Analysis of a nonautonomous epidemic model with density dependent birth rate

Applied Mathematical Modelling 34 (2010) 866–877

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Analysis of a nonautonomous epidemic model with density dependent birth rate Junli Liu a,b,*, Tailei Zhang b a b

School of Science, Xi’an Polytechnic University, Xi’an 710048, PR China Department of Mathematics, Xi’an Jiaotong University, Xi’an 710049, PR China

a r t i c l e

i n f o

Article history: Received 15 November 2008 Received in revised form 1 June 2009 Accepted 22 July 2009 Available online 28 July 2009 Keywords: Nonautonomous SIRS model Density dependent birth rate Permanence Extinction

a b s t r a c t In this paper, a nonautonomous SIRS epidemic model with density dependent birth rate is proposed and studied. Threshold conditions for the permanence and extinction of the disease are established. Some new threshold values of integral form are obtained. We prove that the disease is permanent if R0 > 0, and extinct if R1 6 0 or R2 < 0. For the periodic and almost periodic cases, these threshold conditions act as sharp threshold values for the permanence and extinction of the disease. Global asymptotic stability of periodic solution for the periodic system is derived. Some examples are given to illustrate the main results of this paper. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction In modeling the spread of infectious disease through the population, it is usually assumed that the total population size is constant. More recent studies consider an epidemic model with density dependent birth and death rates to describe disease transmission in population (see [1–5] etc.). In order to make the model more realistic, Yoshida and Hara [6] proposed the following SIR model

8 bSIðthÞ dS rN rN > > > dt ¼  NðthÞ  ðl þ ð1  aÞ K ÞS þ ðb  a K ÞN; > > < dI bSIðthÞ ¼ NðthÞ  ðl þ ð1  aÞ rN ÞI  cI; dt K > dR rN > ¼ cI  ðl þ ð1  aÞ K ÞR: > > dt > :

ð1:1Þ

In this model, the birth and death rates are assumed as density dependent, incubation time during which the infectious agents develop in the vector are incorporated. And also it was assumed that the total number of the population is governed by a logistic equation. Stability of an endemic equilibrium is investigated in terms of the basic reproduction number. The nonautonomous phenomenon is so prevalent in the real life that many epidemiological problems can be modeled by nonautonomous systems of nonlinear differential equations [7–18], which should be more realistic than autonomous differential equations. One case is the spread of infectious childhood diseases, where it has been argued that the school system induces a time-heterogeneity in the per capita infection rate because of the interruption of the infections chain during vacations or the inclusion of new individuals at the beginning of each school year [19]. * Corresponding author. Present address: School of Science, Xi’an Polytechnic University, Xi’an 710048, PR China. E-mail address: [email protected] (J. Liu). 0307-904X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2009.07.004

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J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

In [20], the authors gave a nonautonomous model

8 dS > ¼ lðtÞ  bðtÞSI  lðtÞS þ dðtÞR; > dt > > < dI ¼ bðtÞSI  lðtÞI  cðtÞI; dt dR > ¼ cðtÞI  ðlðtÞ þ dðtÞÞR; > > dt > : NðtÞ ¼ SðtÞ þ IðtÞ þ RðtÞ: They studied the existence and uniqueness of periodic solutions of seasonal epidemiological disease, by using a continuation theorem based on coincidence degree theory, criteria for the existence, uniqueness and global asymptotic stability of the periodic solution of the system are obtained. Given a mathematical model for disease spread, the basic reproduction number or threshold values are very important parameters, which can be used to distinguish the permanence and the extinction of the disease. For autonomous epidemic models, much has been done to calculate these threshold values [6,1–4], but for nonautonomous systems, relatively few works have been done to obtain the corresponding threshold value results. More recently, Teng et al. [21], Zhang and Teng [22] studied the permanence and extinction for the nonautonomous SIRS and SEIRS epidemic models, sufficient conditions for the permanence and extinction of the disease are established. Some new threshold values of the integral form are obtained. Motivated by the works [6,21,20,22], in this paper we will consider a nonautonomous SIRS epidemic model with density dependent birth rate and assume that the total number of the population is governed by a logistic equation. The equations are given by

