existence and local stability conditions

for each equilibrium, and the boundedness of solutions within the set

S = [0, r1/a11]× [0, r2/a22]. If additional restrictions are imposed on the functions

F1 and F2, the discrete model also preserves the monotonicity of the system (2.1)

and the global stability of the coexistence equilibrium. However, it is difficult to

find explicit functions F1 and F2 which satisfy these conditions.

Since the discrete models discussed here are for the Lotka-Volterra competition

model, discrete versions of the Lotka-Volterra predator-prey and cooperative

systems could also be considered in the future.

21


CHAPTER 3

EXTINCTION OR PERSISTENCE OF DISEASE IN STOCHASTIC EPIDEMIC

MODELS

3.1 Introduction

The basic reproduction number, R0, is defined as the expected number of

secondary infections caused by one infectious individual during their infectious

period [10, 40]. This is a well-known threshold for deterministic epidemic models

which is used to determine whether a disease will persist. In general, if R0 > 1, the

disease persists and if R0 ≤ 1, the disease dies out. The basic reproduction number

can be calculated using the next-generation matrix approach of Diekmann et al. [29]

and van den Driessche and Watmough [70, 71]. This method involves linearizing the

system of ordinary differential equations (ODEs) about the disease-free equilibrium

(DFE). Heesterbeek and Roberts [39, 61] derived equivalent thresholds relating to

the control of a specific infectious group i known as type reproduction numbers, Ti.The type reproduction numbers can also be used to predict disease persistence

(Ti > 1) or extinction (Ti < 1).

Continuous-time Markov chain models are the stochastic counterpart of the ODE

model. For CTMC models, time is continuous and the random variables are

discrete. Similar to the basic reproduction number for deterministic epidemic

models, in stochastic epidemic theory there is a threshold which is used to predict

the probability of disease persistence or extinction for the CTMC model. The

stochastic threshold is similar to the basic reproduction number. In fact, Allen and

van den Driessche [9] showed that the stochastic threshold is equivalent to the basic

reproduction number R0. The theory behind the stochastic threshold depends on

continuous-time branching processes. Whittle [73] derived an expression for the

probability of disease persistence for a susceptible-infectious-recovered (SIR) model

provided that the population size is sufficiently large and a small number of

infectious individuals are present. If the initial number of infectious individuals is

I(0) = i, the probability of disease persistence is

1−(

1

R0

)i. (3.1)

22


The result of Whittle is only valid under certain assumptions. In particular, it is

assumed that infectious individuals give ‘birth’ to new infectious individuals

independently of others and that each individual has the same probability of giving

birth. These two assumptions are fundamental in Galton-Watson branching process

theory [16, 38, 44, 55, 60]. Provided that there is a large population size and a small

number of infectious individuals, these assumptions may be realistic and the

approximation (3.1) of Whittle is a good approximation.

Of course, this approximation is only valid for epidemic models with one

infectious class. If there are multiple infectious classes, the probability of disease

persistence depends on the number of infectious individuals in each class. In

particular if there are n infectious classes, Ij(0) = ij infectious individuals in class j

for j = 1, . . . , n, and the probability of extinction for class j is qj, then a well-known

result from multitype branching processes [4, 16, 38, 60] states that the probability

of disease persistence is approximately

1− qi11 qi22 · · · qinn . (3.2)

However, this approximation is only valid under certain restrictions. There have

been several applications of branching process theory to obtain estimates for the

probability of extinction for populations, genetics, cellular processes, and epidemics

on networks [19, 20, 23, 30, 36, 44, 48, 57, 59, 74].

Typically, the deterministic and stochastic thresholds are discussed separately.

The goals of this chapter are to review some classic results on the thresholds for

epidemic models and illustrate a relationship between the deterministic and

stochastic thresholds. Furthermore, these thresholds are calculated for some

well-known epidemic models from the literature. In Section 3.2, some useful

methods and results from probability theory, branching process theory, and the

theory of continuous-time Markov chains will be discussed. In Section 3.3, the

dynamics of an epidemic model with one infectious class will be summarized and

Whittle’s approximation (3.1) for the probability of disease persistence is derived.

In Section 3.4, the deterministic and stochastic thresholds are calculated for

epidemic models with multiple infectious classes and new expressions are derived for

the probability of disease persistence.

23


3.2 Mathematical Methods

A common method for calculating the basic reproduction number is the

next-generation matrix approach. This method will be discussed here briefly. For

more details on the next-generation matrix approach, see Diekmann et al. [29] and

van den Driessche and Watmough [70, 71].

Let ~I = (I1, . . . , In)T denote the vector of infectious individuals so that Ij is the

number of infectious individuals in class j. Linearizing the system of differential

equations about the DFE gives

d~I

dt= J~I = (F − V )~I,

where J = F − V is the Jacobian matrix evaluated at the DFE. The matrix F

contains all terms which represent new infections and the matrix V contains all

remaining terms such as the rates of recovery, death, or transition between

infectious classes. The next-generation matrix is the matrix FV −1 and the basic

reproduction number is the spectral radius of this matrix [70, 71]. That is,

R0 = ρ(FV −1).

Sufficient conditions were stated by van den Driessche and Watmough [70, 71] to

guarantee that the DFE is locally asymptotically stable if R0 < 1. The basic

reproduction number may be interpreted biologically as the asymptotic

per-generation growth rate [29, 43].

The type reproduction numbers may also be calculated from the next-generation

matrix using the method of Heesterbeek and Roberts [39, 61]. This method will be

described breifly. Let K = FV −1 denote the next-generation matrix, ej denote the

jth unit column vector, and Pj denote the n× n matrix with all zero entries except

for the diagonal element pjj = 1. In the first generation, the number of infections of

type j is eTj Kej = kjj. In the second generation, the number of infections of type j

is eTj K[(I− Pj)K]ej, where I is the n× n identity matrix. In generation n, the

number of infections of type j is eTj K[(I− Pj)K]n−1ej. In general, the type

24


reproduction number for the control of a single infectious group Ij is

Tj = eTj K∞∑k=1

[(I− Pj)K]kej = eTj K[I− (I− Pj)K]−1ej,

provided that ρ((I− Pj)K) < 1 and K is irreducible [39, 61]. In the case of one

infectious group, n = 1, the type reproduction number equals the basic reproduction

number, T1 = R0. If there are n = 2 infectious groups, there is a type reproduction

number for each group. They are

T1 = k11 +k12k21

1− k22

,

T2 = k22 +k12k21

1− k11

.

The assumption that K is irreducible requires k12k21 6= 0, and in order for Ti to

exist, kjj < 1 for j = 1, 2.

Roberts and Heesterbeek [61] illustrated that 1−R0 and 1− Ti are either both

positive, negative, or equal to zero. Allen and Lahodny [8] showed that for two

infectious groups, one of the following relationships holds:

Ti < R0 < 1 or Ti > R0 > 1 or Ti = R0 = 1. (3.3)

This relationship between Ti and R0 makes sense biologically. The inequality

T1 > R0 > 1 implies that if the basic reproduction number is greater than one, then

more effort is needed to control a single infectious group than to control both

groups. These deterministic thresholds can be extended to models with periodic or

heterogeneous environments [17, 18, 43].

Next we summarize some results from the theory of multitype branching

processes. The focus will be on results relating to the probability of extinction

[16, 44, 45, 55, 60]. For more details on branching processes and their applications,

consult the following references [16, 30, 36, 38, 44, 45, 48, 55].

Let ~I(t)|t ∈ [0,∞) be a collection of discrete random vectors where~I(t) = (I1(t), . . . , In(t))T . For simplicity, the same notation is used for the random

variables as for the deterministic variables. Assume that individuals of type i give

25


‘birth’ to individuals of type j and that the number of offspring produced by a type

i individual is independent of the number of offspring produced by other type i

individuals or individuals of type j 6= i. Let Yjinj=1 denote the offspring random

variables for a type i individual. That is, Yji is the number of type j offspring

produced by one type i individual for i = 1, . . . , n. Denote the probability that one

type i individual produces kj type j individuals as

Pi(k1, . . . , kn) = ProbY1i = k1, . . . , Yni = kn.

Given Ii(0) = 1 and Ij(0) = 0 for j 6= i, define the offspring probability generating

function (pgf) for type i as

fi(x1, . . . , xn) =∞∑

kn=0

· · ·∞∑k1=0

Pi(k1, . . . , kn)xk11 · · · xknn .

Note that fi always has a fixed point at ~1 = (1, . . . , 1). That is, fi(1, . . . , 1) = 1 for

i = 1, . . . , n.

Assume that each offspring pgf fi is not simple. That is, fi is not a linear function

of the variables xj and fi(0, . . . , 0) 6= 0. Explicitly,

fi(x1, . . . , xn) 6=n∑j=1

ajxj.

The expectation matrix M = [mji] is a nonnegative n× n matrix such that the

element mji is the expected number of type j offspring produced by a type i

individual. That is,

mji =∂fi∂xj

∣∣∣∣~x=~1

.

Assume that the matrix M is irreducible.

Under the assumptions that fi is not simple and M is irreducible, the spectral

radius of the expectation matrix, ρ(M), determines whether the probability of

extinction is less than or equal to one [16, 38, 60]. If ρ(M) ≤ 1, the probability of

disease extinction is one

limt→∞

Prob~I(t) = ~0 = 1.

26


If ρ(M) > 1, there exists a unique fixed point (q1, . . . , qn) ∈ (0, 1)n of the offspring

pgfs, fi(q1, . . . , qn) = qi for i = 1, . . . , n, such that the probability of disease

extinction is

P0 = limt→∞

Prob~I(t) = ~0 = qi11 · · · qinn < 1,

where ij = Ij(0) [16, 60]. This expression follows from the assumption that the

offspring random variables Yji are independent.

3.3 Single Infectious Group

Consider the classic SIR epidemic model where S, I, and R denote the number of

susceptible, infectious, and recovered individuals, respectively. Let N = S + I +R

denote the total population size. The model consists of three ODEs:

S = Λ− dS − βSI

N,

I =βSI

N− (d+ γ + α)I,

R = γI − dR,

where S(0) > 0, I(0) > 0, and R(0) = 0. The parameter Λ is the immigration/birth

rate, βSI/N is the rate at which susceptible individuals are infected, d is the natural

death rate, γ is the recovery rate, and α is the disease-related death rate. The DFE

is (S, 0, 0) = (Λ/d, 0, 0). The basic reproduction number for the SIR model is

R0 =β

d+ γ + α.

The same basic reproduction number applies to SIS and SIRS models as well. This

formula for the basic reproduction number follows directly from the differential

equation for I when S ≈ N . If R0 ≤ 1, then solutions approach the DFE and if

R0 > 1, then solutions approach a unique endemic equilibrium [49].

The corresponding CTMC SIR model can be defined in terms of the transitions

that occur in the stochastic process ~X(t) = (S(t), I(t), R(t)), t ∈ [0,∞) during some

infinitesimally small time ∆t, and the corresponding transition probabilities,

Prob ~X(t+ ∆t) = ~b| ~X(t) = ~a = P (~a,~b)∆t+ o(∆t).

27


The random variables S, I, and R are discrete-valued and continuous in time. For

simplicity, the same notation is used for the random variables and the deterministic

variables. The transitions and rates are given in Table 3.1.

Table 3.1. Transitions and rates for the CTMC SIR epidemic model.

Description State transition ∆ ~X(t) Rate

Birth of S (S, I, R)→ (S + 1, I, R) ΛDeath of S (S, I, R)→ (S − 1, I, R) dSInfection (S, I, R)→ (S − 1, I + 1, R) βSI/NRecovery (S, I, R)→ (S, I − 1, R + 1) γIDeath of I (S, I, R)→ (S, I − 1, R) (d+ α)IDeath of R (S, I, R)→ (S, I, R− 1) dR

To derive a stochastic threshold for disease extinction, the Markov process

summarized in Table 3.1 is approximated near the DFE. The dynamics of I are

considered when the susceptible population size S is approximately S = Λ/d and

the initial number of infectious individuals I(0) is small. Assume that S(t) = S,

R(t) = 0, and the events associated with I(t) are independent. The assumption that

the events are independent leads to a branching process [16, 38, 44, 55, 60]. An

approximation for the probability of disease extinction

P0 = limt→∞

ProbI(t) = 0,

can be obtained from the offspring probability generating function (pgf) for I.

The offspring pgf for infectious individuals is defined when the initial number of

infectious individuals is one, I(0) = 1. The probability of a successful transmission

for an infectious individual is β/(β + d+ γ + α) which leads to a total of two

infectious individuals, x2, and the probability of a death or recovery is

(d+ γ + α)/(β + d+ γ + α) which results in zero infectious individuals, x0 = 1. The

offspring pgf for I is

f(x) =βx2 + d+ γ + α

β + d+ γ + α.

For other examples of constructing offspring pgfs, see [16, 38, 44, 48, 55]. The

28


expected number of offspring per infectious individual is given by

m = f ′(1) =2β

β + d+ γ + α.

For a small initial number of infectious individuals, the branching process either hits

zero (disease extinction) or increases rapidly (disease persistence). The occurrence

of these outcomes depends on the value of m. If m ≤ 1, then

limt→∞

ProbI(t) = 0 = 1,

and if m > 1, there exists a unique fixed point q ∈ (0, 1) of the offspring pgf f ,

f(q) = q, such that

P0 = limt→∞

ProbI(t) = 0 = qi,

where i = I(0) is the initial number of infectious individuals [16, 38, 44, 55, 60]. The

three cases in which m < 1, m = 1, and m > 1 are called the subcritical, critical,

and supercritical cases, respectively.

