# Impact of heterogeneity on the dynamics of an SEIR epidemic model.

**ABSTRACT** An SEIR epidemic model with an arbitrarily distributed exposed stage is revisited to study the impact of heterogeneity on the spread of infectious diseases. The heterogeneity may come from age or behavior and disease stages, resulting in multi-group and multi-stage models, respectively. For each model, Lyapunov functionals are used to show that the basic reproduction number R0 gives a sharp threshold. If R0 ≤ 1, then the disease-free equilibrium is globally asymptotically stable and the disease dies out from all groups or stages. If R0 > 1, then the disease persists in all groups or stages, and the endemic equilibrium is globally asymptotically stable.

**0**Bookmarks

**·**

**89**Views

- Citations (0)
- Cited In (1)

- Ricerche di Matematica 01/2013; 62:161-181.

Page 1

MATHEMATICAL BIOSCIENCES

AND ENGINEERING

Volume 9, Number 2, April 2012

doi:10.3934/mbe.2012.9.393

pp. 393–411

IMPACT OF HETEROGENEITY ON THE DYNAMICS OF

AN SEIR EPIDEMIC MODEL

Zhisheng Shuai and P. van den Driessche

Department of Mathematics and Statistics, University of Victoria

Victoria, B.C., V8W 3R4, Canada

(Communicated by Jia Li)

Abstract. An SEIR epidemic model with an arbitrarily distributed exposed

stage is revisited to study the impact of heterogeneity on the spread of infec-

tious diseases. The heterogeneity may come from age or behavior and disease

stages, resulting in multi-group and multi-stage models, respectively. For each

model, Lyapunov functionals are used to show that the basic reproduction

number R0 gives a sharp threshold. If R0 ≤ 1, then the disease-free equilib-

rium is globally asymptotically stable and the disease dies out from all groups

or stages. If R0> 1, then the disease persists in all groups or stages, and the

endemic equilibrium is globally asymptotically stable.

1. Introduction. Heterogeneity exists in many aspects of disease transmission

processes [1, 2, 12, 33], for example, heterogeneous spatial distribution of host

populations, heterogeneous susceptibility among age groups, heterogeneous social

behavior among groups for sexually transmitted diseases, and multi-hosts for many

diseases such as West Nile virus and Avian influenza. Different types of hetero-

geneous epidemic models and attempts to understand the impact of heterogene-

ity on the disease transmission have appeared in the literature; see, for example,

[19, 22, 33, 37]. It has been found that heterogeneity sometimes does not alter the

dynamical structure of epidemic models, while sometimes it produces more com-

plicated dynamical behavior than found in homogeneous models. For example, if

the basic reproduction number R0completely determines the dynamics of homo-

geneous epidemic models in which the disease eventually either dies out or persists

at a positive level, then the associated heterogeneous models might have the same

dichotomy: there exists either a globally attracting disease-free equilibrium or a

globally attracting endemic equilibrium. On the other hand, if the original homo-

geneous model has rich dynamical behavior such as oscillations, then the associated

heterogeneous models have much richer dynamics such as a switch of oscillations

from one group to the other [41].

In Section 2, we formulate a basic susceptible-exposed-infectious-removed (SEIR)

epidemic model in which an arbitrarily distributed exposed stage is assumed. Us-

ing this basic model as a building block, we consider two different heterogeneous

models: multi-group models for heterogeneity in a host population (Section 3) and

multi-stage models for heterogeneity of infectious individuals (Section 4). For both

2000 Mathematics Subject Classification. Primary: 92D30; Secondary: 34K20.

Key words and phrases. SEIR model, heterogeneity, multi-group model, multi-stage model,

global stability, Lyapunov functional.

393

Page 2

394ZHISHENG SHUAI AND P. VAN DEN DRIESSCHE

models, we establish their global dynamics and demonstrate that the heterogene-

ity does not alter the dynamical structure of the SEIR model with an arbitrarily

distributed exposed stage.

Our proof for the global stability of endemic equilibria utilizes global Lyapunov

functionals that are motivated by the work in [20, 21, 34, 35] and the graph-theoretic

approach for large-scale systems developed in [10, 11, 29]. This graph-theoretic

approach, based on Kirchhoff’s Matrix Tree Theorem, has previously only been

applied to one study of delay epidemic models, namely, the multi-group model

in [30]. We demonstrate that, for the two types of heterogeneous delay epidemic

models considered, this graph-theoretic approach can be successfully applied by

choosing an appropriate weight matrix.

2. Basic model. Since our main goal is to investigate the impact of heterogeneity

on dynamics of epidemic models, we start with a relatively simple model and then

build on this to formulate and analyze different heterogeneous models. We now

formulate a simple epidemic model for a homogeneous population that includes an

arbitrarily distributed exposed stage, while two different heterogeneous models will

be formulated in Sections 3 and 4. Let S(t),E(t),I(t), and R(t) be the numbers

of individuals in the susceptible, exposed, infectious, and removed compartments,

respectively, with the total population N(t) = S(t) + E(t) + I(t) + R(t). Suppose

that A > 0 represents the constant recruitment, m > 0 represents the natural

mortality rate, and α ≥ 0 represents the mortality rate due to the disease. The rate

of change of N(t) is

N?(t) = A − mN(t) − αI(t).

Assuming mass action for the disease transmission and letting β > 0 denote the

effective contact rate, the rate of change of S(t) is

(2.1)

S?(t) = A − βS(t)I(t) − mS(t). (2.2)

Let P(t) denote the fraction of exposed individuals remaining in the exposed class

t units after entering the exposed class. Throughout we assume the following prop-

erties of P(t), which are biologically reasonable.