8 dS ¼ bðtÞSI  lðtÞS þ ðbðtÞ  rðtÞN ÞN þ dðtÞR; > > KðtÞ > dt > < dI ¼ bðtÞSI  lðtÞI  cðtÞI; dt > dR > > dt ¼ cðtÞI  ðlðtÞ þ dðtÞÞR; > : NðtÞ ¼ SðtÞ þ IðtÞ þ RðtÞ:

ð1:2Þ

Here, the population is divided into susceptible, infective and recovered individual whose numbers are denoted by S, I, R, respectively. The number of the total population is denoted by N ¼ S þ I þ R. Function bðtÞ is the transmission rate of infected individual at time t, functions bðtÞ and lðtÞ are the natural birth and death rates at time t, respectively, functions cðtÞ and dðtÞ are the instantaneous per capita rates of leaving the infection stage and removed stage at time t, respectively. rðtÞ ¼ bðtÞ lðtÞ is the intrinsic growth rate at time t and KðtÞ is the carrying capacity of the population at time t. In this paper, our purpose is to study the permanence and extinction of the disease for model (1.2). We will establish the sufficient conditions of the permanence and extinction of the disease, and give some new threshold values of the integral form. We will prove that the disease will be permanent if R0 > 0, and extinct if R1 6 0 or R2 < 0. When system (1.2) reduces to periodic or almost periodic case, the basic reproduction number is obtained (see Corollaries 4.4 and 4.5 given in Section 4), this threshold value acts as a sharp threshold for the permanence and extinction of the disease. This paper is organized as follows. Section 2 gives some assumptions for system (1.2), positivity and boundedness of solutions are obtained. Section 3 deals with the permanence and extinction of system (1.2). In Section 4, we conclude some corollaries, and in particular, we give the global stability for the periodic model (1.2). Some examples are included in the final part to verify our analytical results. 2. Notations and preliminaries For any solution ðSðtÞ; IðtÞ; RðtÞÞ of system (1.2), the initial value is given by

Sð0Þ > 0;

Ið0Þ > 0;

Rð0Þ P 0:

ð2:1Þ

For convenience, we first give some definitions with respect to persistence, permanence and extinction for the infective I in system (1.2). If lim inf t!1 IðtÞ > 0, then we say that the infective I are strongly persistent. If there are positive constants v 1 , v 2 such that

v 1 6 lim inf IðtÞ 6 lim sup IðtÞ 6 v 2 ; t!1 t!1

then we say that the infective I are permanent. If limt!1 IðtÞ ¼ 0, then we say that the infective I go extinct. For system (1.2), we make some assumptions. ðH1 Þ Functions bðtÞ, KðtÞ > 0, bðtÞ, lðtÞ, rðtÞ, cðtÞ and dðtÞ are nonnegative, continuous and bounded on Rþ ¼ ½0; þ1Þ. (H2 ) There are positive constants xi > 0 ði ¼ 1; 2; 3; 4Þ such that

lim inf t!1

Z t

tþx1

bðsÞds > 0;

lim inf t!1

Z t

tþx2

lðsÞds > 0

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J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

and

lim inf t!1

Z

tþx3

rðsÞds > 0;

lim inf

Z

t!1

t

tþx4

rðsÞ ds > 0: KðsÞ

t

Remark 2.1. We can prove that assumption (H2 ) is equivalent to

lim inf t;s!1

1 t

and

lim inf t;s!1

1 t

Z

t

bðq þ sÞds > 0; 0

Z

lim inf t;s!1

t

rðq þ sÞds > 0; 0

lim inf t;s!1

1 t 1 t

Z

t

lðq þ sÞds > 0

0

Z

t

0

rðq þ sÞ ds > 0: Kðq þ sÞ

In particular, when system (1.2) reduces to xperiodic system, that is, bðtÞ, bðtÞ, lðtÞ, rðtÞ, cðtÞ, dðtÞ and KðtÞ > 0 are all nonnegative continuous periodic functions with period x > 0, then assumptions (H2 ) is equivalent to the following cases:

l > 0; r > 0 and

b > 0;

r K

> 0;

where, for any continuous periodic function f with period x > 0, we denote by f the average value of f ðtÞ, i.e., f ¼ x1

Rx 0

f ðtÞdt.