For the SIR model, it is easy to see that R0 < 1 (R0 = 1 or R0 > 1) if and only if

m < 1 (m = 1 or m > 1). Moreover, in the supercritical case m > 1, the fixed point

of the offspring pgf is q = 1/R0. Therefore, the approximation of Whittle [73] is

valid. For I(0) = i and R0 > 1, the probability of disease persistence is given by

1− (1/R0)i.

In the next section, epidemic models with multiple infectious groups are

considered. Deterministic and stochastic thresholds for disease extinction are

calculated.

3.4 Multiple Infectious Groups

In this section, some well-known deterministic epidemic models are considered.

These models include SEIR, vector-host, treatment, and stage-structured models

[26, 70]. The corresponding CTMC models are derived and offspring pgfs are

defined for the infectious groups. The probability of disease extinction is given for

each model and numerical examples are presented for each model.

29


3.4.1 SEIR Model

The classic SEIR model is similar to the SIR model with the exception that there

is an exposed or latent group E, where 1/ν is the average latent period. The model

consists of four ODEs:

S = Λ− dS − βSI

N,

E =βSI

N− νE − dE,

I = νE − (d+ γ + α)I,

R = γI − dR.

The basic reproduction number is calculated using the next-generation matrix

approach for the E and I groups [70, 71]. The next-generation matrix is

FV −1 =

βν

(ν + d)(d+ γ + α)

β

d+ γ + α

0 0

.The basic reproduction number is the spectral radius of this matrix,

R0 =βν

(ν + d)(d+ γ + α).

Since FV −1 is reducible, the type reproduction numbers cannot be defined. Thus, it

is not possible to control the disease by controlling only E or I. Both E and I must

be controlled in order to achieve disease extinction. The dynamics of the SEIR

model are well-known [53]. If R0 ≤ 1, the DFE (S, 0, 0, 0) = (Λ/d, 0, 0, 0) is globally

asymptotically stable and if R0 > 1, a unique endemic equilibrium exists and is

globally asymptotically stable [53].

For the CTMC model, let ~X(t) = (S(t), E(t), I(t), R(t)) be a discrete random

vector. The state transitions and rates for the CTMC SEIR model are given in

Table 3.2.

Offspring pgfs for the multitype branching process can be defined for the variables

E and I. Assume that S(0) = S, R(0) = 0, and the initial population size N(0) ≈ S

30


Table 3.2. Transitions and rates for the CTMC SEIR epidemic model.Description State transition Rate

Birth of S (S,E, I, R)→ (S + 1, E, I, R) ΛDeath of S (S,E, I, R)→ (S − 1, E, I, R) dSInfection (S,E, I, R)→ (S − 1, E + 1, I, R) βSI/NE becomes I (S,E, I, R)→ (S,E − 1, I + 1, R) νEDeath of E (S,E, I, R)→ (S,E − 1, I, R) dERecovery (S,E, I, R)→ (S,E, I − 1, R + 1) γIDeath of I (S,E, I, R)→ (S,E, I − 1, R) (d+ α)IDeath of R (S,E, I, R)→ (S,E, I, R− 1) dR

is sufficiently large. Given E(0) = 1 and I(0) = 0, the offspring pgf for E is

f1(x1, x2) =νx2 + d

ν + d

and given E(0) = 0 and I(0) = 1, the offspring pgf for I is

f2(x1, x2) =βx1x2 + d+ γ + α

β + d+ γ + α.

The expectation matrix for this branching process is

M =

0β

β + d+ γ + αν

ν + d

β

β + d+ γ + α

.The pgfs f1 and f2 are not simple and the matrix M is irreducible. According to the

Jury conditions [3], ρ(M) < 1 iff

trace(M) < 1 + det(M) < 2. (3.4)

The second inequality in the Jury conditions (3.4) is clearly satisfied since

det(M) < 0. The first inequality is satisfied iff R0 < 1. That is,

ρ(M) < 1 iff R0 < 1. (3.5)

31


If R0 > 1, the fixed point of the offspring pgfs is calculated by setting fi(q1, q2) = qi

for i = 1, 2 and solving for qi. The unique solution (q1, q2) ∈ (0, 1)2 is

q1 =ν

ν + d

1

R0

+d

ν + d, (3.6)

q2 =1

R0

. (3.7)

The probability of disease extinction is approximately P0 = qi11 qi22 , where E(0) = i1

and I(0) = i2. In the case I(0) = 1 and E(0) = 0, the same result as Whittle [73] is

obtained for the probability of disease persistence. The probability q1 can be

interpreted epidemiologically. Given one exposed individual, that individual either

dies with probability d/(ν + d) or survives with probability ν/(ν + d) to become

infectious. Then the infectious individual successfully transmits the disease with

probability q2 = 1/R0. Note that q2 < q1. This makes sense biologically since the

disease is more likely to persist if individuals are already infectious rather than just

exposed to the disease.

A numerical example illustrates that the approximate probability of disease

extinction, P0, is in close agreement with an estimate obtained from simulation of

sample paths. Consider the parameter values Λ = 1, d = 0.005, β = 0.25, ν = 0.1,

γ = 0.05, and α = 2d = 0.01. For these values, R0 = 3.66 and ρ(M) = 1.35. The

stable endemic equilibrium for the deterministic model is

(S, E, I, R) = (48.5, 7.21, 11.1, 111).

There is a disease outbreak before the stabilization at the endemic equilibrium. The

results are given in Figure 3.1.

Using the formulas (3.6) and (3.7) for the fixed point of the offspring pgfs,

(q1, q2) = (0.3076, 0.2730). The probability of disease extinction, P0, is compared to

the estimate obtained from examining the proportion of sample paths (out of

10,000) for which the sum E(t) + I(t) hits zero (disease extinction) prior to reaching

an endemic size of 20. If the sum E(t) + I(t) is greater than 20, it is considered an

outbreak. The results are given in Table 3.3.

32


0 50 100 150 2000

20

40

60

80

100

120

140

160

180

200

Time t

S, E

, I, R

SEIR

Figure 3.1. The ODE solution and one sample path of the CTMC SEIR model.Parameter values are Λ = 1, d = 0.005, β = 0.25, ν = 0.1, γ = 0.05, and α = 2d =0.01 with initial conditions S(0) = 200, E(0) = 0, I(0) = 2, and R(0) = 0. Thedisease persists with probability 1− P0 = 0.926.

3.4.2 Vector-Host Model

Consider a vector-host model where S is the number of susceptible hosts, I is the

number of infectious hosts, H = S + I is the total number of hosts, M is the

number of susceptible vectors, and V is the number of infectious vectors. Other

vector-host models have been used to model malaria and dengue fever, where

Table 3.3. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC SEIR model. Parameter valuesare Λ = 1, d = 0.005, β = 0.25, ν = 0.1, γ = 0.05, and α = 2d = 0.01 with initialconditions S(0) = 200, E(0) = e0, I(0) = i0, and R(0) = 0.

e0 i0 P0 Approx.

1 0 0.3076 0.30850 1 0.2730 0.28631 1 0.0840 0.08902 0 0.0946 0.09320 2 0.0745 0.0741

33


mosquitoes are the vector [27, 32]. A simple vector-host model has the form:

S = Λ− dS + γI − βhSV

H,

I =βhSV

H− (d+ γ)I,

M = Γ− µM − βmMI

H,

V =βmMI

H− µV.

The transmission rate from vector to host is βhSV/H and from host to vector is

βmMI/H. The parameters βh and βm are defined as βh = ab and βm = ac, where a

is the number of bites per vector per unit time, b is the per-bite vector to host

transmission probability, and c is the per-bite host to vector transmission

probability. The DFE is (S, 0, M , 0) = (Λ/d, 0,Γ/µ, 0). The basic reproduction

number is the spectral radius of the next-generation matrix,

FV −1 =

0βhµ

βmd+ γ

M

H0

.That is,

R0 =

√βhβm

µ(d+ γ)

M

H=

√βhβm

µ(d+ γ),

where βm = βmM/H. The type reproduction numbers are the square of the basic

reproduction number [61],

Ti = R20

for i = 1, 2. The same amount of control is needed for either the vector or host

population. Note that the inequalities in (3.3) are valid.

For the corresponding CTMC model, let ~X(t) = (S(t), I(t),M(t), V (t)) be a

discrete random vector. The transitions and corresponding rates for the CTMC

vector-host model are given in Table 3.4.

Assume that S(0) = S = H and M(0) = M are sufficiently large. Given I(0) = 1

34


Table 3.4. State transitions and rates for the CTMC vector-host model.Description State transition Rate

Host birth (S, I,M, V )→ (S + 1, I,M, V ) ΛDeath of S (S, I,M, V )→ (S − 1, I,M, V ) dSHost infection (S, I,M, V )→ (S − 1, I + 1,M, V ) βhSV/HHost recovery (S, I,M, V )→ (S + 1, I − 1,M, V ) γIDeath of I (S, I,M, V )→ (S, I − 1,M, V ) dIVector birth (S, I,M, V )→ (S, I,M + 1, V ) ΓDeath of M (S, I,M, V )→ (S, I,M − 1, V ) µMVector infection (S, I,M, V )→ (S, I,M − 1, V + 1) βmMI/HDeath of V (S, I,M, V )→ (S, I,M, V − 1) µV

and V (0) = 0, the offspring pgf for I is

f1(x1, x2) =βmx1x2 + d+ γ

βm + d+ γ,

and given I(0) = 0 and V (0) = 1, the offspring pgf for V is

f2(x1, x2) =βhx1x2 + µ

βh + µ.

The expectation matrix for this branching process is

M =

βm

βm + d+ γ

βhβh + µ

βm

βm + d+ γ

βhβh + µ

.The offspring pgfs are not simple and the matrix M is irreducible. The spectral

radius of M is

ρ(M) =βm

βm + d+ γ+

βhβh + µ

.

It is simple to verify that ρ(M) < 1 iff R0 < 1.

35


If R0 > 1, the unique fixed point (q1, q2) ∈ (0, 1)2 of the offspring pgfs is given by

q1 =(βh + µ)(d+ γ)

βh(βm + d+ γ)=

βm

βm + d+ γ

1

T1

+d+ γ

βm + d+ γ, (3.8)

q2 =µ(βm + d+ γ)

βm(βh + µ)=

βhβm + µ

1

T1

+µ

βh + µ. (3.9)

The expression for q1 can be interpreted epidemiologically. Given one infectious

host, that host can either transmit the disease to a susceptible vector with

probability βm/(βm + d+ γ) or die or recover before transmission with probability

(d+ γ)/(βm + d+ γ). If the transmission is successful, then the probability of

transmission from host to host is 1/T1. Similarly, the probability q2 has an

epidemiological meaning. Given one infectious vector, that vector can either

transmit the disease to a susceptible host with probability βh/(βh + µ) or die with

probability µ/(βh + µ). If the transmission is successful, then the probability of

transmission from an infectious vector to a susceptible vector is 1/T1.

Using the formulas for q1 and q2, if I(0) = i0 and V (0) = v0, the probability of

disease extinction is

P0 = qi01 qv02 =

(d+ γ

βh

)i0 ( µ

βm

)v0 ( βh + µ

βm + d+ γ

)i0−v0.

The same expression, except for notation, was obtained by Bartlett [21].

The probability of disease extinction, P0, is illustrated in a numerical example.

Consider the parameter values Λ = 0.5, d = 0.005, γ = 0.1, Γ = 500, µ = 0.5, and

βm = βh = 0.2. For these values, R0 = 2.76, Ti = 7.62, and ρ(M) = 1.24. The DFE

and endemic equilibrium for the ODE model are given by S = 100, M = 1000, and

(S, I , M , V ) = (17.5, 83.3, 750, 250). One sample path of the CTMC vector-host

model for which an outbreak occurs is illustrated in Figure 3.2.

Using the formulas (3.8) and (3.9) for the fixed point of the offspring pgfs,



10,000) for which the sum I(t) + V (t) hits zero (disease extinction) prior to reaching

an endemic size of 50. The results are given in Table 3.5.

36


0 50 1000

10

20

30

40

50

60

70

80

90

100

110

Time t

S, I

SI

0 50 1000

200

400

600

800

1000

1200

Time t

M, V

MV

Figure 3.2. The ODE solution and one sample path of the CTMC vector-host model.Parameter values are Λ = 0.5, d = 0.005, γ = 0.1, Γ = 500, µ = 0.5, and βm = βh =0.2 with initial conditions S(0) = 100, I(0) = 2, M(0) = 1000, and V (0) = 0. Thedisease persists with probability 1− P0 = 0.970.

3.4.3 Stage-Structured Model

Consider a stage-structured model with m stages of infection:

S = Λ− dS −m∑k=1

βkSIkN

,

I1 =m∑k=1

βkSIkN

− (ν1 + d1)I1,

Ii = νi−1Ii−1 − (νi + di)Ii, i = 2, 3, . . . ,m,

R = νmIm − dR,

Table 3.5. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC vector-host model. Parametervalues are Λ = 0.5, d = 0.005, γ = 0.1, Γ = 500, µ = 0.5, and βm = βh = 0.2 withinitial conditions S(0) = 100, I(0) = i0, M(0) = 1000, and V (0) = v0.

i0 v0 P0 Approx.

1 0 0.1746 0.17620 1 0.7518 0.75731 1 0.1312 0.13422 0 0.0305 0.03320 2 0.5652 0.5619

37


where N = S +∑m

i=1 Ii +R. Susceptible individuals are infected and then progress

through m stages of infection before recovery, νm = γ. Natural death or

disease-related death may occur in each stage of the infection, di = d+ αi. The

DFE is S = Λ/d. This model, except for notation, was considered by van den

Driessche and Watmough [70]. The basic reproduction number was calculated by

van den Driessche and Watmough [70]. The next-generation matrix is FV −1 where

F =

β1 β2 · · · βm

0 0 · · · 0...