(H) P(t) is nonincreasing, piecewise continuous with possibly finitely many jumps,

satisfies P(0) = 1 and limt→∞P(t) = 0, and the mean latent period ω =

?∞

The number of exposed individuals can be expressed by the integral

?t

where E(0) is the number exposed at t = 0. Differentiating (2.3) gives

?t

Here the integral is in the Riemann-Stieltjes sense, and dtP(t−u) =dP(t−u)

ever the derivative exists. It can be verified that E(t) given in (2.3) is the unique

solution of (2.4) with the initial condition E(0). Assuming that the recovery rate

is γ, γ ≥ 0, the rate of change of I(t) is

?t

0P(u)du is finite.

E(t) = E(0)e−mt+

0

βS(u)I(u)e−m(t−u)P(t − u) du, (2.3)

E?(t) = βS(t)I(t) − mE(t) +

0

βS(u)I(u)e−m(t−u)dtP(t − u) du.(2.4)

dt

when-

I?(t) = −

0

βS(u)I(u)e−m(t−u)dtP(t − u) du − (m + α + γ)I(t). (2.5)

Page 3

IMPACT OF HETEROGENEITY ON DYNAMICS OF AN EPIDEMIC MODEL 395

If γ > 0, then 1/γ is the average infectious period; whereas if γ = 0, then there is

no removed compartment. Substituting (2.1), (2.2), (2.4), and (2.5) into R?(t) =

N?(t) − S?(t) − E?(t) − I?(t) leads to

R?(t) = γI(t) − mR(t).

Therefore, the epidemic model can be written as the system

S?(t)=A − βS(t)I(t) − mS(t),

E?(t)= βS(t)I(t) − mE(t) +

?t

γI(t) − mR(t),

?t

0

βS(u)I(u)e−m(t−u)dtP(t − u) du,

I?(t)=

−

0

βS(u)I(u)e−m(t−u)dtP(t − u) du − (m + α + γ)I(t),

R?(t)=

(2.6)

with nonnegative initial conditions. Model (2.6) includes as special cases several

earlier models, such as the standard SEIR ordinary differential equation (ODE)

model [21, 24] and the SEIR model with a discrete delay [17, 39]. Epidemic models

such as (2.6) with an arbitrarily distributed exposed stage have been studied in

the literature; see, for example, [8, 14, 15]. Recently, a model of this type, but

including the possibility of disease relapse, has been proposed in [40, 43] to study

the transmission and spread of some infectious disease such as herpes, and its global

dynamics have been completely investigated in [31, 40]. The model in [31, 40, 43]

can be regarded as a generalization of our model (2.6), and thus the stability results

there can be immediately applied to our model (2.6) by setting the relapse rate to

zero. We outline some of their results in the following; their proofs and more detailed

study can be found in [31, 40].

The existence, uniqueness, and continuity of solutions of system (2.6) follow from

the standard theory of Volterra integro-differential equations, see, for example, [36,

p.338]. The feasible region

?

is positively invariant with respect to (2.6). Let

(2.6) always has a disease-free equilibrium (DFE) P0 = (S0,0,0,0) in D, where

S0=A

m.

Let

?∞

and note that with a positive mean latent period,

?∞

Define the basic reproduction number as

D =(S,E,I,R) ∈ R4??? S,E,I,R ≥ 0, S + E + I + R ≤A

m

?

◦

D denote the interior of D. System

Q = −

0

e−muduP(u) du, (2.7)

0 < Q = 1 −

0

mP(u)e−mudu < 1.

R0=

βQS0

m + α + γ,

which completely determines the stability of the DFE. We refer the reader to [3, 7,

40, 42, 43] for biological interpretation of R0.

Page 4

396 ZHISHENG SHUAI AND P. VAN DEN DRIESSCHE

Proposition 2.1 (van den Driessche et al. [40], Liu et al. [31]). The following

results hold for system (2.6).

(1) If R0≤ 1, then the DFE is globally asymptotically stable in D.

(2) If R0> 1, then the DFE is unstable.

In order to study the dynamical behavior of (2.6) when R0> 1, the following

limiting system (see [36, p.176]) has been studied as a special case in [31]:

S?(t)=A − βS(t)I(t) − mS(t),

?t

I?(t)=

−

−∞

R?(t)= γI(t) − mR(t).

For any κ ∈ (0,m), define a Banach space of fading memory type (e.g., see [4] and

references therein)

?

E?(t)= βS(t)I(t) − mE(t) +

?t

−∞

βS(u)I(u)e−m(t−u)dtP(t − u) du,

βS(u)I(u)e−m(t−u)dtP(t − u) du − (m + α + γ)I(t),

(2.8)

Cκ=φ ∈ ((−∞,0],R)

??? φ(s)eκsis uniformly continuous on (−∞,0],

and sup

s≤0|φ(s)|eκs< ∞

?

(2.9)

with norm ||φ||κ= sups≤0|φ(s)|eκs. For ψ ∈ C(R,R) and t > 0, let ψt∈ Cκbe

such that ψt(s) = ψ(t + s),s ∈ (−∞,0]. Consider system (2.8) in the phase space

X = Cκ× R × Cκ× R.

Let E(0),R(0) ≥ 0 and φ,ψ ∈ Cκ such that φ(s) ≥ 0,ψ(s) ≥ 0 for all s ∈

(−∞,0]. For any solution (St,E(t),It,R(t)) of system (2.8) with initial conditions

(φ,E(0),ψ,R(0)), standard theory of functional differential equations [16] implies

that St,It∈ Cκfor all t > 0. The set

∆ =(S(·),E,I(·),R) ∈ X

(2.10)

?

??? S(s) ≥ 0,I(s) ≥ 0,s ∈ (−∞,0],E,R ≥ 0,

S(0) + E + I(0) + R ≤A

m

?

(2.11)

is positively invariant for system (2.8). Let

When R0> 1, the limiting system (2.8) has a unique endemic equilibrium (EE)

P∗= (S∗,E∗,I∗,R∗) in∆. Here, S∗=S0

provided Q < 1, I∗=m

of the EE is established by constructing a Lyapunov functional, giving the following

result.