When system (1.2) reduces to almost periodic system, that is, bðtÞ, bðtÞ, lðtÞ, rðtÞ, cðtÞ, dðtÞ and KðtÞ > 0 are all nonnegative continuous almost periodic functions, then assumption (H2 ) is equivalent to the following cases:

mðbÞ > 0;

r > 0; mðrÞ > 0 and m K

mðlÞ > 0;

where, for any continuous almost periodic function f , we denote by mðf Þ the average value of f ðtÞ, i.e., mðf Þ ¼ limt!1 1t Rt f ðtÞdt. 0 Consider the following nonautonomous nonlinear equation:

dz ¼ zðpðtÞ  qðtÞzÞ: dt

ð2:2Þ

In our model (1.2), the total population NðtÞ satisfies the following equation:

  dN N ¼ rðtÞN 1  ; dt KðtÞ rðtÞ then pðtÞ ¼ rðtÞ, qðtÞ ¼ KðtÞ . For Eq. (2.2), we have the following results (see [23–26]).

Lemma 2.1. Suppose that assumptions ðH1 Þ, ðH2 Þ hold. Then (a) There is a constant M > 0 such that for any positive solution zðtÞ of (2.2), lim supt!1 zðtÞ 6 M. (b) There exist positive constants m and M such that for any positive solution zðtÞ of (2.2),

m < lim inf zðtÞ 6 lim sup zðtÞ < M: t!1

(c) If

t!1

lim supt!1 pðtÞ qðtÞ

< 1; then for any positive solution zðtÞ of (2.2),

 m  M p p < lim inf zðtÞ 6 lim sup zðtÞ < ; t!1 q q t!1 where

 m p pðtÞ ¼ lim inf ; t!1 q qðtÞ

 M p pðtÞ ¼ lim sup : q qðtÞ t!1

And the constant M in (a) can be choosen by M ¼ lim supt!1 ðpðtÞ Þ < 1: qðtÞ (d) limt!1 ðz1 ðtÞ  z2 ðtÞÞ ¼ 0 for any two positive solutions z1 ðtÞ and z2 ðtÞ of (2.2). (e) When Eq. (2.2) is x periodic, then (2.2) has a unique positive xperiodic solution z ðtÞ, which is globally uniformly attractive. For convenience, we denote

a ¼ sup bðtÞ;

c ¼ sup cðtÞ;

tP0

tP0

d ¼ sup lðtÞ:

Since for our model pðtÞ ¼ rðtÞ, qðtÞ ¼

tP0 rðtÞ , KðtÞ

then the following result is true.

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J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

Lemma 2.2. Suppose that assumptions ðH1 Þ, ðH2 Þ hold. Then any solution ðSðtÞ; IðtÞ; RðtÞÞ of system (1.2) with initial condition (2.1) is nonnegative and uniformly bounded on ½0; þ1Þ. 3. Permanence and extinction In this section, we will discuss the permanence and extinction of the disease in system (1.2), and give sufficient conditions that guarantee the permanence and extinction of the disease. Let N ðtÞ ¼ SðtÞ þ IðtÞ þ RðtÞ with initial value N ð0Þ ¼ Sð0Þ þ Ið0Þ þ Rð0Þ, then N ðtÞ is a solution of Eq. (2.2). Then system (1.2) becomes

8 dIðtÞ ¼ bðtÞðN  ðtÞ  IðtÞ  RðtÞÞIðtÞ  ðlðtÞ þ cðtÞÞIðtÞ; > > < dt dRðtÞ ¼ cðtÞIðtÞ  ðlðtÞ þ dðtÞÞRðtÞ; dt > > : dN ðtÞ ¼ rðtÞð1  N ðtÞÞN ðtÞ: dt

ð3:1Þ

KðtÞ

Denote function

bðt; uÞ ¼ bðtÞu  ðlðtÞ þ cðtÞÞ rðtÞ and z ðtÞ be some fixed solution of equation (2.2) with initial value z ð0Þ > 0 and pðtÞ ¼ rðtÞ, qðtÞ ¼ KðtÞ . Then we first give the permanence theorem. 

Theorem 3.1. Suppose that assumptions, ðH1 Þ, ðH2 Þ hold. Then the following results are equivalent for system (3.1). (a) Infective I are permanent. (b) Infective I are strong persistent. (c) There is a constant k > 0 such that.