.... . .

...

0 0 · · · 0

,

and

V −1 =

1

ν1 + d1

0 · · · 0

ν1

(ν1 + d1)(ν2 + d2)

1

ν2 + d2

· · · 0

......

. . ....

ν1ν2 · · · νm−1

(ν1 + d1)(ν2 + d2) · · · (νm + dm)

ν2 · · · νm−1

(ν2 + d2) · · · (νm + dm)· · · 1

νm + dm

.

Thus,

R0 =β1

ν1 + d1

+β2ν1

(ν1 + d1)(ν2 + d2)+ · · ·+ βmν1 · · · νm−1

(ν1 + d1) · · · (νm + dm).

For the corresponding CTMC stage-structured model, let~X(t) = (S(t), I1(t), . . . , Im(t), R(t)) be a discrete random vector. The state

transitions and rates are given in Table 3.6.

Assume that S(0) = S is sufficiently large and R(0) = 0. Given Ii(0) = 1 and

Ij(0) = 0 for j 6= i, the offspring pgf for Ii is

fi(x1, . . . , xm) =βix1xi + νixi+1 + di

βi + νi + di, i = 1, . . . ,m− 1.

38


Table 3.6. State transitions and rates for the CTMC stage-structured model.Description State transition Rate

Birth of S S → S + 1 ΛDeath of S S → S − 1 dS

Infection (S, I1)→ (S − 1, I1 + 1)m∑k=1

βkSIk/N

Death of Ij Ij → Ij − 1 djIjProgression (Ij, Ij+1)→ (Ij − 1, Ij+1 + 1) νjIjRecovery (Im, R)→ (Im − 1, R + 1) νmImDeath of R R→ R− 1 dR

Given Im(0) = 1 and Ij(0) = 0 for j = 1, . . . ,m− 1, the offspring pgf for Im is

fm(x1, . . . , xm) =βmx1xm + νm + dmβm + νm + dm

.

The expectation matrix for this branching process is an m×m matrix

M =

2β1

β1 + ν1 + d1

β2

β2 + ν2 + d2

· · · βmβm + νm + dm

ν1

β1 + ν1 + d1

β2

β2 + ν2 + d2

· · · 0

0ν2

β2 + ν2 + d2

· · · 0

......

. . ....

0 0 · · · βmβm + νm + dm

.

In the case of two infectious groups, m = 2, the expectation matrix is

M =

2β1

β1 + ν1 + d1

β2

β2 + ν2 + d2ν1

β1 + ν1 + d1

β2

β2 + ν2 + d2

.The 2× 2 matrix M satisfies ρ(M) < 1 iff the Jury conditions (3.4) are satisfied. It

is a tedious but straightforward calculation to derive conditions for the second

39


inequality of the Jury conditions:

det(M) < 1 iffβ1

ν1 + d1

< 1.

Moreover, conditions for the first inequality of the Jury conditions are

trace(M) < 1 + det(M) iff R0 < 1.

Thus

ρ(M) < 1 iff R0 < 1.

Unfortunately, even in the case m = 2, if R0 > 1 an analytical expression for the

fixed point (q1, q2) of the offspring pgfs cannot be obtained. The fixed point and

probability of extinction are calculated numerically.

Consider the parameter values Λ = 1, d = 0.005, β1 = 0.4, β2 = 0.1, ν1 = 0.2,

ν2 = 0.05, d1 = 0.02, and d2 = 0.01. For these values, R0 = 3.33 and ρ(M) = 1.52.

The DFE value is S = 200 and the endemic equilibrium is

(S, I1, I2, R) = (53.7, 3.33, 11.1, 111). The fixed point of the offspring pgfs is



10,000) for which the sum I1(t) + I2(t) hits zero (disease extinction) prior to

reaching an endemic size of 20. If the sum I1(t) + I2(t) is greater than 20, it is

considered an endemic. The results are given in Table 3.7. The ODE solution and

one sample path for the CTMC stage-structured model are plotted in Figure 3.3.

The initial conditions are S(0) = S = 200, I1(0) = I2(0) = 1, and R(0) = 0.

3.4.4 Treatment Model

A model for the treatment of tuberculosis (TB) was developed by Castillo-Chavez

and Feng [26]. Tuberculosis is a bacterial infection and antibiotics are used to treat

infected individuals. When treatment methods are either inadequate or incomplete,

there is a possibility that antibiotic resistance will develop [26]. The model of

Castillo-Chavez and Feng [26] considers two strains of TB, drug-sensitive and

drug-resistant strains. For this model, there are six groups: susceptible individuals

40


Table 3.7. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC staged-progression model. Pa-rameter values are Λ = 1, d = 0.005, β1 = 0.4, β2 = 0.1, ν1 = 0.2, ν2 = 0.05,d1 = 0.02, and d2 = 0.01 with initial conditions S(0) = 200, I1(0) = i1, I2(0) = i2,and R(0) = 0.

i1 i2 P0 Approx.

1 0 0.1943 0.19320 1 0.4268 0.42951 1 0.0829 0.08392 0 0.0378 0.03870 2 0.1822 0.1836

S, treated individuals T , exposed individuals E1 and E2, and infectious individuals

I1 and I2. The subscript i = 1 denotes the drug-sensitive strain of tuberculosis and

i = 2 denotes the drug-resistant strain. Susceptible individuals are exposed to either

the sensitive or resistant strain and enter the exposed or latent group E1 or E2,

respectively. After some latent period, exposed individuals become infectious. There

is disease-related death for both strains, di = d+ αi. Antibiotic treatment is not

effective for the resistant strain. Thus, exposed and infectious individuals with the

sensitive strain are treated at rates r1 and r2, respectively. The treatment of

exposed individuals may be inadequate, pr2, or incomplete which results in

antibiotic resistance, qr2. Therefore treatment is only effective in some proportion of

individuals (1− p− q). Treated individuals, T , can become infected with either

strain, but are less likely to become infected with the resistant strain than

susceptible individuals, βT ≤ β1. Except for notation, the model of Castillo-Chavez

41


0 20 40 60 80 1000

50

100

150

200

Time t

S, I 1, I

2, R

SI1I2R

Figure 3.3. The ODE solution and one sample path of the CTMC staged-progressionmodel. Parameter values are Λ = 1, d = 0.005, β1 = 0.4, β2 = 0.1, ν1 = 0.2,ν2 = 0.05, d1 = 0.02, and d2 = 0.01 with initial conditions S(0) = 200, I1(0) = 1,I2(0) = 1, and R(0) = 0. The disease persists with probability 1− P0 = 0.917.

and Feng [26] has the form:

S = Λ− dS − β1SI1

N− β2SI2

N,

E1 =β1SI1

N+βTTI1

N− β2E1I2

N− (d+ ν1 + r1)E1 + pr2I1,

I1 = ν1E1 − (d1 + r2)I1,

E2 = β2S + E1 + T

NI2 + qr2I1 − (d+ ν2)E2,

I2 = ν2E2 − d2I2,

T = −βTTI1

N− β2TI2

N+ r1E1 + (1− p− q)r2I1 − dT.

Let N = S + E1 + I1 + E2 + I2 + T denote the total population size. The DFE is

given by S = Λ/d. The failure of treatment is not considered as a new infection [70].

Let ~X(t) = (E1(t), E2(t), I1(t), I2(t))T . The system of ODEs for the latent and

42


infectious groups can be linearized about the DFE:

d ~X(t)

dt= (F − V ) ~X(t),

where

F =

0 0 β1 0

0 0 0 β2

0 0 0 0

0 0 0 0

and V =

d+ ν1 + r1 0 −pr2 0

0 d+ ν2 −qr2 0

−ν1 0 d1 + r2 0

0 −ν2 0 d2

.

The next-generation matrix is

FV −1 =

β1ν1

(d+ ν1 + r1)(d1 + r2)− ν1pr2

0 ∗ 0

β2ν1ν2qr2

d2(d+ ν2)[(d+ ν1 + r1)(d1 + r2)− ν1pr2]

β2ν2

d2(d+ ν2)∗ ∗

0 0 0 0

0 0 0 0

,

where “*” denotes a nonzero entry but does not affect the value ρ(FV −1). Thus,

R0 = max

β1ν1

(d+ ν1 + r1)(d1 + r2)− ν1pr2

,β2ν2

d2(d+ ν2)

= maxR1,R2.

The threshold R1 was derived by van den Driessche and Watmough [70]. For a more

in-depth analysis of this model, see [26].

For the corresponding CTMC treatment model, let

~X(t) = (S(t), E1(t), E2(t), I1(t), I2(t), T (t))

be a discrete random vector. The state transitions and rates for the CTMC

stage-structured model are given in Table 3.8.

Assume that S(0) = S is sufficiently large and T (0) = 0. Given E1(0) = 1 and

43


Table 3.8. State transitions and rates for the CTMC treatment model.Description State transition Rate

Birth of S S → S + 1 ΛDeath of S S → S − 1 dSDrug-sensitive infection (S,E1)→ (S − 1, E1 + 1) β1SI1/NDrug-resistant infection (S,E2)→ (S − 1, E2 − 1) β2SI2/NResistant to infection of E1 (E1, E2)→ (E1 − 1, E2 + 1)Latent to infectious (E1, I1)→ (E1 − 1, I1 + 1) ν1E1

Treatment of E1 (E1, T )→ (E1 − 1, T + 1) r1E1

Death of E1 E1 → E1 − 1 dE1

Death of I1 I1 → I1 − 1 d1I1

Effective treatment of I1 (I1, T )→ (I1 − 1, T + 1) (1− p− q)r2I1

Inadequate treatment of I1 (E1, I1)→ (E1 + 1, I1 − 1) pr2I1

Treatment resistance I1 (I1, E2)→ (I1 − 1, E2 + 1) qr2I1

Drug-sensitive infection of T (E1, T )→ (E1 + 1, T − 1) βTTI1/NDrug-resistant infection of T (E2, T )→ (E2 + 1, T − 1) βTTI2/NDeath of T T → T − 1 dTDeath of E2 E2 → E2 − 1 dE2

Latent to infectious (E2, I2)→ (E2 − 1, I2 + 1) ν2E2

Death of I2 I2 → I2 − 1 d2I2

I1(0) = E2(0) = I2(0) = 0, the offspring pgf for E1 is

f1(x1, x2, x3, x4) =ν1x2 + d+ r1

ν1 + d+ r1

.

Similarly, the offspring pgfs for I1, E2, and I2 are

f2(x1, x2, x3, x4) =β1x1x2 + pr2x1 + qr2x3 + d1 + (1− p− q)r2

β1 + d1 + r2

,

f3(x1, x2, x3, x4) =ν2x4 + d

ν2 + d,

f4(x1, x2, x3, x4) =β2x3x4 + d2

β2 + d2

.

44


Note that the offspring pgfs are not simple. The expectation matrix is

M =

0β1 + pr2

β1 + d1 + r2

0 0

ν1

ν1 + d+ r1

β1

β1 + d1 + r2

0 0

0qr2

β1 + d1 + r2

0β2

β2 + d2

0 0ν2

d+ ν2

β2

β2 + d2

=

[M1 0

∗ M2

],

where M1 and M2 are the 2× 2 matrices in the upper left and lower right corners of

M. Apply the Jury conditions (3.4) to each of the submatrices M1 and M2. Clearly,

trace(Mi) > 0 and det(Mi) < 1 for i = 1, 2. Now

1 + det(M1)− trace(M1) = 1− ν1(β1 + pr2)

(ν1 + d+ r1)(β1 + d1 + r2)− β1

β1 + d1 + r2

=(d1 + r2)(ν1 + d+ r1)− ν1(β1 + pr2)

(ν1 + d+ r1)(β1 + d1 + r2).

Thus, trace(M1) < 1 + det(M1) if and only if

ν1(β1 + pr2)

(d1 + r2)(ν1 + d+ r1)< 1.

This inequality holds iff R1 < 1. Similarly, trace(M2) < 1 + det(M2) iff R2 < 1. By

the Jury conditions, ρ(Mi) < 1 iff Ri < 1. It follows that ρ(M) > 1 iff R0 > 1.

Since the matrix M is reducible, there is not necessarily a unique fixed point

(q1, q2, q3, q4) ∈ (0, 1)4 of the offspring pgfs for R0 > 1 (ρ(M) > 1). In fact, there are

four fixed points of the offspring pgfs lying in (0, 1]4 One of the fixed points is

(q1, q2, q3, q4) = (1, 1, 1, 1) which always exists. The existence of the remaining three

fixed points depends on the values R1 and R2. If R1 > 1, the first fixed point is

q1 =ν1

ν1 + d+ r1

1

R1

+d+ r1

ν1 + d+ r1

,

q2 =1

R1

,

q3 = 1,

q4 = 1.

(3.10)

45


If R2 > 1, there are two more fixed points in (0, 1]4. The second fixed point is

q1 = 1,

q2 = 1,

q3 =ν2

d+ ν2

1

R2

+d

d+ ν2

,

q4 =1

R2

.

(3.11)

The third fixed point is

q1 =ν1q2

ν1 + d+ r1

+d+ r1

ν1 + d+ r1

,

q3 =ν2

d+ ν2

1

R2

+d

d+ ν2

,

q4 =1

R2

,

(3.12)

and q2 is the unique solution in (0, 1) of the quadratic equation Ax2 +Bx+ C = 0

where

A =β1ν1

(β1 + d1 + r2)(ν1 + d+ r1),

B = −β1ν1 + (d1 + r2)(ν1 + d+ r1)− pr2ν1

(β1 + d1 + r2)(ν1 + d+ r1),

C(q3) =pr2(d+ r1) + [qr2q3 + d1 + (1− p− q)r2][ν1 + d+ r1]

(β1 + d1 + r2)(ν1 + d+ r1).