◦

∆ be the interior of ∆.

◦

R0=m+α+γ

βQ

β(R0−1) > 0. The global stability

> 0,E∗= (1−Q)(A

m−S∗) > 0

β(R0−1) > 0, and R∗=γ

Proposition 2.2 (Liu et al. [31]). If R0 > 1, then the EE of system (2.8) is

globally asymptotically stable in∆. As a consequence, all solutions of system (2.6)

◦

D approach the EE of (2.8) if R0> 1.

When P(t) = e−?twith ? > 0, system (2.6) becomes an ODE model, for which

the global dynamics have been established by Li and Muldowney [24] using the

theory of compound matrices and by Korobeinikov and Maini [21] using the method

of Lyapunov functions.When P(t) is a step function, system (2.6) becomes a

◦

starting in

Page 5

IMPACT OF HETEROGENEITY ON DYNAMICS OF AN EPIDEMIC MODEL 397

delay differential equation model [39]. Propositions 2.1 and 2.2 establish the global

dynamics of the model (2.6) with an arbitrarily exposed stage.

3. Multi-group model. System (2.6) assumes homogeneous mixing of individuals

in the host population, and this is certainly unrealistically simple. Heterogeneity in

the host population can result from different contact patterns such as those among

children and adults for childhood diseases (e.g., measles and mumps), or different

behavior (e.g., numbers of sexual partners for some sexually transmitted infections).

In this section, we extend model (2.6) to the situation in which the population is

divided into n groups according to different contact patterns or differential infec-

tivity. Each group is further partitioned into four compartments: Si,Ei,Ii, and Ri,

denoting the number of susceptible, exposed, infectious, and removed individuals

in group i, respectively. For 1 ≤ i,j ≤ n, the disease effective contact rate between

compartments Siand Ij is denoted by βij, so that the new infection occurring in

the i-th group is given by

n

?

Hence, the n-group model associated with (2.6) can be written as the following

system with the stated assumptions:

j=1

βijSi(t)Ij(t).

S?

i(t) = Ai−

n

?

j=1

βijSi(t)Ij(t) − miSi(t),

E?

i(t) =

n

?

j=1

βijSi(t)Ij(t) − miEi(t) +

?t

i(t) = γiIi(t) − miRi(t),

Assume that Ai > 0,mi > 0,γi ≥ 0,αi ≥ 0, and each Pi satisfies assumption

(H) in Section 2. The contact matrix B = (βij) is assumed to be nonnegative

and irreducible; thus any two groups i and j have a direct or indirect route of

transmission.

Model (3.1) has been previously seen in [44]. In the special case when Pi,1 ≤

i ≤ n, are gamma distributions, the global dynamics have been established in [44]

by using a linear chain trick. However, the linear chain trick fails to apply to

model (3.1) with general distributions Pi. In this section we demonstrate that the

graph-theoretic approach developed in [10, 11, 29] can be successfully applied to

construct suitable Lyapunov functionals, and thus prove the global stability of the

endemic equilibrium for model (3.1) with general distributions Pi. This model differs

from the multi-group delay epidemic model in [30] in the delay terms: arbitrarily

distributed exposed stages are modeled in (3.1) while age structure is modeled in

[30]. Model (3.1) can also be applied to study heterogeneous spatial distribution of

the host population with implicit migration among groups; see, for example, [33].

For model (3.1) and the other heterogeneous model in Section 4, the existence,

uniqueness, and continuity of solutions follow as for the basic model (2.6) from [36,

p.338]. It can be easily verified that every solution of (3.1) with nonnegative initial

conditions remain nonnegative. From the first equation of (3.1), it follows that

n

?

j=1

?t

0

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du,

I?

i(t) = −

n

?

j=1

0

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du − (mi+ αi+ γi)Ii(t),

R?

i = 1,2,...,n.(3.1)

Page 6

398ZHISHENG SHUAI AND P. VAN DEN DRIESSCHE

S?

equations of (3.1) together yields

i(t) ≤ Ai− miSi(t), and thus limsupt→∞Si(t) ≤

Ai

mifor all i. Adding the four

(Si(t) + Ei(t) + Ii(t) + Ri(t))?≤ Ai− mi(Si(t) + Ei(t) + Ii(t) + Ri(t)),

which implies that, for each i, limsupt→∞(Si(t) + Ei(t) + Ii(t) + Ri(t)) ≤

Therefore, the feasible region

?

Ai

mi.

Dg=(S1,E1,I1,R1,...,Sn,En,In,Rn) ∈ R4n??? Si+ Ei+ Ii+ Ri≤Ai

mi,

Si,Ei,Ii,Ri≥ 0, 1 ≤ i ≤ n

◦

Dgdenote the interior of Dg.

1,0,0,0,...,S0

?

is positively invariant with respect to system (3.1). Let

System (3.1) always admits a disease-free equilibrium P0= (S0

0,0) with S0

miin Dg. Let

n,0,

i=Ai

Qi= −

?∞

0

e−miuduPi(u) du. (3.2)

It can be verified that 0 < Qi< 1 for all i. With ρ denoting the spectral radius,

define the basic reproduction number as the spectral radius of the n × n matrix

Y =

mi+αi+γi

, that is,

??

The following result establishes that R0is a threshold value for the DFE.

Theorem 3.1. The following results hold for system (3.1) with R0given by (3.3).

(1) If R0≤ 1, then the DFE is globally asymptotically stable in Dg.

(2) If R0> 1, then the DFE is unstable.

Proof. Since the variables Eiand Rido not appear in the equations of Siand Ii

of (3.1), we can first study the following reduced system consisting of variables Si

and Ii

?

I?

i(t) = −

j=1

0

i = 1,2,...,n.

?

βijQiS0

i

?

R0= ρ(Y ) = ρ

βijQiS0

mi+ αi+ γi

i

??