R0 ¼ lim inf t!1

Z

tþk

bðs; z ðsÞÞds > 0:

ð3:2Þ

t

Proof 1. Firstly, we prove that the number R0 is independent of the choice of z ðtÞ. In fact, Lemma 2.1 implies that for any  > 0 small enough and any solution zðtÞ of equation (2.2) with initial value zð0Þ > 0, there exists a T > 0 such that as t P T,

z ðtÞ   6 zðtÞ 6 z ðtÞ þ ;

z ðtÞ P m:

Hence,

bðt; z ðtÞ  Þ 6 bðt; zðtÞÞ 6 bðt; z ðtÞ þ Þ: For t P T, we obtain

lim inf t!1

Z

tþk

bðs; z ðsÞ þ Þds 6 R0 þ ka:

t

and

lim inf t!1

Z

tþk

bðs; z ðsÞ  Þds P R0  ka:

t

By the arbitrariness of , we finally obtain

lim inf t!1

Z

tþk

bðs; zðsÞÞds ¼ R0 :

t

This shows that R0 is independent of the choice of z ðtÞ. Therefore,

lim inf t!1

Z

tþk

bðs; N ðsÞÞds > 0

We now prove ðcÞ ) ðaÞ: By assumptions ðH1 Þ, ðH2 Þ and (3.3), we can choose small enough positive constants there exist T 1 > 0 and g1 > 0 satisfying

Z

tþx2

t

Z

t

and

ð3:3Þ

t

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh < g1 ;

1 , 2 , then ð3:4Þ

tþk

bðs; N ðsÞ  k1  2 Þds > g1

ð3:5Þ

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J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

N ðtÞ  k1  2 P m;

N  ðtÞ 6 M

ð3:6Þ

for all t P T 1 , where k ¼ 1 þ cx2 . Firstly, we will prove

lim sup IðtÞ > 1

ð3:7Þ

t!1

for any solution of (3.1). Suppose that (3.7) is not true, then there exist a solution ðIðtÞ; RðtÞ; N  ðtÞÞ of (3.1) and T 2 > T 1 such that IðtÞ 6 1 for all t P T 2 . If RðtÞ P 2 for all t P T 2 , then from the second equation of system (3.1), we have

RðtÞ  RðT 2 Þ ¼

Z

t

cðhÞIðhÞ  ðlðhÞ þ dðhÞÞRðhÞdh 6

T2

Z

t

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh

T2

for all t P T 2 . By (3.4), it follows that RðtÞ ! 1 as t ! 1. This is a contradiction. Hence, there is a Rðs1 Þ < 2 . In the following, we will prove

s1 P T 2 such that

RðtÞ 6 2 þ cx2 1

ð3:8Þ

for all t P s1 . If it is not true, then there is a s2 > s1 satisfying Rðs2 Þ > 2 þ cx2 1 . Hence, there must be a s3 2 ðs1 ; s2 Þ such that Rðs3 Þ ¼ 2 and RðtÞ > 2 for all t 2 ðs3 ; s2 Þ. Choose an integer p P 0 such that s2 2 ½s3 þ p x2 ; s3 þ ðp þ 1Þx2 Þ. Integrating the second equation of system (3.1) from s3 to s2 , and using (3.4), we obtain

2 þ cx2 1 < Rðs2 Þ ¼ Rðs3 Þ þ

Z s2

cðhÞIðhÞ  ðlðhÞ þ dðhÞÞRðhÞdh 6 2 þ

Z s2

s3

¼ 2 þ

j¼p Z s3 þjx2 X j¼1

6 2 þ

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh þ

s3 þðj1Þx2

Z s2

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh

s3

Z s2

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh

s3 þpx2

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh 6 2 þ

s3 þpx2

Z s2

cðhÞ1 dh 6 2 þ cx2 1 :

s3 þpx2

This is a contradiction. Hence, (3.8) is valid. From this, we conclude that there exists a T 3 > T 2 such that (3.8) is true for all t P T3. For t P T 3 , integrating the first equation in (3.1) from T 3 to t, we have

IðtÞ P IðT 3 Þ exp

Z

t

 bðs; N ðsÞ  k1  2 Þds :

T3

From (3.5), we have that lim supt!1 IðtÞ ¼ 1. This contradicts with the boundedness of IðtÞ. From this contradiction, we finally conclude that lim supt!1 IðtÞ > 1 . Secondly, we will prove that there is a constant v 1 > 0 such that

lim inf IðtÞ P v 1 :