Numerical examples will illustrate which of the three fixed points is used to

determine the probability of disease extinction. Fix the parameter values Λ = 1,

d = 0.005, βT = 0.25β2, r1 = r2 = 0.05, p = 0.5, q = 0.1, ν1 = 0.5, ν2 = 0.1, and

d1 = d2 = 1.5d. Only the transmission rates β1 and β2 will be varied. In each

example, the probability of disease extinction P0 is calculated and compared with

the numerical approximation based on the proportion of sample paths (out of

10,000) for which the infected population E1(t) + I1(t) + E2(t) + I2(t) hits zero

before reaching a size of 20. If the infected population exceeds 20, it is considered

an endemic. In each of the examples, assume S(0) = S and T (0) = 0.

For the first example, let β1 = 0.1 and β2 = 0.005. Then R1 = 2.58 and

46


R2 = 0.635. For the deterministic model, there exists a stable endemic equilibrium

(S, E1, I1, E2, I2, T ) = (69.8, 1.83, 15.9, 1.53, 20.3, 72.4). The fixed point (3.10),

(q1, q2, q3, q4) = (0.4489, 0.3883, 1, 1) is used to calculate the probability of

extinction. For a small initial number of individuals infected with the drug sensitive

strain, E1(0) and I1(0), the probability of extinction and the numerical

approximation for the branching process are calculated. The results are given in

Table 3.9. The extinction probability does not depend on the drug-resistant strain

for small initial values. Moreover, if only the drug resistant strain is present,

E1(0) = 0 and I1(0) = 0, the probability of extinction is one.

Table 3.9. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC treatment model. Parametervalues are Λ = 1, d = 0.005, β1 = 0.1, β2 = 0.005, βT = 0.25β2, r1 = r2 = 0.05,p = 0.5, q = 0.1, ν1 = 0.5, ν2 = 0.1, and d1 = d2 = 1.5d with initial conditionsS(0) = S = 200, T (0) = 0, E2(0) = 0, I2(0) = 0, E1(0) = e1, and I1(0) = i1.

e1 i1 P0 Approx.

1 0 0.4489 0.44580 1 0.3883 0.39611 1 0.1743 0.18172 0 0.2015 0.21140 2 0.1507 0.1554

For the second example, let β1 = 0.1 and β2 = 0.04. Then R1 = 2.58 and


(S, E1, I1, E2, I2, T ) = (28.7, 0, 0, 8.16, 109, 0). The fixed points

(q1, q2, q3, q4) = (0.4489, 0.3883, 1, 1),

(q1, q2, q3, q4) = (0.3922, 0.3253, 0.2351, 0.1969),

are used to calculate the probability of extinction. The proportion of sample paths

for which the total number of infectives hits zero, (Approx.a), is calculated and

compared to the probability of extinction calculated from the fixed point (3.12), Pa0.

In addition, the proportion of sample paths for which the drug sensitive strain hits

zero (Approx.b = E1(t) + I1(t) = 0) is compared to the probability of extinction

47


calculated from the point (3.11), Pb0. The results for several initial conditions are

summarized in Table 3.10. The ODE solution and one sample path of the CTMC

treatment model are plotted in Figure 3.4.

Table 3.10. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC treatment model. Parametervalues are Λ = 1, d = 0.005, β1 = 0.1, β2 = 0.04, βT = 0.25β2, r1 = r2 = 0.05,p = 0.5, q = 0.1, ν1 = 0.5, ν2 = 0.1, and d1 = d2 = 1.5d with initial conditionsS(0) = S = 200, T (0) = 0, E1(0) = e1, I1(0) = i1, E2(0) = e2, and I2(0) = i2.Probability Pa0 is calculated from fixed point (3.12) and probability Pb0 is calculatedfrom fixed point (3.11).

e1 i1 e2 i2 Pa0 Approx.a Pb0 Approx.b

1 0 0 0 0.3922 0.3893 0.4489 0.44710 1 0 0 0.3253 0.3233 0.3883 0.38801 1 0 0 0.1178 0.1270 0.1743 0.17900 0 1 0 0.2351 0.2336 1 10 0 0 1 0.1969 0.1956 1 10 0 1 1 0.0463 0.0472 1 11 0 1 0 0.0852 0.0934 0.4489 0.44760 1 0 1 0.0640 0.0688 0.3383 0.3953

For the third example, let β1 = 0.02 and β2 = 0.04. Then R1 = 0.515 and


(S, E1, I1, E2, I2, T ) = (69.8, 1.83, 15.9, 1.53, 20.3, 72.4). The fixed point (3.12),

(q1, q2, q3, q4) = (0.8307, 0.8120, 0.2351, 0.1969) is used to calculate the probability of

extinction. The results are given in Table 3.11.

3.5 Discussion

Deterministic thresholds such as the basic reproduction number or type

reproduction numbers are important in mathematical epidemiology. These

thresholds provide information about the persistence or extinction of an infectious

disease, the amount of control needed to eradicate a particular disease, and which

parameters should be adjusted in order to control the spread of disease. The

deterministic thresholds are well-known [29, 39, 61, 70, 71]. However, much less is

known about stochastic thresholds with the exception of the approximation (3.1) of

48


0 200 4000

20

40

60

80

100

120

140

160

180

200

Time t

S, E

1, I1

0 200 4000

50

100

150

Time tT,

E2, I

2

SE1I1

TE2I2

Figure 3.4. The ODE solution and one sample path of the CTMC treatment model.Parameter values are Λ = 1, d = 0.005, β1 = 0.1, β2 = 0.04, βT = 0.25β2, r1 = r2 =0.05, p = 0.5, q = 0.1, ν1 = 0.5, ν2 = 0.1, and d1 = d2 = 1.5d ν1 = 0.2, ν2 = 0.05,d1 = 0.02, and d2 = 0.01. The reproduction numbers are R1 = 2.58 and R2 = 5.08.The disease persists with probability 1− P0 = 0.936.

Table 3.11. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC treatment model. Parametervalues are Λ = 1, d = 0.005, β1 = 0.02, β2 = 0.04, βT = 0.25β2, r1 = r2 = 0.05,p = 0.5, q = 0.1, ν1 = 0.5, ν2 = 0.1, and d1 = d2 = 1.5d with initial conditionsS(0) = S = 200, T (0) = 0, E1(0) = e1, I1(0) = i1, E2(0) = e2, and I2(0) = i2.

e1 i1 e2 i2 P0 Approx.

1 0 0 0 0.8307 0.83560 1 0 0 0.8120 0.81151 1 0 0 0.6745 0.67700 0 1 0 0.2351 0.23830 0 0 1 0.1969 0.19250 0 1 1 0.0463 0.04701 0 1 0 0.1953 0.19240 1 0 1 0.1599 0.1670

49


Whittle [73]. If a small initial number of infectious individuals are introduced into a

large susceptible population, then an estimate for the probability of disease

persistence or extinction can be obtained from multitype branching processes

approximations.

New expressions for the probability of extinction were derived for several

well-known models. For each of the models discussed, the estimate for the

probability of disease extinction, P0, was shown to be in strong agreement with the

numerical approximation based on computing sample paths. Furthermore, new

relationships between the deterministic and stochastic thresholds were derived.

50


CHAPTER 4

STOCHASTIC MULTI-PATCH EPIDEMIC MODELS

4.1 Introduction

Infectious diseases in humans or domestic and wild animals such as influenza,

tuberculosis, SARS, and foot-and-mouth disease can be easily transmitted from

region to region. Factors such as spatial connectivity and the dispersal of

individuals through travel can significantly affect the likelihood that a disease will

persist in a given region. For instance, a disease may not be prevalent in rural areas,

but if individuals from an urban center travel to these more remote regions, there

may be outbreaks in remote areas. Alternatively, the reverse may be the case where

the disease is present in rural areas and is spread to urban areas [68]. Similarly, the

spread of disease can occur at an accelerated rate in social or environmental

‘hotspots’ such as airports, schools, or common water sources [15, 22, 47]. Thus, it

is important to account for factors such as dispersal and environmental

heterogeneity in epidemic models.

The role of dispersal on disease dynamics has been examined for general epidemic

models [5, 12, 13, 14, 24, 41, 72] and models focused on particular diseases such as

tuberculosis [25], malaria [33], and influenza [66, 42]. Many of these models involve

dividing a region into multiple patches and allowing dispersal of susceptible or

infectious individuals between these patches [5, 41, 42, 67, 72]. Allen, Kirupaharan,

and Wilson [7] considered a discrete-time two-patch SIS epidemic model. One patch

was considered high-risk, having a patch reproduction number greater than one, and

the other patch was low-risk with a patch reproduction number less than one. In

the absence of movement, the disease persisted only in the high-risk patch. When

the dispersal of susceptible and infectious individuals was allowed between the two

patches, an endemic equilibrium was reached in both patches. Furthermore, when

only infectious individuals were allowed to move between the two patches the

disease did not persist in either patch; all susceptible individuals eventually moved

into the low-risk patch and the high-risk patch became empty. Similar results were

obtained by Allen et al. [5].

The role of movement also has important implications for disease control. For

51


instance, it may be possible to stop the spread of a disease by conducting border

checks and restricting the movement of infectious individuals or restricting the

movement to and from high-risk patches where a disease is more prevalent [14]. In

the case of influenza, van den Driessche et al. [42] showed that restricting the

dispersal of infectious individuals from low to high-risk patches helps control the

disease. However, for certain parameter values, restricting the dispersal of infectious

individuals from high to low-risk patches could have negative effects on disease

control [42]. Moreover, banning all travel of infectious individuals from high to

low-risk patches could result in the low-risk patch becoming disease-free, while the

high-risk patch becomes even more disease-prevalent [42]. Similarly, Ruan, Wang,

and Levin [65] showed that the spread of SARS can be contained by implementing

border screening for infectious individuals. On the other hand, screening at the

borders is only effective in identifying individuals exhibiting symptoms. Gao and

Ruan [33] state that the dispersal of exposed or latent individuals can contribute to

the spread of disease and that ineffective border screening may adversely affect

disease transmission.

Our goals are to develop stochastic multi-patch epidemic models that are

analytically tractable and to use these models to examine how dispersal between

patches and the location of an outbreak affect the probability of disease persistence

or extinction. These results have important implications for disease control. In the

next two sections, models with multiple patches with and without demographics are

studied.

4.2 Multi-Patch SIS, SIR, and SIRS Models without Demographics

In this section multi-patch SIS, SIR, and SIRS models will be considered. In

Section 4.2.1, deterministic (ODE) multi-patch models are introduced and the

dynamics in each patch are discussed when there is no movement between patches.

Then the dynamics of the models are explored when there is movement between the

patches. Next, stochastic versions of these multi-patch models are introduced. In

Section 4.2.2, CTMC models are considered and some results from continuous-time

multitype branching process theory are used to derive an expression for the

approximate probability of disease extinction. In Section 4.2.3, SDE patch models

52


are derived from the corresponding ODE models. Finally, in Section 4.2.4, the

differences between the deterministic and stochastic patch models are explored

numerically.

4.2.1 Deterministic Models

Consider a multi-patch SIS model, where n ≥ 2 is the number of patches,

Ω = 1, 2, . . . , n, and Sj(t) and Ij(t) denote the number of susceptible and

infectious individuals in patch j at time t ≥ 0, respectively. For ease of notation, let

Sj = Sj(t) and Ij = Ij(t). Denote the total population in patch j by Nj = Sj + Ij.

The model takes the form

Sj = −βjSjIjNj

+ γjIj +∑k∈Ω

(dskjSk − dsjkSj), (4.1)

Ij =βjSjIjNj

− (γj + αj)Ij +∑k∈Ω

(dikjIk − dijkIj), (4.2)

where Sj(0) > 0 and Ij(0) ≥ 0 for all j ∈ Ω. The parameters dskj ≥ 0 and dikj ≥ 0

represent the degree of movement from patch k to patch j by susceptible and

infectious individuals, respectively. The parameters βj > 0, γj ≥ 0, and αj ≥ 0

represent the rate of infection, the recovery rate, and the disease-related death rate

in patch j, respectively. This model is similar to the multi-patch SIS model of Allen,

Bolker, Lou, and Nevai [5] with the exception that there are disease-related deaths.

Summing the equations for Sj and Ij in (4.1)-(4.2), it follows that for each j ∈ Ω

Nj = −αjIj +∑k∈Ω

(dskjSk − dsjkSj + dikjIk − dijkIj). (4.3)

Let N denote the total population size in all patches,

N =∑j∈Ω

Nj. (4.4)

It follows from equation (4.3) that

N = −∑j∈Ω

αjIj. (4.5)

53


Note that if there are no disease-related deaths, αj = 0 for each j ∈ Ω, then

N = N(0) is a constant.

It will be useful to introduce some notation regarding the dispersal parameters

dskj and dikj. Define the connectivity matrices as

Ds = [dskj] and Di = [dikj]. (4.6)

In the case of two patches,

Ds =

[ds11 ds21

ds12 ds22

]and Di =

[di11 di21

di12 di22

].

For each patch j ∈ Ω, define the patch reproduction number as

R0j =βj

γj + αj. (4.7)

If γj = αj = 0, then let R0j =∞. If R0j > 1, then patch j is said to be a high-risk

patch. Otherwise, it is called a low-risk patch [5].

The following theorem summarizes the dynamics in each patch when there is no

movement.

Theorem 4.1. If Ds = Di = O (zero matrices), the asymptotic solution for patch j

in the multi-patch SIS epidemic model depends on R0j.