. (3.3)

S?

i(t) = Ai−

n

j=1

?t

βijSi(t)Ij(t) − miSi(t),

n

?

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du − (mi+ αi+ γi)Ii(t),

(3.4)

We prove that the equilibrium (S0

asymptotically stable by the method of Lyapunov functions. Then our stability

results for the reduced system (3.4) can be extended to system (3.1) by the theory

of asymptotically autonomous systems.

Since B = (βij) is irreducible, the nonnegative matrix Y = (

irreducible, and Y has a positive left eigenvector (w1,w2,...,wn) corresponding to

the spectral radius ρ(Y ) = R0> 0. Motivated by [10], let

ci=

mi+ αi+ γi

1,0,S0

2,0,...,S0

n,0) of system (3.4) is globally

βijQiS0

mi+αi+γi) is also

i

wi

> 0,

Page 7

IMPACT OF HETEROGENEITY ON DYNAMICS OF AN EPIDEMIC MODEL399

and define

Qi(r) = −

?∞

r

e−miuduPi(u) du.(3.5)

From (3.2), it follows that Qi(0) = Qi. Consider a Lyapunov functional for system

(3.4)

L =

n

?

i=1

ci

?

Qi

?

Si− S0

i− S0

ilnSi

S0

i

?

+ Ii+

n

?

j=1

?t

0

βijQi(r)Si(t − r)Ij(t − r) dr

?

.

Observe that L ≥ 0, and L = 0 if and only if Si = S0

0 ≤ r ≤ t. Before we differentiate L along the solution of system (3.4), we follow

the idea in [34, 35] and using integration by parts obtain

i,Ii(r) = 0 for all i and

∂

∂t

??t

0

Qi(r)Si(t − r)Ij(t − r) dr

?t

?t

= Qi(t)Si(0)Ij(0) − Qi(r)Si(t − r)Ij(t − r)

?t

?t

Now differentiating L along the solution of system (3.4), using (3.6) and the non-

increasing property of Pi (see assumption (H)), and setting I = (I1,I2,...,In)T

give

?

= Qi(t)Si(0)Ij(0) +

0

Qi(r)∂

∂t(Si(t − r)Ij(t − r)) dr

Qi(r)(−1)∂

= Qi(t)Si(0)Ij(0) +

0

∂r(Si(t − r)Ij(t − r)) dr

???

t

r=0

+

0

Si(t − r)Ij(t − r)e−mirdrPi(r) dr

= QiSi(t)Ij(t) +

0

Si(t − r)Ij(t − r)e−mirdrPi(r) dr. (3.6)

L?|(3.4) =

n

?

i=1

ci

?

Qi

?

Ai−

n

?

?

j=1

βijSi(t)Ij(t) − miSi(t) − Ai

?t

n

?

βijSi(t − r)Ij(t − r)e−mirdrPi(r) dr

S0

Si(t)+ miS0

i

i

+

n

?

j=1

βijS0

iIj(t)

−

n

?

j=1

0

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du

− (mi+ αi+ γi)Ii(t) +

?t

n

?

− (mi+ αi+ γi)Ii(t)

j=1

βijQiSi(t)Ij(t)

+

n

?

j=1

0

?

=

i=1

ciQiAi

?

2 −Si(t)

S0

i

−

?

S0

Si(t)

i

?

+

n

?

i=1

ci

?

n

?

j=1

βijQiS0

iIj(t)

Page 8

400 ZHISHENG SHUAI AND P. VAN DEN DRIESSCHE

=

n

?

i=1

ciQiAi

?

2 −Si(t)

S0

i

−

S0

Si(t)

i

?

+

n

?

i=1

wi

mi+ αi+ γi

?

n

?

j=1

βijQiS0

iIj(t) − (mi+ αi+ γi)Ii(t)

?

=

n

?

n

?

i=1

ciQiAi

?

?

2 −Si(t)

S0

i

−

S0

Si(t)

i

?

?

+ (w1,w2,...,wn)(Y I − I)

=

i=1

ciQiAi

2 −Si(t)

S0

i

−

S0

Si(t)

i

+ (ρ(Y ) − 1)(w1,w2,...,wn)I

≤ 0, if R0≤ 1. (3.7)

Let

K = {(S1,I1,...,Sn,In) | L?|(3.4)= 0},

and ? be the largest invariant set in K. We now show ? = {(S0

From (3.7) and ci> 0, L?|(3.4)= 0 implies that 2−Si(t)

and t ≥ 0, and thus Si(t) ≡ S0

?n

Therefore, ? = {(S0

Theorem 3.4.7] or [13, Theorem 5.3.1]), it follows that (S0

solutions of system (3.4) whose initial conditions satisfy 0 ≤ Si(0) + Ii(0) ≤

By Lemma A.1 in Appendix A, it follows that (S0

for system (3.4) since there exists a nonnegative monotone increasing function a(r)

such that (A.2) holds. Therefore, for the reduced system (3.4), the equilibrium

(S0

n,0) is globally asymptotically stable when R0 ≤ 1. Using the fact

that Si(t) → S0

autonomous systems, Ei(t) → 0 and Ri(t) → 0 as t → ∞. Therefore, the DFE P0

for system (3.1) is globally asymptotically stable in Dgwhen R0≤ 1.

If R0> 1 and I ?= 0, it follows that

(ρ(Y ) − 1)(w1,w2,...,wn)I > 0,

which implies that by continuity L?|(3.4)> 0 in a small enough neighborhood of P0

inDg. Therefore, P0is unstable when R0> 1.

To analyze the dynamical behavior of (3.1) when R0> 1, we consider the limiting

system of (3.1)

1,0,...,S0

n,0)}.

S0

i

−

S0

Si(t)= 0 for all 1 ≤ i ≤ n

i

i=

Ai

mi. Hence, the first equation of (3.4) gives

j=1βijS0

each j, there exists i ?= j such that βij ?= 0, and thus Ij(t) = 0 for all t ≥ 0.