ð3:9Þ

t!1

From (3.4)–(3.6) and (H2 ), we have that there exist T P T 1 , P > 0 and g > 0 such that

Z

tþa

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh < M;

ð3:10Þ

bðs; N ðsÞ  k1  2 Þds > g

ð3:11Þ

t

Z

tþa

t

for every a P P, t P T. Choose an integer K 0 > 0 large enough such that

exp

Z

tþK 0 Pþa0

 Z bðs; N ðsÞ  k1  2 Þds P eK 0 g exp

t

tþK 0 Pþa0

 bðs; N ðsÞ  k1  2 Þds > 1

ð3:12Þ

tþK 0 P

for any a0 2 ½0; PÞ. From (3.7), for any t 0 P 0, we claim that it is impossible that IðtÞ 6 1 for all t P t0 . For this claim, we will discuss the following two possibilities. (i) IðtÞ P 1 for all large t; (ii) IðtÞ oscillates about 1 for all large t. Finally, we will show that IðtÞ P 1 eðcþdÞðK 0 þ2ÞP ,v 1 as t is large enough. Clearly, we only need to consider case ðiiÞ. Let t1 and t 2 be sufficiently large such that

Iðt 1 Þ ¼ Iðt 2 Þ ¼ 1 ; IðtÞ < 1

for all t 2 ðt1 ; t 2 Þ:

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J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

If t2  t 1 6 ðK 0 þ 2ÞP, then

_ ¼ bðtÞðN  ðtÞ  IðtÞ  RðtÞÞIðtÞ  ðlðtÞ þ cðtÞÞIðtÞ P ðc þ dÞIðtÞ and Iðt1 Þ ¼ 1 ; IðtÞ which implies IðtÞ P 1 eðcþdÞðK 0 þ2ÞP for all t 2 ½t1 ; t2 . If t2  t 1 > ðK 0 þ 2ÞP, then it is clear that IðtÞ P 1 eðcþdÞðK 0 þ2ÞP for all t 2 ½t1 ; t1 þ ðK 0 þ 2ÞP. If RðtÞ P 2 for all t 2 ½t1 ; t1 þ P, then

Rðt 1 þ PÞ ¼ Rðt 1 Þ þ

Z

t 1 þP

cðhÞIðhÞ  ðlðhÞ þ dðhÞÞRðhÞdh 6 M þ

Z

t1

t 1 þP

cðhÞ1  ðlðhÞ þ dðhÞÞ2 dh < 0:

ð3:13Þ

t1

This is a contradiction. Hence, there is a t 2 ½t 1 ; t 1 þ P such that RðtÞ < 2 . From (3.8), we can obtain

for all t 2 ½ t; t 2 :

RðtÞ 6 2 þ cx2 1

ð3:14Þ

Obviously, as t 2 ½t 1 ; t 1 þ 2P,

IðtÞ P 1 eðcþdÞ2P ,v 2 > v 1 :

ð3:15Þ

We claim that IðtÞ P v 1 for all t 2 ½t 1 þ ðK 0 þ 2ÞP; t2 . If it is not true, then there is a T 0 P 0 such that Iðt 1 þ ðK 0 þ 2ÞP þ T 0 Þ ¼ v 1 and IðtÞ P v 1 on ½t1 ; t1 þ ðK 0 þ 2ÞP þ T 0 . Let t0 ¼ t 1 þ ðK 0 þ 2ÞP þ T 0 , T 0 ¼ nP þ b0 , where n P 0 is an integer, b0 2 ½0; PÞ. Then the derivative of IðtÞ along solutions of (3.1) satisfies

_ ¼ bðtÞðN  ðtÞ  IðtÞ  RðtÞÞIðtÞ  ðlðtÞ þ cðtÞÞIðtÞ P bðt; N  ðtÞ  k1  2 ÞIðtÞ IðtÞ for all t 2 ½t1 þ 2P; t 2 . Integrating the above inequality from t1 þ 2P to t 0 , we further have

Iðt0 Þ P Iðt 1 þ 2PÞ exp P v 2 exp

Z

t0

bðt; N ðtÞ  k1  2 Þdt



t 1 þ2P

Z

t 1 þ2PþK 0 Pþb0



bðt; N ðtÞ  k1  2 Þdt



exp

t 1 þ2P

Z

t 1 þðK 0 þ2ÞPþb0 þnP

! 