(i) If R0j > 1, then

limt→∞

(Sj(t), Ij(t)) =

(Nj(0)

R0j,(

1− 1R0j

)Nj(0)

), αj = 0

(0, 0), αj > 0.

(ii) If R0j ≤ 1, then

limt→∞

(Sj(t), Ij(t)) =

(Nj(0), 0), αj = 0

(Cj, 0), αj > 0,

for some Cj where 0 ≤ Cj < Nj(0).

54


Proof. For fixed j ∈ Ω, let ij = Ij/Nj. For simplicity, we omit the subscript j and

write i = I/N and R0 = β/(γ + α). Then

i = i[β − γ − α− (β − α)i], (4.8)

and

N = −αiN. (4.9)

The differential equation (4.8) can be expressed as

i = i(a− bi), (4.10)

where a = β − γ − α and b = β − α.

If R0 > 1 and α = 0, then N = N(0) is a constant and a, b > 0. It follows that

limt→∞

i(t) = 1− 1

R0

.

Since N is a constant, the result in part (i) follows.

If R0 > 1 and α > 0, then a, b > 0. It follows that

limt→∞

i(t) = 1− γ

β − α> 0.

Thus, i(t) is bounded below by a positive constant for sufficiently large t. It follows

from (4.9) that

limt→∞

N(t) = 0.

Thus, I(t)→ 0 and S(t)→ 0 as t→∞.

If R0 ≤ 1 and α = 0, then N = N(0) is a constant and a ≤ 0. It follows that

limt→∞

i(t) = 0.

Then part (ii) follows.

If R0 ≤ 1 and α > 0, it follows from (4.8) that

limt→∞

i(t) = 0.

55


So I(t)→ 0 as t→∞. Since N(t) is decreasing, S(t)→ C for some

0 ≤ C < N(0).

Next, we define a basic reproduction number for the patch model (4.1)-(4.2). Let

F = diag[βj], (4.11)

V = diag

[γj + αj +

∑k∈Ω

dijk

]−Di, (4.12)

where Di is the dispersal matrix for infectives. The eigenvalues of F − V have

negative real part if and only if the spectral radius of FV −1 is less than one [70].

The basic reproduction number is the spectral radius of FV −1 [70],

R0 = ρ(FV −1). (4.13)

The next theorem states that if R0 < 1, the infection dies out. A result similar to

Allen et al. [5] will be applied.

Theorem 4.2. If R0 < 1, then the multi-patch SIS epidemic model (4.1)-(4.2)

satisfies I(t)→ 0 as t→∞.

Proof. Suppose R0 < 1. For each fixed j ∈ Ω,

Ij ≤

(βj − γj − αj −

∑k∈Ω

dijk

)Ij +

∑k∈Ω

dikjIk.

That is,

I ≤ (F − V )I,

where F and V are defined by (4.11) and (4.12) and I = (I1, . . . , In)T . Consider the

initial-value problem

X = (F − V )X,

X(0) = I(0).

Then X(t) = I(0)e(F−V )t. Since R0 < 1, the eigenvalues of F − V have negative real

part and since F − V has nonnegative off-diagonal elements, the solution X(t) can

56


be compared with the solution I(t) [51]. By the comparison principle, since

X(t)→ 0 as t→∞, I(t)→ 0 as t→∞ [51].

Corollary 4.1. For fixed j ∈ Ω, if R0j < 1 and dijk = dikj = 0 for k 6= j, then in

model (4.1)-(4.2), Ij(t)→ 0 as t→∞ .

Proof. Fix j ∈ Ω and suppose R0j < 1. Then

Ij ≤ (βj − γj − αj)Ij < 0.

It follows that Ij is a strictly decreasing function which is bounded below by zero

and so the conclusion follows.

Consider a multi-patch SIR or SIRS model, where n ≥ 2 is the number of patches

and Ω = 1, 2, . . . , n. As before, Sj(t), Ij(t), Rj(t) denote the number of

susceptible, infectious, and recovered individuals in patch j at time t ≥ 0,

respectively. For simplicity, let Sj = Sj(t), Ij = Ij(t), and Rj = Rj(t). Let

Nj = Sj + Ij +Rj denote the total population size in patch j. The model takes the

form

Sj = −βjSjIjNj

+ νjRj +∑k∈Ω


Ij =βjSjIjNj

− (γj + αj)Ij +∑k∈Ω


Rj = γjIj − νjRj +∑k∈Ω

(drkjRk − drjkRj), (4.16)

where Sj(0) > 0, Ij(0) ≥ 0, and Rj(0) = 0 for each j ∈ Ω. The notation used here is

the same as for the multi-patch SIS model (4.1)-(4.2). Here, drkj ≥ 0 represents the

degree of movement from patch k to patch j by recovered individuals and νj ≥ 0 is

the rate of loss of immunity for recovered individuals in patch j. It is assumed that

γj > 0 for each j ∈ Ω. The patch reproduction numbers remain the same as for the

SIS model,

R0j =βj

γj + αj.

In addition, Theorem 4.2 and Corollary 4.1 apply, which is stated here as a corollary.

57


Corollary 4.2. (i) If R0 < 1, then in model (4.14)-(4.16), I(t)→ 0 as t→∞.

(ii) For fixed j ∈ Ω, if R0j < 1 and dijk = dikj for k 6= j, then in model

(4.14)-(4.16), Ij(t)→ 0 as t→∞.

The following theorem summarizes the dynamics of the SIR and SIRS models in

each patch when there is no movement. The connectivity matrix Dr = [drkj] is

defined as in (4.6).

Theorem 4.3. If Ds = Di = Dr = O (zero matrices), the asymptotic solution for

patch j in the SIR/SIRS epidemic model depends on R0j.

(i) If R0j > 1, then

limt→∞

(Sj(t), Ij(t), Rj(t)) =

(Sj, Ij, Rj), γj > 0, νj > 0, αj = 0

(Aj, 0, Cj), γj > 0, νj = 0, αj = 0

(0, 0, 0), γj > 0, νj > 0, αj > 0

(Bj, 0, Ej), γj > 0, νj = 0, αj > 0

where Sj = N(0)/R0, Ij = N(0)ν(R0 − 1)/R0(γ + ν),

Rj = N(0)γ(R0 − 1)/R0(γ + ν), Aj, Bj, Cj, Ej > 0, Aj + Cj = Nj(0), and

Bj + Ej < Nj(0).

(ii) If R0j ≤ 1, then

limt→∞

(Sj(t), Ij(t), Rj(t)) =

(Nj(0), 0, 0), γj > 0, νj > 0, αj = 0

(Aj, 0, Cj), γj > 0, νj = 0, αj = 0

(Bj, 0, 0), γj > 0, νj > 0, αj > 0

(Bj, 0, Ej), γj > 0, νj = 0, αj > 0

where 0 ≤ Aj, Bj < Nj(0), 0 < Cj ≤ Nj(0), 0 < Ej < Nj(0), Aj + Cj = Nj(0),

and Bj + Ej < Nj(0).

Proof. For fixed j ∈ Ω, let sj = Sj/Nj, ij = Ij/Nj, and rj = Rj/Nj. For simplicity,

omit the subscript j and write s = S/N , i = I/N , r = R/N , and R0 = β/(γ + α).

58


Then s+ i+ r = 1 and r = 1− s− i. So the dynamics depend on s and i. Now

N = −αiN, (4.17)

and

s = (α− β)si+ ν(1− s− i),

i = i(βs− γ − α + αi).

Consider the SIRS model with ν > 0. Allen et al. [6] showed that the system of

differential equations for s and i has a unique globally stable positive equilibrium

(s, i) with s+ i < 1 iff β > γ + α. When β > γ + α, the equilibrium (1, 0) is a saddle

point with stable manifold on the s-axis. Also, when β ≤ γ + α, (1, 0) is globally

stable. The origin is a saddle with stable manifold the i-axis and unstable manifold

the s-axis. The preceeding result shows that for α = 0 when the population size is

constant, either the population approaches a DFE (R0 ≤ 1) or an endemic

equilibrium (R0 > 1). In the case α > 0, the population size is decreasing and either

the population becomes disease-free (R0 ≤ 1) with a reduced population size or

reaches extinction (R0 > 1). Now consider the SIR model with ν = 0. In this case,

all equilibria lie on the s-axis (i = 0). Hence, in either case R0 > 1 or R0 ≤ 1,

i(t)→ 0 which implies I(t)→ 0.

4.2.2 Markov Chain Models

Consider a corresponding CTMC multi-patch SIS, SIR, or SIRS model. Let

~X(t) = (S1(t), I1(t), . . . , Sn(t), In(t))T ,

~X(t) = (S1(t), I1(t), R1(t), . . . , Sn(t), In(t), Rn(t))T

be discrete random vectors for the corresponding CTMC SIS and SIR/SIRS models,

respectively. For simplicity, the same notation is used for the random variables and

the deterministic variables. The state transitions and rates for the CTMC SIS and

SIR/SIRS models are given in Tables 4.1 and 4.2.

As in Section 3.2, under certain restrictions, offspring pgfs for the multitype

59


Table 4.1. State transitions and rates for the CTMC multi-patch SIS epidemic model.Description State transition Rate

Infection in patch j (0, . . . , Sj − 1, Ij + 1, . . . , 0) βjSjIj/Nj

Recovery in patch j (0, . . . , Sj + 1, Ij − 1, . . . , 0) γjIjDeath of Ij (0, . . . , Ij − 1, . . . , 0) αjIjSusceptible dispersal from j to k (0, . . . , Sj − 1, . . . , Sk + 1, . . . , 0) dsjkSjInfectious dispersal from j to k (0, . . . , Ij − 1, . . . , Ik + 1, . . . , 0) dijkIj

Table 4.2. State transitions and rates for the CTMC multi-patch SIR/SIRS epidemicmodel.

Description State transition Rate

Infection in patch j (0, . . . , Sj − 1, Ij + 1, . . . , 0) βjSjIj/Nj

Recovery in patch j (0, . . . , Ij − 1, Rj + 1, . . . , 0) γjIjDeath of Ij (0, . . . , Ij − 1, . . . , 0) αjIjLoss of immunity in patch j (0, . . . , Sj + 1, Ij, Rj − 1, . . . , 0) νjRj

Susceptible dispersal from j to k (0, . . . , Sj − 1, . . . , Sk + 1, . . . , 0) dsjkSjInfectious dispersal from j to k (0, . . . , Ij − 1, . . . , Ik + 1, . . . , 0) dijkIjRecovered dispersal from j to k (0, . . . , Rj − 1, . . . , Rk + 1, 0, . . . , 0) drjkRj

branching processes can be defined for each random variable Ij, j ∈ Ω. Assume that

individuals of type j, Ij, give ‘birth’ to individuals of type k, Ik, and that the

number of offspring produced by a type j individual does not depend on the number

of offspring produced by other individuals of type j or k 6= j. Moreover, assume

that the initial population in each patch is sufficiently large, Sj(0) ≈ Nj(0). Then

continuous-time multitype branching process theory can be used to approximate the

probability of extinction.

As in Chapter 3, the method of Allen and Lahodny [8] is applied. Let

Sj(0) ≈ Nj(0), Ij(0) = 1 and Ik(0) = 0 for all k 6= j. The offspring pgf for Ij is

fj(x1, . . . , xn) =

βjx2j + γj + αj +

∑k∈Ω\j

dijkxk

βj + γj + αj +∑

k∈Ω\j

dijk. (4.18)

60


The term βj/(βj + γj + αj +∑

j∈Ω\j dijk) represents the probability that a

susceptible individual becomes infectious which results in two type j infectious

individuals, x2j . The term (γj + αj)/(βj + γj + αj +

∑j∈Ω\j d

ijk) represents the

probability that an infectious individual is lost due to recovery or death resulting in

zero infectious individuals, x0j . The term dijk/(βj + γj +αj +

∑j∈Ω\j d

ijk) represents

the probability of movement from patch j to k resulting in one type k infectious

individual and zero type j infectious individuals, xkx0j . The offspring pgfs for Ij are

the same for the SIS, SIR, and SIRS models.

The expectation matrix M = [mkj] is a nonnegative n× n matrix such that

mkj =

2βj


k∈Ω\j

dijk, k = j

dijk


k∈Ω\j

dijk, k 6= j.

(4.19)

Explicitly, the expectation matrix has the form:

M =

2β1

A1

di21

A2

· · · din1

Andi12

A1

2β2

A2

· · · din2

An...

.... . .

...

di1nA1

di2nA2

· · · 2β2

An

,

where Aj = βj + γj + αj +∑

k∈Ω\j dijk for j = 1, . . . , n. Note that if dijk > 0, then

M is irreducible.

Applying branching process theory and properties of M -matrix theory, it can be

shown that the spectral radius of the expectation matrix M determines whether the

probability of extinction is less than or equal to one [4, 9]. Let

D = diag


k∈Ω\j

dijk

, (4.20)

61


an n× n diagonal matrix. It follows that

[M− I]D = F − V (4.21)

where I is the n× n identity matrix and F and V are defined as in (4.11) and

(4.12), respectively. Applying the theory of branching processes [16, 30, 38, 45, 60],

it follows that the continuous-time multitype branching process is subcritical,

critical, or supercritical if the spectral abscissa (the real part of the largest

eigenvalue) of F − V is less than, equal to, or greater than zero. This is equivalent

to R0 < 1 (= 1, > 1) which is in turn equivalent to ρ(M) < 1 (= 1, > 1) [9].

For fixed j ∈ Ω, assume that Nj(0) is sufficiently large, aj ∈ N is sufficiently

small, R0 > 1 (ρ(M) > 1), and M is irreducible. If in the ODE model there is a

significant increase in the number of infectives (an outbreak), then given

Sj(0) ≈ Nj(0) and Ij(0) = aj the probability of an outbreak is approximately

1− qa11 · · · qann ,

where qi = fi(q1, . . . , qn) and qi ∈ (0, 1).