1,0,...,S0

iIj(t) = 0 for all t ≥ 0 and 1 ≤ i ≤ n. By the irreducibility of B, for

n,0)}. Using the LaSalle-Lyapunov Theorem (see [23,

1,0,...,S0

n,0) attracts all

Ai

mi.

1,0,...,S0

n,0) is locally stable

1,0,...,S0

iand Ii(t) → 0 as t → ∞ along with the theory of asymptotically

◦

S?

i(t) = Ai−

n

?

j=1

βijSi(t)Ij(t) − miSi(t),

E?

i(t) =

n

?

j=1

βijSi(t)Ij(t) − miEi(t) +

?t

i(t) = γiIi(t) − miRi(t),

n

?

j=1

?t

−∞

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du,

I?

i(t) = −

n

?

j=1

−∞

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du − (mi+ αi+ γi)Ii(t),

R?

i = 1,2,...,n. (3.8)

Page 9

IMPACT OF HETEROGENEITY ON DYNAMICS OF AN EPIDEMIC MODEL401

For any κi∈ (0,mi), the Banach space of fading memory type Cκican be defined

similarly as given in (2.9). We study system (3.8) in the phase space

Xg=

n

?

i=1

(Cκi× R × Cκi× R).

It can be easily verified that

?

Si(s),Ii(s) ≥ 0,s ∈ (−∞,0],Si(0) + Ei+ Ii(0) + Ri≤Ai

∆g=(S1(·),E1,I1(·),R1,...,Sn(·),En,In(·),Rn) ∈ Xg

??? Ei,Ri≥ 0,

mi, 1 ≤ i ≤ n

?

is positively invariant with respect to system (3.8). Let

The following result establishes the existence and uniqueness of an endemic equi-

librium for system (3.8).

◦

∆gbe the interior of ∆g.

Lemma 3.2. If R0> 1, then system (3.8) has a unique endemic equilibrium P∗=

(S∗

∆g.

1,E∗

1,I∗

1, R∗

1,...,S∗

n,E∗

n,I∗

n,R∗

n) in

◦

Proof. From the equilibrium equations of (3.8),

Ai−

n

?

j=1

βijSiIj− miSi= 0, (3.9)

and

n

?

j=1

βijSiIj−mi+ αi+ γi

Qi

Ii= 0. (3.10)

Consider the auxiliary system

S?

i

=Ai−

n

?

n

?

βijSiIj−mi+ αi+ γi

j=1

βijSiIj− miSi,

I?

i

=

j=1

Qi

Ii.

(3.11)

The global dynamics of system (3.11) have been studied in [10], where it is shown

that if R0 = ρ(Y ) = ρ

(S∗

S∗

tions of Eiand Riin (3.8), it follows that E∗

for each i. Therefore, if R0> 1, then P∗= (S∗

unique endemic equilibrium for system (3.8).

?

βijQiS0

mi+αi+γi

i

?

> 1, then a unique endemic equilibrium

1,I∗

n,I∗

1,...,S∗

n) solving the equations (3.9) and (3.10) if R0> 1. Using the equilibrium equa-

n,I∗

n) exists. This is, there exists a unique positive solution (S∗

1,I∗

1,...,

i=

1,E∗

?n

1,I∗

j=1βijS∗

iI∗

mi

1,...,S∗

j(1−Qi)

and R∗

n,E∗

i=

γiI∗

mi

i

1, R∗

n,I∗

n,R∗

n) is a

Using the graph-theoretic approach recently developed in [10, 11, 29], we prove

the global stability of the EE for (3.8).

Theorem 3.3. If R0> 1, then the unique EE of system (3.8) is globally asymptot-

ically stable in∆g. As a consequence, all solutions of (3.1) starting in

the EE of (3.8) if R0> 1.

◦◦

Dgapproach

Page 10

402ZHISHENG SHUAI AND P. VAN DEN DRIESSCHE

Proof. As in the proof of Theorem 3.1, we first study the reduced system

S?

i(t) = Ai−

n

?

?t

j=1

βijSi(t)Ij(t) − miSi(t),

I?

i(t) = −

n

?

j=1

−∞

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du − (mi+ αi+ γi)Ii(t),

i = 1,2,...,n,(3.12)

which is the limiting system of (3.4). In the following, we prove that the equilibrium

(S∗

a suitable Lyapunov functional. Set

1,I∗

1,S∗

2,I∗

2,...,S∗

n,I∗

n) of (3.12) is globally asymptotically stable by constructing

Ui= Qi

?

Si− S∗

i− S∗

ilnSi

S∗

i

?

+ Ii− I∗

i− I∗

ilnIi

I∗

i

,

Wi=

n

?

j=1

?∞

0

βijQi(r)

?

Si(t − r)Ij(t − r) − S∗

iI∗

j− S∗

iI∗

jlnSi(t − r)Ij(t − r)

S∗

iI∗

j

?

dr,

and

Vi= Ui+ Wi.

Here Qi and Qi(r) are defined in (3.2) and (3.5), respectively, and Qi(0) = Qi.

Lyapunov functions as Uihave been successfully applied to epidemic models since

the work in [20, 21], while Lyapunov functionals as Wi have been shown to be

powerful for delay epidemic models [34, 35]. Functional Wiis different from those

in [30, 34, 35] because of different delay terms in model (3.1), but is motivated by

those in [17, 18, 27, 28]. Differentiating Uialong the solution of (3.12) and using

the equilibrium equations (3.9) and (3.10) yield

U?

i|(3.12)=Qi

?

?

n

?

Ai−

n

?

j=1

βijSi(t)Ij(t) − miSi(t) − Ai

S∗

Si(t)+

i

n

?

j=1

βijS∗

iIj(t) + miS∗

i

?

−

n

j=1

?t

?t

?