bðt; N ðtÞ  k1  2 Þdt

> v 2 eng > v 1 ;

t 1 þðK 0 þ2ÞPþb0

a contradiction to Iðt 0 Þ ¼ v 1 . So IðtÞ P v 1 is valid for all t 2 ½t1 ; t2 . Hence, we have

lim inf IðtÞ P v 1 > 0: t!þ1

According to Lemma 2.1, we have that the infective I are permanent. Next, we will prove that ðbÞ ) ðcÞ, which can be obtained by the proof of Theorem 1 in [21]. Since ðaÞ ) ðbÞ is obvious, then the proof is complete. h Remark 3.1. If in system (1.2), bðtÞ, bðtÞ, lðtÞ, rðtÞ, dðtÞ, KðtÞ > 0 and cðtÞ are replaced by nonnegative constants, then system (1.2) becomes an autonomous SIRS system. The basic reproduction number of the resulting system is given by

R0 ¼

bK

lþc

:

It is easy to verify that if R0 > 1, the corresponding autonomous system of system (1.2) has one positive equilibrium which is globally asymptotically stable, and thus the disease is permanent. It is obvious that if lim inf t!1 ½bðtÞN  ðtÞ  ðlðtÞ þ cðtÞÞ > 0; then the infective I is permanent. Now, we are about to give the extinction theorem for system (3.1). Theorem 3.2. Suppose that assumptions ðH1 Þ; ðH2 Þ hold. If there is a constant k > 0 such that

R1 ¼ lim sup t!1

Z

tþk

½bðhÞN  ðhÞ  ðlðhÞ þ cðhÞÞdh 6 0

ð3:16Þ

t

or

R2 ¼ lim sup t!1

1 t

Z

t

½bðhÞN  ðhÞ  ðlðhÞ þ cðhÞÞdh < 0 0

then infective I in system (3.1) become extinct, i.e., limt!1 IðtÞ ¼ 0. Proof 2. From assumption (H2 ), we can choose g > 0 small enough and T 1 > 0 large enough such that

Z

tþx1

bðhÞdh P g

t

for all t P T 1 .

ð3:17Þ

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J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

For any constant 0 <  < 1, we set

Z

0 ¼ min

n

kg 1 2x1 ; 2

o

g > 0. If (3.16) holds, then there exists T 2 P T 1 such that

tþk

bðhÞN  ðhÞ  ðlðhÞ þ cðhÞÞdh 6 0

t

for all t P T 2 . Choose an integer n0 satisfying

Z

tþk0

2 x1 k

bðhÞN ðhÞ  ðlðhÞ þ cðhÞÞ  bðhÞdh 6

t

6 n0 < 2xk 1 þ 1. Set k0 ¼ n0 k, then

Z

tþn0 k

bðhÞN ðhÞ  ðlðhÞ þ cðhÞÞdh 

t

Z

tþ2x1

bðhÞdh 6 n0 0  2g

t

1 6  g,  g0 < 0: 2

ð3:18Þ

From (3.1), we obtain

I0 ðtÞ ¼ bðtÞðN ðtÞ  IðtÞ  RðtÞÞIðtÞ  ðlðtÞ þ cðtÞÞIðtÞ 6 bðtÞðN ðtÞ  IðtÞÞIðtÞ  ðlðtÞ þ cðtÞÞIðtÞ:

ð3:19Þ

If IðtÞ P  for all t P T 2 , then from (3.19), we obtain

IðtÞ 6 IðT 2 Þ exp

Z

t

½bðsÞN ðsÞ  ðlðsÞ þ cðsÞÞ  bðsÞds:

ð3:20Þ

T2

From (3.18), it follows that IðtÞ ! 0 as t ! 1. This is a contradiction with IðtÞ P . Hence, there must be a t 1 P T 2 such that Iðt1 Þ < . Let NðÞ ¼ suptPT 2 fjbðtÞN  ðtÞ  ðlðtÞ þ cðtÞÞj þ bðtÞg, then NðÞ is bounded for each  2 ð0; 1Þ. We claim that