Unfortunately, the size of Nj(0) and aj may depend on the particular parameter

values and obtaining analytical expressions for qi is not possible in most cases. For a

single isolated patch j ∈ Ω with Di = O, M = [mjj] = diag[2βj/(βj + γj + αj)]. It is

easy to verify that mjj > 1 if and only if R0j > 1 and that

qj = 1/R0j = (γj + αj)/βj. The latter result is due to Whittle [73]. To check the

general result about probability of extinction, we compute the proportion out of

10,000 sample paths where the infective population “hits” zero and compare it with

the estimate P0 calculated from the fixed point of the pgfs.

4.2.3 Stochastic Differential Equation Models

In this section, we derive Ito stochastic differential equation (SDE) multi-patch

SIS, SIR, and SIRS models corresponding to the deterministic models. It is first

necessary to define a Wiener process and Ito stochastic integration.

A Wiener process is a continuous stochastic process, W (t)|t ≥ 0, such that

1. W (0) = 0,

62


2. For 0 ≤ t1 ≤ t2 <∞, W (t2)−W (t1) ∼ N(0, t2 − t1),

3. For 0 ≤ t0 ≤ t1 ≤ t2 <∞, the increments W (t1)−W (t0) and W (t2)−W (t1)

are independent [1, 58].

Wiener processes are needed to define the Ito stochastic integral of a random

function. Let f(t,X(t)) be a random function on [a, b]× X(t)|t ≥ 0 such that∫ b

a

E(f 2(t))dt <∞

[1, 58]. If a = t0 < t1 < · · · < tm = b is a partition of [a, b], ∆t = (b− a)/m, and

∆W (ti) = W (ti+1)−W (ti), then the Ito stochastic integral of f is defined as

∫ b

a

f(t)dW (t) = limm→∞

m−1∑i=0

f(ti)∆W (ti),

where the convergence is in the mean square sense [1, 58]. It is now possible to

define an Ito SDE.

An Ito SDE on the interval [0, T ] has the form

dX(t) = f(t,X(t))dt+ g(t,X(t))dW (t), (4.22)

where W (t) is a Wiener process. The functions f and g are called the drift and

diffusion coefficients, respectively [1, 58]. The notation dW (t) in equation (4.22) is

slightly abusive since Wiener processes are nowhere differentiable. Equation (4.22)

is used for notational convenience for the solution X(t) of

X(t) = X(0) +

∫ t

0

f(s,X(s))ds+

∫ t

0

g(s,X(s))dW (s), (4.23)

where t ∈ [0, T ] [1, 58]. The first integral is a Riemann integral and the second

integral is an Ito stochastic integral [1, 58]. The notation X(t) is also for

convenience since X(t) denotes a sample path X(t) = X(t, ω), where ω is an

element of the sample space of X. For more information on the theory and

applications of SDEs, see [1, 4, 58].

63


Next, a procedure will be applied to construct an equivalent Ito SDE model from

a given deterministic model. This procedure, illustrated in [1, 2, 4], is often referred

to as the diffusion approximation method by Kurtz [50] or the chemical Langevin

equation by Gillespie [34]. Given the transitions and rates in Tables 4.1 and 4.2, the

corresponding SDE model has the form

d ~X(t) = ~f( ~X, t)dt+G( ~X, t)d ~W (t), (4.24)

where ~f( ~X, t) = E(∆ ~X(t)) is the drift vector, G( ~X, t) is a diffusion matrix which

satisfies GGT = Σ = E(∆ ~X(t)(∆ ~X(t))T ), and ~W (t) is a vector of independent

Wiener processes.

First consider a SDE two-patch SIS model. The drift vector has the same form as

the deterministic model

~f( ~X, t) =

−β1S1I1

N1

+ γ1I1 − ds12S1 + ds21S2

β1S1I1

N1

− γ1I1 − α1I1 − di12I1 + di21I2

−β2S2I2

N2

+ γ2I2 + ds12S1 − ds21S2

β2S2I2

N2

− γ2I2 − α2I2 + di12I1 − di21I2

,

and the diffusion matrix is given by

G =

[G11 O G13

O G22 G23

],

64


where

Gjj =

−

√βjSjIjNj

√γjIj 0√

βjSjIjNj

−√γjIj −

√αjIj

, j = 1, 2,

G13 =

[−√ds12S1

√ds21S2 0 0

0 0 −√di12I1

√di21I2

],

G23 =

[√ds12S1 −

√ds21S2 0 0

0 0√di12I1 −

√di21I2

].

The explicit form of the stochastic two-patch SIS model is

dS1 =

(−β1S1I1

N1

+ γ1I1 − ds12S1 + ds21S2

)dt−

√β1S1I1

N1

dW1

+√γ1I1dW2 −

√ds12S1dW7 +

√ds21S2dW8,

dI1 =

(β1S1I1

N1

− γ1I1 − α1I1 − di12I1 + di21I2

)dt+

√β1S1I1

N1

dW1

−√γ1I1dW2 −

√α1I1dW3 −

√di12I1dW9 +

√di21I2dW10,

dS2 =

(−β2S2I2

N2

+ γ2I2 + ds12S1 − ds21S2

)dt−

√β2S2I2

N2

dW4

+√γ2I2dW5 +

√ds12S1dW7 −

√ds21S2dW8,

dI2 =

(β2S2I2

N2

− γ2I2 − α2I2 + di12I1 − di21I2

)dt+

√β2S2I2

N2

dW4

−√γ2I2dW5 −

√α2I2dW6 +

√di12I1dW9 −

√di21I2dW10.

Two-patch SIR/SIRS models of the form (4.24) can be constructed in a similar

manner. For the SDE two-patch SIRS model, the drift vector is the same as the

right-hand side of the ODE model (4.14)-(4.16) and the diffusion matrix is

G =

[G11 O G13 G14 G15

O G22 G23 G24 G25

],

65


where

Gjj =

−

√βjSjIjNj

0√νjRj 0

−

√βjSjIjNj

−√γjIj 0 −

√α1I1

0√γjIj −

√νjRj 0

, j = 1, 2,

G13 =

−√ds12S1

√ds21S2

0 0

0 0

,

G23 =

√ds12S1 −

√ds21S2

0 0

0 0

.The 3× 2 submatrices G14, G24, G15, and G25 have a similar form. The stochastic

66


two-patch SIR/SIRS model has the following explicit form:

dS1 =

(−β1S1I1

N1

+ ν1R1 − ds12S1 + ds21S2

)dt−

√β1S1I1

N1

dW1

+√ν1R1dW3 −

√ds12S1dW9 +

√ds21S2dW10,

dI1 =

(β1S1I1

N1

− γ1I1 − α1I1 − di12I1 + di21I2

)dt+

√β1S1I1

N1

dW1

−√γ1I1dW2 −

√α1I1dW4 −

√di12I1dW11 +

√di21I2dW12,

dR1 = (γ1I1 − ν1R1 − dr12R1 + d221R2)dt+

√γ1I1dW2 −

√ν1R1dW3

−√dr12R1dW13 +

√dr21R2dW14,

dS2 =

(−β2S2I2

N2

+ ν2R2 + ds12S1 − ds21S2

)dt−

√β2S2I2

N2

dW5

+√ν2R2dW7 +

√ds12S1dW9 −

√ds21S2dW10,

dI2 =

(β2S2I2

N2

− γ2I2 − α2I2 + di12I1 − di21I2

)dt+

√β2S2I2

N2

dW5

−√γ2I2dW6 −

√α2I2dW8 +

√di12I1dW11 −

√di21I2dW12,

dR2 = (γ2I2 − ν2R2 + dr12R1 − dr21R2)dt+√γ2I2dW6 −

√ν2R2dW7

+√dr12R1dW13 −

√dr21R2dW14.

4.2.4 Numerical Examples

The dynamics of the ODE, CTMC, and SDE multi-patch models will be

illustrated in several numerical examples. The first set of examples considers

two-patch and three-patch SIS epidemic models without disease-related deaths of

the form (4.1)-(4.2). We apply branching process theory to derive the probability of

disease persistence and illustrate the significance of the location of the outbreak. In

the second set of examples we illustrate the difference in extinction behavior

between the deterministic and stochastic models with disease-related deaths. These

latter examples illustrate the effects of finite-time extinction. In a third and final

example, we illustrate a stochastic nine-patch model representative of the analogous

deterministic model of Allen et al. [5].

Consider a two-patch SIS epidemic without disease-related deaths. If

67


ds12 + ds21 > 0, then there exists a unique DFE for the ODE model given by

(S1, 0, S2, 0) =

(ds21N

ds12 + ds21

, 0,ds12N

ds12 + ds21

, 0

), (4.25)

where N = N(0). According to Theorem 4.2, the DFE (4.25) is globally

asymptotically stable if R0 < 1, where R0 is defined as in (4.13) [70]. The

next-generation matrix is

FV −1 =1

γ1γ2 + γ1di21 + γ2di12

[β1(γ2 + di21) β1d

i21

β2di12 β2(γ1 + di12)

]

and the basic reproduction number R0 equals

β1(γ2 + di21) + β2(γ1 + di12) +√

[β1(γ2 + di21)− β2(γ1 + di12)]2 + 4β1β2di12di21

2(γ1γ2 + γ1di21 + γ2di12).

For the corresponding CTMC two-patch model, the offspring pgfs for I1 and I2 are

given by (4.18) and the expectation matrix is given by (4.19). It is straightforward

to show that R0 < 1 iff ρ(M) < 1. If R0 > 1, then the probability of disease

persistence is given by equation (3.2). Unfortunately, if R0 > 1, an analytical

expression for the unique fixed point (q1, q2) ∈ (0, 1)2 of the offspring pgfs cannot be

calculated. The fixed point and probability of disease extinction are found

numerically.

Consider the parameter values β1 = 0.5, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, and

dskj = dikj = 0.1 for k, j = 1, 2 with a total population size of N = 400. For these

values, R01 = 5, R02 = 0.5, R0 = 2.83, and ρ(M) = 1.45. Patch 1 is a high-risk

patch and patch 2 is low-risk. According to Theorem 4.1, in the absense of dispersal

the disease persists in patch 1 and dies out in patch 2. However, when dispersal is

allowed the disease persists in both patches. For the deterministic model, there

exists a locally stable endemic equilibrium (S1, I1, S2, I2) ≈ (68, 132, 161, 39). This

result is consistent with the results of Allen and Kirupaharan [7]. In Table 4.3, the

probability of disease extinction P0 is calculated and compared to the approximation

obtained from the proportion of sample paths (out of 10,000) for which the sum

I1(t) + I2(t) hits zero (disease extinction) before time t = 150. The solution of the

68


ODE and one sample path for the CTMC model are plotted in Figure 4.1.

Table 4.3. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC two-patch SIS model. Parametervalues are β1 = 0.5, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, ds12 = ds21 = 0.1, and di12 = di21 = 0.1with initial conditions I1(0) = i1, S1(0) = 200− i1, I2(0) = i2, and S2(0) = 200− i2.

i1 i2 P0 Approx.

1 0 0.3410 0.34060 1 0.8055 0.80621 1 0.2747 0.27732 0 0.1163 0.11710 2 0.6489 0.6495

Figure 4.1. The ODE solution and one sample path for the CTMC two-patch SISmodel. Parameter values are β1 = 0.5, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, ds12 = ds21 = 0.1,and di12 = di21 = 0.1 with initial conditions S1(0) = 199, I1(0) = 1, S2(0) = 200, andI2(0) = 0. An outbreak occurs with probability 1 − P0 = 0.6590. The locally stableendemic equilibrium is (S1, I1, S2, I2) ≈ (68, 132, 161, 39).

Consider a three-patch SIS epidemic without disease-related deaths. Suppose that

the patches are arranged in a strip so there is no direct dispersal between patches

one and three. Consider the parameter values β1 = 0.5, γ1 = 0.1, β2 = 0.2, γ2 = 0.4,

69


β3 = 0.1, γ3 = 0.4, and dskj = dikj = 0.1 for k 6= 3 and j 6= 3 with a total population

size of N = 450. For these values, R01 = 5, R02 = 0.5, R03 = 0.25, R0 = 2.77, and

ρ(M) = 1.43. Patch 1 is a high-risk patch and patches 2 and 3 are low-risk. For the

deterministic model, there exists a locally stable endemic equilibrium,

(S1, I1, S2, I2, S3, I3) ≈ (53, 97, 126, 24, 144, 6). In Table 4.4, the probability of

disease extinction P0 is calculated and compared to the approximation obtained

from the proportion of sample paths (out of 10,000) for which the sum

I1(t) + I2(t) + I3(t) hits zero before time t = 150. The solution of the ODE and one

sample path for the CTMC model are plotted in Figure 4.2.

Table 4.4. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC three-patch SIS model. Parametervalues are β1 = 0.5, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, β3 = 0.1, γ3 = 0.4, dskj = dikj = 0.1for k 6= 3 and j 6= 3, and ds13 = ds31 = di13 = di31 = 0 with initial conditions I1(0) = i1,S1(0) = 150− i1, I2(0) = i2, S2(0) = 150− i2, I3(0) = i3, and S3(0) = 150− i3.

i1 i2 i3 P0 Approx.

1 0 0 0.3509 0.35890 1 0 0.8407 0.84220 0 1 0.9613 0.96311 1 1 0.2836 0.2901

The results in Tables 4.3 and 4.4 illustrate that the location of an outbreak plays

an important role in the persistence of a disease. The probability of disease

extinction is significantly higher if an outbreak occurs in a low-risk patch.