βijQiS∗

−∞

βijSi(u)Ij(u)e−mi(t−u)dtPi(t − u) du − (mi+ αi+ γi)Ii(t)

+

j=1

−∞

2 −Si(t)

βijSi(u)Ij(u)I∗

i

Ii(t)e−mi(t−u)dtPi(t − u) du + (mi+ αi+ γi)I∗

?

2 −Si(t)Ij(t)

S∗

j

I∗

j

?

i

=QimiS∗

i

S∗

i

−

S∗

Si(t)

i

+

n

?

n

?

j=1

iI∗

j

?

iI∗

−

S∗

Si(t)+Ij(t)

i

−Ii(t)

I∗

i

?

−

j=1

?t

−∞

βijSi(u)Ij(u)1 −

I∗

i

Ii(t)

?

e−mi(t−u)dtPi(t − u) du.(3.13)

Page 11

IMPACT OF HETEROGENEITY ON DYNAMICS OF AN EPIDEMIC MODEL403

For the integral in Wi, integration by parts gives

?∞

?∞

· e−mirdrPi(r) dr.

It follows that

n

?

+S∗

0

Qi(r)∂

∂t

?

Si(t − r)Ij(t − r) − S∗

?

iI∗

j− S∗

iI∗

jlnSi(t − r)Ij(t − r)

S∗

Si(t)Ij(t)

Si(t − r)Ij(t − r)

iI∗

j

?

dr

= QiSi(t)Ij(t) +

0

Si(t − r)Ij(t − r) + S∗

iI∗

jln

?

W?

i|(3.12)

=

j=1

βij

?

QiSi(t)Ij(t) +

?∞

0

?

?

Si(t − r)Ij(t − r)

iI∗

jln

Si(t)Ij(t)

Si(t − r)Ij(t − r)

e−mirdrPi(r) dr

?

.

(3.14)

Adding equations (3.13) and (3.14) yields

?

−Ii(t)

I∗

i

V?

i|(3.12)=QimiS∗

i

2 −Si(t)

?∞

S∗

i

−

S∗

Si(t)

i

?

+

n

?

j=1

βijS∗

iI∗

j

?

Qi

?

2 −

S∗

Si(t)+Ij(t)

i

I∗

j

??

+

0

?Si(t − r)Ij(t − r)I∗

?

?

?∞

?

iI∗

j

I∗

j

i

S∗

iI∗

jIi(t)

+ ln

Si(t)Ij(t)

Si(t − r)Ij(t − r)

· e−mirdrPi(r) dr

n

?

+ lnIi(t)

I∗

i

≤

j=1

βijS∗

iI∗

j

?

Qi

1 −

S∗

Si(t)+ ln

i

S∗

Si(t)

i

?

+ Qi

?Ij(t)

+ lnSi(t − r)Ij(t − r)I∗

S∗

I∗

j

−Ii(t)

I∗

i

− lnIj(t)

I∗

j

?

−

0

?

1 −Si(t − r)Ij(t − r)I∗

S∗

i

iI∗

jIi(t)

i

iI∗

jIi(t)

?

· e−mirdrPi(r) dr

n

?

≤

j=1

βijQiS∗

?Ij(t)

−Ii(t)

I∗

i

− lnIj(t)

I∗

j

+ lnIi(t)

I∗

i

?

, (3.15)

where the following inequalities have been used:

2 −Si(t)

S∗

i

−

S∗

Si(t)≤ 0,

i

1 −

S∗

Si(t)+ ln

i

S∗

Si(t)≤ 0,

i

1 −Si(t − r)Ij(t − r)I∗

S∗

i

iI∗

jIi(t)

+ lnSi(t − r)Ij(t − r)I∗

S∗

i

iI∗

jIi(t)

≤ 0.

Considering (3.15), take the weight matrix W = (wij) with constant entry wij=

βijQiS∗

ci=?

n

?

Set

n

?

iI∗

T ∈Tiw(T) ≥ 0 be as given in (B.1) in Appendix B with (G,W). Then, by

(B.2), the following identity holds

?Ii(t)

j≥ 0 and denote the corresponding weighted digraph as (G,W). Let

i,j=1

ciβijQiS∗

iI∗

j

I∗

i

− lnIi(t)

I∗

i

?

=

n

?

i,j=1

ciβijQiS∗

iI∗

j

?Ij(t)

I∗

j

− lnIj(t)

I∗

j

?

. (3.16)

V =

i=1

ciVi.

Page 12

404ZHISHENG SHUAI AND P. VAN DEN DRIESSCHE

Using (3.15) and (3.16) gives

V?|(3.12)=

n

?

i=1

ciV?

i≤

n

?

i,j=1

ciβijQiS∗

iI∗

j

?Ij(t)

I∗

j

−Ii(t)

I∗

i

− lnIj(t)

I∗

j

+ lnIi(t)

I∗

i

?

= 0.

Therefore, V is a Lyapunov functional for system (3.12). This rules out the possi-

bility that solutions of (3.12) approach the boundary Si= 0 or Ii= 0 in ∆gsince

V → ∞ if Si→ 0 or Ii→ 0. Since B is irreducible, ci> 0 for all i (see Appendix B),

and thus V?|(3.12)= 0 implies that Si= S∗

the proof of Theorem 3.1 the largest invariant set where V?|(3.12)= 0 is the sin-

gleton {(S∗

similar argument as in the proof of Theorem 3.1, it follows that (S∗

is globally asymptotically stable for system (3.12). Thus for the limiting system

(3.8) if R0> 1, then the EE P∗is globally asymptotically stable in

diate consequence of Theorem 7.2 in [36] is that P∗attracts all solutions of (3.1)

◦

Dg.

ifor all i. It can be verified that as in

1,I∗

1,...,S∗

n,I∗

n)}. Therefore, by the LaSalle-Lyapunov Theorem and a

1,I∗

1,...,S∗

n,I∗

n)

◦

∆g. An imme-

in

Theorems 3.1 and 3.3 completely determine the dynamical behavior of the multi-

group model (3.1) under the stated assumptions: if the basic reproduction number

R0≤ 1, then the disease dies out from all groups; if R0> 1, the disease persists at a

positive level in all groups. It follows that the multi-group structure does not alter

the qualitative behavior of the SEIR model (2.6) with an arbitrarily distributed

exposed stage.

4. Multi-stage model. Multi-stage models has been proposed in the literature

to describe the progression of infectious diseases with a long infectious period, for

example, HIV/AIDS. Individuals infected with HIV are highly infectious in the first

few weeks after infection, then remain in an asymptotic stage of low infectiousness

for many years, and become gradually more infectious as their immune systems

become compromised and they progress to AIDS [9, 19, 38]. So infectious individuals

can be categorized into n different stages according to the age of infection. We

include such stage structure into our model (2.6) and arrive at the following system

in which Ii(t) denotes the number of individuals in infection stage i with effective

contact rate βi≥ 0, γi> 0 is the rate of transferring from stage Iito Ii+1(or to

compartment R when i = n), δi≥ 0 is the rate of transferring from stage Iito Ii−1

when i ≥ 2, and αn+1≥ 0 is the mortality due to disease in the removed (AIDS)

compartment:

S?(t) = A −

n

?

j=1

βjS(t)Ij(t) − mS(t),

E?(t) =

n

?

j=1

βjS(t)Ij(t) − mE(t) +

?t

i(t) = γi−1Ii−1(t) − (m + αi+ δi+ γi)Ii(t) + δi+1Ii+1(t),

I?

n(t) = γn−1In−1(t) − (m + αn+ δn+ γn)In(t),

R?(t) = γnIn(t) − (m + αn+1)R(t).

n

?

j=1

?t

0

βjS(u)Ij(u)e−m(t−u)dtP(t − u) du,

I?

1(t) = −

n

?

j=1

0

βjS(u)Ij(u)e−m(t−u)dtP(t − u) du − (m + α1+ γ1)I1(t) + δ2I2(t),

I?

i = 2,3,...,n − 1,

(4.1)

Page 13

IMPACT OF HETEROGENEITY ON DYNAMICS OF AN EPIDEMIC MODEL405

Without loss of generality, assume that there exists k ≥ 1 such that βk> 0, βj= 0,

and δj > 0 for all k < j ≤ n. In fact, if βj = 0 and δj = 0 for all k < j ≤ n,

then the variables Ij,k < j ≤ n, do not appear in the first k + 2 equations and

play the same role as the removed compartment R. In the case when k = n − 1,

that is, βn−1> 0, βn= 0, and δn> 0, the compartment Incan be treated as a

temporally removed compartment and δnIn represents the relapse of diseases, so

the models with disease relapse in [40, 43] become special cases of (4.1). To the

best of our knowledge, model (4.1) is the first multi-stage model in the literature

with an arbitrarily distributed exposed stage.

With assumptions as in the previous sections, it can be verified that

?

Ds=(S,E,I1,...,In,R) ∈ Rn+3??? S,E,R,Ii≥ 0, 1 ≤ i ≤ n,

S + E + R +

n

?

i=1

Ii≤A

m

?

is positively invariant with respect to system (4.1). Let

For system (4.1), there always exists a disease-free equilibrium P0= (S0,0,0,...,0,0)

with S0=A

min Ds.

Writing η1= m + α1+ γ1and ηi= m + αi+ γi+ δi,2 ≤ i ≤ n, define the basic

reproduction number

R0= QS0(β1,β2,...,βn)M−1(1,0,...,0)T,

where Q is defined as in (2.7) and tridiagonal

The following result establishes that R0is a threshold value for the DFE.

Theorem 4.1. The following results hold for system (4.1).

(1) If R0≤ 1, then the DFE is globally asymptotically stable in Ds.

(2) If R0> 1, then the DFE is unstable.

Proof. Since all off-diagonal entries of M are nonpositive and the sum of the entries

in each column of M is positive, M is a nonsingular M-matrix and M−1≥ 0

[5, p.137]. Following [9], let (w1,w2,...,wn) = (β1,β2,...,βn)M−1, giving w1=

(β1,β2,...,βn)M−1(1,0,...,0)Tand R0= w1QS0. As in the proof of Theorem 3.1,

we construct a Lyapunov functional for the reduced system of (4.1)

?

where Q(r) = −?∞

L?|(4.1)=w1QmS0?

≤(w1QS0− 1)(β1,...,βn)I ≤ 0,

◦

Dsdenote the interior of Ds.

(4.2)

M =

η1

−γ1

−δ2

η2

...

...

...

ηn−1

−γn−1

−δn

ηn

.

L = w1QS − S0− S0lnS

S0

?

+

n

?

i=1

wiIi+ w1

n

?

i=1

?t

0

βiQ(r)S(t − r)Ii(t − r) dr,

r

e−muduP(u) du. Differentiating L along the reduced system of

(4.1) and using a result as in (3.6) give

?

2 −Si(t)

S0

i

−

S0

Si(t)

i

+ w1QS0(β1,...,βn)I − (w1,...,wn)MI

if R0≤ 1.

Page 14

406ZHISHENG SHUAI AND P. VAN DEN DRIESSCHE

Here I = (I1(t),I2(t),...,In(t))T. It can be verified that the largest invariant set

where L?|(4.1) = 0 is the singleton of the DFE. Therefore, the DFE is globally

asymptotically stable in Ds if R0 ≤ 1, and unstable if R0 > 1 (using similar

arguments as in the proof of Theorem 3.1).

To study the dynamical behavior of (4.1) when R0> 1, we consider the limiting

system of (4.1)

S?(t) = A −

n

?

j=1

βjS(t)Ij(t) − mS(t),

E?(t) =

n

?

j=1

βjS(t)Ij(t) − mE(t) +

?t

i(t) = γi−1Ii−1(t) − ηiIi(t) + δi+1Ii+1(t),

I?

n(t) = γn−1In−1(t) − ηnIn(t),

R?(t) = γnIn(t) − (m + αn+1)R(t).