The next set of examples illustrates differences between the extinction dynamics

of deterministic and stochastic multi-patch models with disease-related mortality. In

particular, these examples show that there is finite-time extinction for the stochastic

(CTMC and SDE) models, and for the deterministic models the infectious

population asymptotically approaches zero without ever hitting zero. The difference

in extinction behavior between deterministic and stochastic models is well-known.

When population sizes are small in deterministic models, they can rebound but this

is not the case in stochastic models because extinction occurs for small population

sizes. In 1991, Mollison [56] coined the term “atto-fox” to denote the fraction of one

fox (10−18 of a fox), an unrealistically small fraction reached by the predator

70


Figure 4.2. The ODE solution and one sample path for the CTMC three-patch SISmodel. Parameter values are β1 = 0.5, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, β3 = 0.1,γ3 = 0.4, dskj = dikj = 0.1 for k 6= 3 and j 6= 3, and ds13 = ds31 = di13 = di31 = 0 withinitial conditions S1(0) = 149, I1(0) = 1, S2(0) = 150, I2(0) = 0, S3(0) = 150, andI3(0) = 0. An outbreak occurs with probability 1 − P0 = 0.6491. The locally stableendemic equilibrium is (S1, I1, S2, I2, S3, I3) ≈ (53, 97, 126, 24, 144, 6).

population in a deterministic predator-prey system.

Consider a two-patch SIS epidemic with a relatively high rate of disease-related

mortality compared to the rate of recovery. Consider the parameter values β1 = 0.8,

γ1 = 0.1, β2 = 0.3, γ2 = 0.4, α1 = α2 = 0.5, and dskj = dikj = 0.1 for k, j = 1, 2 with

an initial population size of N(0) = 400. For these values, R01 = 4/3, R02 = 1/3,

R0 = 1.17, and ρ(M) = 1.075. There are significant differences between the

dynamics of the deterministic and stochastic models. For the ODE model, the total

population size in both patches asymptotically approaches zero. For the CTMC

model, the number of infectious individuals hits zero in a finite amount of time. In

the absence of infectious individuals, the number of susceptible individuals remains

constant and is not driven to zero. The solution of the ODE and one sample path

for the CTMC model are plotted in Figure 4.3. Using the same parameter values

and initial conditions, a similar result holds for the SDE two-patch SIS model. The

results are plotted in Figure 4.4.

71


Figure 4.3. The ODE solution and one sample path for the CTMC two-patch SISmodel. Parameter values are β1 = 0.8, γ1 = 0.1, β2 = 0.3, γ2 = 0.4, α1 = α2 = 0.5,ds12 = ds21 = 0.1, and di12 = di21 = 0.1 with initial conditions S1(0) = 195, I1(0) = 5,S2(0) = 185, and I2(0) = 15.

Figure 4.4. The ODE and SDE solutions for the two-patch SIS model. Parametervalues are β1 = 0.8, γ1 = 0.1, β2 = 0.3, γ2 = 0.4, α1 = α2 = 0.5, ds12 = ds21 = 0.1,and di12 = di21 = 0.1 with initial conditions S1(0) = 195, I1(0) = 5, S2(0) = 185, andI2(0) = 15.

72


Next consider a two-patch SIRS model with disease-related deaths. Consider the

parameter values β1 = 0.8, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, ν1 = ν2 = 0.1,

α1 = α2 = 0.4, and dskj = dikj = drkj = 0.1 for k, j = 1, 2 with an initial population

size of N(0) = 400. For these values, R01 = 1.6, R02 = 0.25, R0 = 1.36, and

ρ(M) = 1.15. Again, the total population asymptotically approaches zero for the

deterministic model. For the CTMC model, the number of infectious individuals

hits zero in a finite amount of time. In the absense of infectious individuals,

susceptible individuals remain susceptible and recovered individuals lose their

immunity driving the number of recovered individuals in each patch to zero. The

solution of the ODE and one sample path for the CTMC model are plotted in

Figure 4.5. A similar result holds for the SDE model, graphed in Figure 4.6.

Figure 4.5. Solution of the ODE and one sample path for the CTMC two-patch SIRSmodel. Parameter values are β1 = 0.8, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, ν1 = ν2 = 0.1,α1 = α2 = 0.4, and dskj = dikj = drkj = 0.1 for k, j = 1, 2 with initial conditionsS1(0) = 195, I1(0) = 5 R1(0) = 0, S2(0) = 185, I2(0) = 15, and R2(0) = 0.

In two final examples, ODE and SDE nine-patch SIS models without

disease-related mortality are simulated and compared. Consider a nine-patch SIS

model where the patches are arranged in a 3× 3 grid and dispersal only occurs

between two patches if they share an adjacent edge. The patches are numbered

consecutively from left to right and top to bottom with the patch in the (1, 1)

73


Figure 4.6. The ODE and SDE solutions for the two-patch SIRS model. Parametervalues are β1 = 0.8, γ1 = 0.1, β2 = 0.2, γ2 = 0.4, ν1 = ν2 = 0.1, α1 = α2 = 0.4, anddskj = dikj = drkj = 0.1 for k, j = 1, 2 with initial conditions S1(0) = 195, I1(0) = 5R1(0) = 0, S2(0) = 185, I2(0) = 15, and R2(0) = 0.

position being patch 1 and the patch in the (3, 3) position being patch 9. The

deterministic model is the same as the nine-patch SIS model discussed by Allen et al.

[5]. The focus of this investigation will be on the corresponding stochastic models.

The dispersal rates for susceptible individuals are small and the infectious dispersal

rates are much larger. For j = 2, 4, 6, 8, let βj = 0.3 and γj = 0.2. For j = 1, 3, 7, 9,

let βj = 0.2 and γj = 0.4. For the center patch, j = 5, let β5 = 0.6 and γ5 = 0.3. If

two patches j and k share an edge, then dskj = dsjk = 0.01 and dikj = dijk = 0.1.

Assume that the initial population size in each patch is 100, Sj(0) + Ij(0) = 100 for

j = 1, . . . , 9, so the total population size is N = 900. For these values, the patch

reproduction numbers are R0j = 1.5 for j = 2, 4, 6, 8, R0j = 0.5 for j = 1, 3, 7, 9, and

R05 = 2. The basic reproduction number is R0 = 1.27. For the ODE and SDE

nine-patch models, the disease persists in all patches. In the high-risk patches, there

is a greater number of infectives and a reduced number of susceptibles. Susceptible

individuals move to the low risk patches [5]. The solution of the ODE model and a

sample path of the SDE model are plotted in Figure 4.7. In the branching process

approximation of the CTMC model, with a small number of infectious individuals

74


there is a positive probability of disease extinction without an outbreak in any

patch, provided ρ(M) > 1. This probability can be computed from the fixed point of

the offspring pgfs. In this example, we did not compute this probability.

Figure 4.7. Solutions of the ODE and SDE nine-patch SIS models. Parameter valuesare βj = 0.3, γj = 0.2 for j = 2, 4, 6, 8, βj = 0.2, γj = 0.4, for j = 1, 3, 7, 9, β5 = 0.6,and γ5 = 0.3 with initial conditions S5(0) = 95, I5(0) = 5, and Sj(0) = 100, Ij(0) = 0for j 6= 5.

Now consider a nine-patch SIS model with parameter values βj = 0.1, γj = 0.4 for

j = 1, 2, 3, 4, 7, βj = 0.1, γj = 0.2 for j = 5, 6, 8, and β9 = 0.5 and γ9 = 0.1. If two

patches j and k share an edge, then dsjk = dskj = 0.01, and dijk = dikj = 0.1. Again,

suppose that Sj(0) + Ij(0) = 100 for j = 1, . . . , 9 so the total population size is

N = 900. For these values, the patch reproduction numbers are R0j = 0.25 for

j = 1, 2, 3, 4, 7, R0j = 0.5 for j = 5, 6, 8, and R09 = 5. The basic reproduction

number is R0 = 2.01. Suppose at outbreak occurs in patch 9, a high-risk patch.

Choose the initial conditions I9(0) = 1, S9(0) = 99, and for all j 6= 9, Sj(0) = 100

and Ij(0) = 0. For the ODE and SDE nine-patch models, the results are similar.

The disease persists in patch 9, the high-risk patch, but there is a significant

reduction in the total population size. There is very little infection in the low-risk

patches. See Figure 4.8. Although it is not clear in the SDE model, in the

branching process approximation of the CTMC model, with ρ(M) > 1, there is a

75


positive probability of disease extinction.

Figure 4.8. Solutions of the ODE and SDE nine-patch SIS models. Parameter valuesare βj = 0.1, γj = 0.4 for j = 1, 2, 3, 4, 7, βj = 0.1, γj = 0.2 for j = 5, 6, 8, andβ9 = 0.5 and γ9 = 0.1 with initial conditions S9(0) = 99, I9(0) = 1, and Sj(0) = 100,Ij(0) = 0 for j 6= 9.

4.3 Multi-Patch SIS, SIR, and SIRS Models with Demographics

In this section multi-patch SIS, SIR, and SIRS models with demographics will be

considered. The development and analysis of these models is similar to the analysis

of the multi-patch models without demographics. However, there are some

significant differences in that with immigration and births, disease-related deaths

cannot drive the total population to extinction.

4.3.1 Deterministic Models

Consider a multi-patch SIS model with demographics, where n ≥ 2 is the number

of patches and Ω = 1, 2, . . . , n. The notation is the same as for the multi-patch

76


SIS model without demographics (4.1)-(4.2). The model takes the form

Sj = Λj − µjSj −βjSjIjNj

+ γjIj +∑k∈Ω


Ij =βjSjIjNj

− (γj + αj + µj)Ij +∑k∈Ω


with Sj(0) > 0 and Ij(0) ≥ 0 for all j ∈ Ω. The parameters Λj > 0 and µj > 0

represent the immigration rate and natural death rate in patch j, respectively.

Summing the equations for Sj and Ij in (4.26)-(4.27), it follows that for each

j ∈ Ω

Nj = Λj − µjNj − αjIj +∑k∈Ω

(dskjSk − dsjkSj + dikjIk − dijkIj). (4.28)

Let N denote the total population size in all patches,

N =∑j∈Ω

Nj. (4.29)

It follows from equation (4.28) that

N =∑j∈Ω

(Λj − µjNj − αjIj) >∑j∈Ω

Λj −∑j∈Ω

(µj + αj)Nj ≥∑j∈Ω

Λj − mjN, (4.30)

where mj = maxj∈Ω(µj + αj). Thus, N is always positive, it is impossible to drive

the total population to zero.

For each patch j ∈ Ω, define the patch reproduction number as

R0j =βj

γj + µj + αj. (4.31)

The following theorem summarizes the dynamics in each patch when there is no

movement or disease-related death.

Theorem 4.4. Let Ds = Di = O (zero matrices) and suppose αj = 0. Then

(i) If R0j ≤ 1, then

limt→∞

(Sj(t), Ij(t)) =

(Λj

µj, 0

).

77


(ii) If R0j > 1, then

limt→∞

(Sj(t), Ij(t)) =

(Nj

R0j

, Nj −Nj

R0j

)where Aj +Bj = Λj/µj.

Proof. For fixed j ∈ Ω, let ij = Ij/Nj. For simplicity, omit the subscript j and write

i = i/N and R0 = β/(γ + µ). Then

N = Λ− µN, (4.32)

and

i = i

(β − γ − Λ

N− βi

). (4.33)

The solution of (4.32) is

N(t) =Λ

µ+

(N(0)− Λ

µ

)e−µt.

Since µ > 0,

limt→∞

N(t) =Λ

µ.

Thus, for all 0 < ε < µ there exists T sufficiently large so that for t ≥ T

Λ

µ(1 + ε/µ)≤ N(t) ≤ Λ

µ(1− ε/µ).

Then equation (4.33) is bounded for t ≥ T :

i(β − γ − µ− ε− βi) ≤ i ≤ i(β − γ − µ+ ε− βi).

If R0 ≤ 1, then β − γ − µ ≤ 0. Thus,

i ≤ i(ε− βi).

78


Since ε > 0, by the comparison principle, the limit i = limt→∞ i(t) is bounded:

0 ≤ i ≤ ε

β.

Since ε > 0 is arbitrary, i(t)→ 0. Thus,

limt→∞

(S(t), I(t)) =

(Λ

µ, 0

).

If R0 > 1, then β − γ − µ > 0 and β > 0. By the comparison principle, if R0 > 1

and ε is sufficiently small, then the limit i = limt→∞ i(t) is bounded:

β − γ − µ− εβ

≤ i ≤ β − γ − µ+ ε

β.

Since ε is arbitrary, i(t)→ 1− 1/R0. Thus,

limt→∞

(S(t), I(t)) =

(N

R0

, N − N

R0

).

Next, a basic reproduction number is defined for the patch model (4.26)-(4.27).

Let

F = diag[βj], (4.34)

V = diag

[γj + αj + µj +

∑k∈Ω

dijk

]−Di, (4.35)

where Di is the dispersal matrix for infectives. The basic reproduction number is

the spectral radius of FV −1 [70],

R0 = ρ(FV −1). (4.36)

The next theorem states that if R0 < 1, then the infection dies out. The proof is

similar to the proofs of Theorem 4.2 and Corollary 4.1.

Theorem 4.5. (i) If R0 < 1, then the multi-patch SIS epidemic model with

79


demographics (4.26)-(4.27) satisfies I(t)→ 0 as t→∞.

(ii) For fixed j ∈ Ω, if R0j < 1 and dijk = dikj = 0 for k 6= j, then in model

(4.26)-(4.27), Ij(t)→ 0 as t→∞.