We consider system (4.3) in the phase space

n

?

j=1

?t

−∞

βjS(u)Ij(u)e−m(t−u)dtP(t − u) du,

I?

1(t) = −

n

?

j=1

−∞

βjS(u)Ij(u)e−m(t−u)dtP(t − u) du − η1I1(t) + δ2I2(t),

I?

i = 2,3,...,n − 1,

(4.3)

Xs= Cκ× R × (Cκ)n× R,

where κ ∈ (0,m) and Cκis given in (2.9). It can be verified that

∆s=(S(·),E,I1(·),...,In(·),R) ∈ Xs

E,R ≥ 0, S(0) + E + I1(0) + ··· + In(0) + R ≤A

?

??? S(s),Ii(s) ≥ 0,1 ≤ i ≤ n,s ∈ (−∞,0],

m

?

is positively invariant with respect to system (4.3). Let

The existence and uniqueness of an endemic equilibrium for (4.3) can be established

using an argument as in the proof of Lemma 3.2 and a result for the multi-stage

ODE model in [9, Theorem 3.1].

◦

∆sbe the interior of ∆s.

Lemma 4.2. If R0> 1, then system (4.3) has a unique endemic equilibrium P∗=

(S∗,E∗,I∗

∆s.

1,...,I∗

n,R∗) in

◦

The global stability of the EE can be established by choosing an appropriate

weight matrix as in Section 3.

Theorem 4.3. If R0> 1, then the EE of system (4.3) is globally asymptotically

stable in∆s. As a consequence, all solutions of (4.1) starting in

EE of (4.3) if R0> 1.

Proof. Choose the weight matrix W = (wij) as given by

◦◦

Dsapproach the

wij=

β2QS∗I∗

βjQS∗I∗

γi−1I∗

δi+1I∗

0

2+ δ2I∗

2

if i = 1,j = 2

if i = 1,j ?= 2

if i ≥ 2,j = i − 1

if 2 ≤ i ≤ n − 1,j = i + 1

otherwise.

j

i−1

i+1

Page 15

IMPACT OF HETEROGENEITY ON DYNAMICS OF AN EPIDEMIC MODEL 407

Let ci=?

all k < j ≤ n, it follows that W is irreducible. As a consequence, ci> 0 for all i

(see Appendix B). A Lyapunov functional

T ∈Tiw(T) be as given in (B.1) in Appendix B with the weighted digraph

(G,W). Since βk> 0, βj= 0, δj> 0, and γi> 0 for some k ≥ 1, all 1 ≤ i ≤ n, and

V =c1Q

?

S − S∗− S∗lnS∗

?∞

S

?

+

n

?

i=1

ci

?

Ii− I∗

i− I∗

ilnIi

I∗

i

?

ilnS(t − r)Ii(t − r)

S∗I∗

+ c1

n

?

i=1

0

βiQ(r)

?

S(t − r)Ii(t − r) − S∗I∗

i− S∗I∗

i

?

dr.

can be constructed for system (4.3). Using similar derivations as in the proof of

Theorem 3.3 and identity (B.2), differentiating V along the limiting system (4.3)

gives

V?=c1QmS∗?

2 −S(t)

S∗−

?∞

S∗

S(t)

?

+ c1Q

n

?

j=1

βjS∗I∗

j

?

+ lnS(t − r)Ij(t − r)I∗

S∗I∗

1 −

S∗

S(t)−S(t)

S∗

?

−

n

?

j=1

c1βjS∗I∗

j

0

?

?

1 −S(t − r)Ij(t − r)I∗

S∗I∗

1

jI1(t)

?Ij(t)

+ 1 −Ii−1(t)I∗

I∗

1

jI1(t)

?

?

· e−mirdrPi(r) dr +

n

j=1

c1QβjS∗I∗

j

I∗

j

−I1(t)

I∗

1

− lnIj(t)

I∗

j

+ lnI1(t)

I∗

1

+

n

?

n−1

?

c1QβjS∗I∗

i=2

ciγi−1I∗

i−1

?Ii−1(t)

?Ii+1(t)

?Ij(t)

?Ii−1(t)

?Ii+1(t)

?Ij(t)

I∗

i−1

−Ii(t)

I∗

i

i

i−1Ii(t)

?

?

+

i=1

ciδi+1I∗

i+1

I∗

i+1

−Ii(t)

I∗

i

+ 1 −Ii+1(t)I∗

I∗

i

i+1Ii(t)

≤

n

?

j=1

j

I∗

j

−I1(t)

I∗

1

− lnIj(t)

I∗

j

+ lnI1(t)

I∗

1

?

+

n

?

n−1

?

n

?

i=2

ciγi−1I∗

i−1

I∗

i−1

−Ii(t)

I∗

i

− lnIi−1(t)

I∗

i−1

+ lnIi(t)

I∗

i

?

?

+

i=1

ciδi+1I∗

i+1

I∗

i+1

−Ii(t)

I∗

i

− lnIi+1(t)

I∗

i+1

+ lnIi(t)

I∗

i

=

i,j=1

=0.

ciwij

I∗

j

−Ii(t)

I∗

i

− lnIj(t)

I∗

j

+ lnIi(t)

I∗

i

?

By similar arguments as in the proof of Theorem 3.3, it follows that for system (4.3)

P∗is globally asymptotically stable in ∆s. As a consequence, all solutions of (4.1)

◦

Dsapproach P∗as the time goes to infinity.

◦

starting in

The global dynamics of (4.1) are completely determined by Theorems 4.1 and

4.3. Biologically, if the basic reproduction number R0≤ 1, then the disease dies

out eventually. If R0> 1, then the disease persists in all stages of infection. As