Consider a multi-patch SIR or SIRS model with demographics, where n ≥ 2 is the

number of patches and Ω = 1, 2, . . . , n. The notation is the same as for the

multi-patch SIR or SIRS model without demographics. The model takes the form

Sj = Λj − µjSj −βjSjIjNj

+ νjRj +∑k∈Ω


Ij =βjSjIjNj

− (γj + αj + µj)Ij +∑k∈Ω


Rj = γjIj − µjRj − νjRj +∑k∈Ω

(drkjRk − drjkRj), (4.39)

where Sj(0) > 0, Ij(0) ≥ 0, and Rj(0) = 0 for each j ∈ Ω. The patch reproduction

numbers remain the same as for the SIS model with demographics,

R0j =βj

γj + µj + αj.

In addition, Theorem 4.5 applies which is stated here as a corollary.

Corollary 4.3. (i) If R0 < 1, then the multi-patch SIR or SIRS epidemic model

with demographics (4.37)–(4.39) satisfies I(t)→ 0 as t→∞.

(ii) For fixed j ∈ Ω, if R0j < 1 and dijk = dikj = 0 for k 6= j, then in model

(4.37)–(4.39), Ij(t)→ 0 as t→∞.

The next theorem summarizes the dynamics of the SIR and SIRS models with

demographics in each patch when there is no movement or disease-related death.

The following results will be used in the proof of Theorem 4.6. Consider the

following systems:

x = f(t, x), (4.40)

y = g(y), (4.41)

80


where f and g are continuous and locally Lipschitz in x in Rn and solutions exist for

all positive time. The equation (4.40) is said to be asymptotically autonomous with

limit equation (4.41) if f(t, x)→ g(x) as t→∞ uniformly for x in Rn [37].

Corollary 4.4. ([37], p. 851) If solutions of system (4.40) are bounded and the

equilibrium e of the limit system (4.41) is globally asymptotically stable, then any

solution x(t) of system (4.40) satisfies x(t)→ e as t→∞.

Theorem 4.6. Let Ds = Di = Dr = O (zero matrices) and suppose αj = 0. Then

(i) If R0j ≤ 1, then

limt→∞

(Sj(t), Ij(t), Rj(t)) =

(Λ

µ, 0, 0

).

(ii) If R0j > 1, then

limt→∞

(Sj(t), Ij(t), Rj(t)) = (S, I , R),

where S, I , R > 0 and S + I + R = Λ/µ.

Proof. For fixed j ∈ Ω, let sj = Sj/Nj, ij = Ij/Nj, and rj = Rj/Nj. For simplicity,

omit the subscript j and write s = S/N , i = I/N , r = R/N , and R0 = β/(γ + α).

Then s+ i+ r = 1 and r = 1− s− i. So the dynamics depend on s and i. First,

N = Λ− µN. (4.42)

It follows that

limt→∞

N(t) =Λ

µ.

The differential equations for s and i are:

s =Λ

N(1− s)− βsi+ ν(1− s− i),

i = i

(βs− γ − Λ

N

).

(4.43)

Since N(t)→ Λ/µ as t→∞, system (4.43) is asymptotically autonomous with the

limit system

s = µ(1− s)− βsi+ ν(1− s− i),i = i(βs− γ − µ).

(4.44)

81


By Corollary 4.4, the limiting equilibrium of (4.43) is the same as the limiting

equilibrium of (4.44). Applying Dulac’s criterion [3] to (4.44) with a Dulac function

D(s, i) = 1/i, shows that there are no periodic solutions in the interior of R2+.

Indeed, let

f(s, i) =s

i=

µ(1− s) + ν(1− s− i)i

− βs,

g(s, i) =i

i= βs− γ − µ.

It follows that∂f

∂s+∂g

∂i= −µ+ ν

i− β.

This expression is negative in the interior of R2+. The i-nullclines of (4.44) are i = 0

and s = 1/R0, and the s-nullcline is

i =(µ+ ν)(1− s)

βs+ ν.

If R0 > 1, then there exists a globally asymptotically stable endemic equilibrium:

s =1

R0

,

i =(µ+ ν)(R0 − 1)

β +R0ν,

r =(β −R0)(R0 − 1)

R0(β +R0ν).

If R0 ≤ 1, then the only equilibrium is (1, 0, 0). By way of contradiction, suppose

that solutions do not approach this equilibrium. That is, the ω-limit set of (4.44)

does not contain (1, 0, 0). By the Poincare-Bendixon criterion [46], the positive orbit

of (4.44) is periodic or the ω-limit set is periodic, a contradiction since there are no

periodic solutions in R2+. Thus, (1, 0, 0) is a stable equilibrium.

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4.3.2 Markov Chain Models

Consider a corresponding CTMC multi-patch SIS, SIR, or SIRS model with

demographics. Let

~X(t) = (S1(t), I1(t), . . . , Sn(t), In(t))T ,

~X(t) = (S1(t), I1(t), R1(t), . . . , Sn(t), In(t), Rn(t))T

be discrete random vectors for the SIS and SIR/SIRS models respectively. For

simplicity, the same notation is used for the deterministic variables and stochastic

random variables. The transitions and corresponding rates for ~X(t) are listed in

Tables 4.5 and 4.6.

Table 4.5. State transitions and rates for the CTMC multi-patch SIS epidemic modelwith demographics.


Immigration in patch j (0, . . . , Sj + 1, . . . , 0) Λj

Death of Sj (0, . . . , Sj − 1, . . . , 0) µjSjInfection in patch j (0, . . . , Sj − 1, Ij + 1, . . . , 0) βjSjIj/Nj

Recovery in patch j (0, . . . , Sj + 1, Ij − 1, . . . , 0) γjIjDeath of Ij (0, . . . , Ij − 1, . . . , 0) (αj + µj)IjSusceptible dispersal from j to k (0, . . . , Sj − 1, . . . , Sk + 1, . . . , 0) dsjkSjInfectious dispersal from j to k (0, . . . , Ij − 1, . . . , Ik + 1, . . . , 0) dijkIj

Let Sj(0) ≈ Nj(0), Ij(0) = 1, and Ik(0) = 0 for all k 6= j. The offspring pgf for Ij

is similar to the branching process without demographics. That is,

fj(x1, . . . , xn) =

βjx2j + γj + αj + µj +

∑k∈Ω\j

dijkxk

βj + γj + αj + µj +∑

k∈Ω\j

dijk. (4.45)

The offspring pgfs for Ij are the same for the SIS, SIR, and SIRS models.

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Table 4.6. State transitions and rates for the CTMC multi-patch SIR/SIRS epidemicmodel with demographics.


Immigration in patch j (0, . . . , Sj + 1, . . . , 0) Λj

Death of Sj (0, . . . , Sj − 1, . . . , 0) µjInfection in patch j (0, . . . , Sj − 1, Ij + 1, . . . , 0) βjSjIj/Nj

Recovery in patch j (0, . . . , Ij − 1, Rj + 1, . . . , 0) γjIjDeath of Ij (0, . . . , Ij − 1, . . . , 0) (αj + µj)IjDeath of Rj (0, . . . , Rj − 1, 0, . . . , 0) µjRj

Loss of immunity in patch j (0, . . . , Sj + 1, Ij, Rj − 1, . . . , 0) νjRj

Susceptible dispersal from j to k (0, . . . , Sj − 1, . . . , Sk + 1, . . . , 0) dsjkSjInfectious dispersal from j to k (0, . . . , Ij − 1, . . . , Ik + 1, . . . , 0) dijkIjRecovered dispersal from j to k (0, . . . , Rj − 1, . . . , Rk + 1, . . . , 0) drjkRj

The expectation matrix M = [mkj] is a nonnegative n× n matrix such that

mkj =

2βj


k∈Ω\j

dijk, k = j

dijk


k∈Ω\j

dijk, k 6= j.

(4.46)

4.3.3 Stochastic Differential Equation Models

Consider the corresponding SDE multi-patch SIS, SIR, and SIRS models with

demographics. Given the transitions and rates in Tables 4.5 and 4.6, the

corresponding SDE models have the form (4.24). The explicit form of the stochastic

84


two-patch SIS model is

dS1 =

(Λ1 − µ1S1 −

β1S1I1

N1

+ γ1I1 − ds12S1 + ds21S2

)dt+

√Λ1dW1

−√µ1S1dW2 −

√β1S1I1

N1

dW5 +√γ1I1dW6 −

√ds12S1dW13 +

√ds21S2dW14,

dI1 =

(β1S1I1

N1

− γ1I1 − α1I1 − µ1I1 − di12I1 + di21I2

)dt+

√β1S1I1

N1

dW5

−√γ1I1dW6 −

√α1I1dW9 −

√µ1I1dW10 −

√di12I1dW15 +

√di21I2dW16,

dS2 =

(Λ2 − µ2S2 −

β2S2I2

N2

+ γ2I2 + ds12S1 − ds21S2

)dt+

√Λ2dW3

−√µ2S2dW4 −

√β2S2I2

N2

dW7 +√γ2I2dW8 +

√ds12S1dW13 −

√ds21S2dW14,

dI2 =

(β2S2I2

N2

− γ2I2 − α2I2 − µ2I2 + di12I1 − di21I2

)dt+

√β2S2I2

N2

dW7

−√γ2I2dW8 −

√α2I2dW11 −

√µ2I2dW12 +

√di12I1dW15 −

√di21I2dW16.

Stochastic two-patch SIR/SIRS models with demographics of the form (4.24) can be

constructed in a similar manner.

4.3.4 Numerical Examples

Consider a two-patch SIS epidemic with demographics and disease-related deaths.

The unique DFE for the ODE model given by

(S1, 0, S2, 0) =

(Λ1(µ2 + ds21) + Λ2d

s21

µ1µ2 + µ1ds21 + µ2ds12

, 0,Λ1d

s12 + Λ2(µ1 + ds12)

µ1µ2 + µ1ds21 + µ2ds12

, 0

). (4.47)

According to Theorem 4.5, this DFE is globally asymptotically stable if R0 < 1,

where R0 is given by (4.36) [70]. For this example, the next-generation matrix is

FV −1 =1

A1A2 − di12di21

[β1A2 β1d

i21

β2di12 β2A1

],

85


where A1 = γ1 + α1 + µ1 + di12 and A2 = γ2 + α2 + µ2 + di21 and the basic

reproduction number R0 equals

β1A2 + β2A1 +√

(β1A2 − β2A1)2 + 4β1β2di12di21

2(A1A2 − di12di21)

.

Consider the following parameter values for the two-patch SIS model with

demographics: Λ2 = Λ2 = 2, µ1 = µ2 = 0.01, β1 = 0.5, γ1 = 0.2, β2 = 0.2, γ2 = 0.5,

and dskj = dikj = 0.1 for k, j = 1, 2. For these values, R01 = 2.38, R02 = 0.39,

R0 = 1.73, and ρ(M) = 1.25. Patch 1 is a high-risk patch and patch 2 is low-risk. In

Table 4.7, the probability of disease extinction P0 is calculated and compared to the

approximation obtained from the proportion of sample paths (out of 10,000) for

which the sum I1(t) + I2(t) hits zero (disease extinction) before time t = 150. The

solution of the ODE and one sample path for the CTMC model are plotted in

Figure 4.9. Using the same parameter values, the solution of the ODE and SDE

models are plotted in Figure 4.10.

i1 i2 P0 Approx.

1 0 0.5735 0.57480 1 0.9008 0.90301 1 0.5166 0.51862 0 0.3289 0.32150 2 0.8114 0.8119

Table 4.7. Probability of disease extinction P0 and numerical approximation (Ap-prox.) based on 10,000 sample paths of the CTMC two-patch SIS model with demo-graphics. Parameter values are Λ1 = Λ2 = 2, µ1 = µ2 = 0.01, β1 = 0.5, γ1 = 0.2,β2 = 0.2, γ2 = 0.5, and dskj = dikj = 0.1 for k, j = 1, 2 with initial conditions I1(0) = i1,S1(0) = 200− i1, I2(0) = i2, and S2(0) = 200− i2.

4.4 Discussion

Multi-patch epidemic models with and without demographics have been derived.

For the deterministic models, in the absense of dispersal, the dynamics in each

patch depend on the patch reproduction number. In addition, a basic reproduction

number was defined for the system. If R0 < 1, the disease is driven to extinction.

86


Figure 4.9. The ODE solution and one sample path for the CTMC two-patch SISmodel with demographics. Parameter values are Λ1 = Λ2 = 2, µ1 = µ2 = 0.01,β1 = 0.5, γ1 = 0.2, β2 = 0.2, γ2 = 0.5, and dskj = dikj = 0.1 for all k, j = 1, 2 withinitial conditions S1(0) = 199, I1(0) = 1, S2(0) = 200, and I2(0) = 0. An outbreakoccurs with probability 1− P0 = 0.4265.

Figure 4.10. Solution of the ODE and SDE two-patch SIS models with demographics.Parameter values are Λ1 = Λ2 = 2, µ1 = µ2 = 0.01, β1 = 0.5, γ1 = 0.2, β2 = 0.2,γ2 = 0.5, and dskj = dikj = 0.1 for k, j = 1, 2 with initial conditions S1(0) = 195,I1(0) = 5, S2(0) = 195, and I2(0) = 5.

87


However, if R0 > 1, the disease may or may not persist depending on the dispersal

rates. Furthermore, stochastic multi-patch epidemic models were constructed and

branching process theory was used to determine the probability of disease

persistence or extinction. The relationship between the basic reproduction number,

R0, and the spectral radius of the expectation matrix, ρ(M), with respect to the

threshold value one has been illustrated in Chapter 3 for the case of two patches.

Numerical results show the effects of dispersal and the location of an outbreak on

the probability of disease persistence or extinction. In addition, the numerical

results illustrate a difference between the deterministic and stochastic multi-patch

models with regard to finite-time extinction.

88


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