IðtÞ 6  expðNðÞk0 Þ

ð3:21Þ

for all t P t 1 . Otherwise, then there exists a t2 > t1 , such that Iðt 2 Þ >  expðNðÞk0 Þ. Hence, there exists a t3 2 ðt 1 ; t2 Þ such that Iðt3 Þ ¼  and IðtÞ >  for all t 2 ðt3 ; t2 Þ. Let p be a nonnegative integer such that t 2 2 ðt3 þ pk0 ; t3 þ ðp þ 1Þk0 , then from (3.18) and (3.19), we have

 expðNðÞk0 Þ < Iðt2 Þ 6 Iðt3 Þ exp

Z

t2

bðtÞN ðtÞ  ðlðtÞ þ cðtÞÞ  bðtÞdt 6 epg0 expðNðÞk0 Þ 6  expðNðÞk0 Þ;

t3

which leads to a contradiction. Hence, inequality (3.21) holds. Furthermore, since  can be arbitrarily small, we conclude that IðtÞ ! 0 as t ! 1. Suppose that (3.17) holds. Then there exist d > 0 and T 0 > 0 such that

1 t

Z

t

bðhÞN  ðhÞ  ðlðhÞ þ cðhÞÞdh < d

ð3:22Þ

0

for all t P T 0 . From (3.20) we directly obtain

IðtÞ 6 IðT 0 Þ exp

Z

t

½bðsÞN ðsÞ  ðlðsÞ þ cðsÞÞds

T2

for all t P T 0 . From (3.22), IðtÞ ! 0 as t ! 1. This completes the proof of Theorem 3.2. h Remark 3.2. For the corresponding autonomous system of system (3.1), the conditions (3.16) and (3.17) reduces to

R0 ¼

bK

lþc

6 1 and R0 ¼

bK

lþc

< 1;

respectively. If R0 6 1, the corresponding autonomous system has one disease-free equilibrium, which is globally asymptotically stable, i.e., the disease will go to extinct when R0 6 1. Also it is easy to proof that if lim supt!1 ½bðtÞN  ðtÞ  ðlðtÞ þ cðtÞÞ < 0; then the infective I go extinct. 4. Some corollaries In this section, we will give some corollaries according to the results in Theorems 3.1 and 3.2. Corollary 4.1. Suppose that assumptions, (H1 ), (H2 ) hold, then the following results are true. (i) If there is a constant k > 0 such that

lim inf t!1

Z

tþk

bðs; K m Þds > 0;

t

then the infective I of system (3.1) are permanent. (ii) If there is a constant k > 0 such that

lim sup t!1

Z t

tþk

bðs; K M Þds 6 0;

J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

or

lim sup t!1

1 t

Z

t

873

bðs; K M Þds < 0;

0

then the infective I of system (3.1) will be extinct. Corollary 4.2. Suppose that assumptions (H1 ), (H2 ) hold, then the disease is permanent if R01 > 1, and the disease is extinct if R 01 6 1, where

R01 ¼

ðbz Þ0 ðl þ cÞ

0

;

Here,

ðbz Þ0 ¼ lim inf t!1

R01 ¼ Z t

ðl þ cÞ0 ¼ lim inf

tþk1

Z

t!1

ðbz Þ0 : ðl þ cÞ0

bðsÞz ðsÞds;

ðbz Þ0 ¼ lim sup t!1

tþk2

ðlðsÞ þ cðsÞÞds;

Z

tþk2

ðl þ cÞ0 ¼ lim sup t!1

t

bðsÞz ðsÞds;

t

Z

tþk1

ðlðsÞ þ cðsÞÞds:

t

z ðtÞ be some fixed solution of equation (2.2) with initial value z ð0Þ > 0, k i > 0 ði ¼ 1; 2Þ are some constants. Corollary 4.3. Suppose that assumptions (H1 ), (H2 ) hold, then the disease is permanent if R 02 > 1, and the disease is extinct if R 02 6 1, where

R02 ¼ Here,

ðbz Þ0 ðl þ cÞ

0

;

R02 ¼

Z 1 t bðs þ sÞz ðs þ sÞds; ðbz Þ0 ¼ lim sup bðsÞz ðsÞds; t;s!1 t 0 t!1 0 Z Z 1 t 1 t ðlðsÞ þ cðsÞÞds; ðl þ cÞ0 ¼ lim sup ðlðs þ sÞ þ cðs þ sÞÞds: ðl þ cÞ0 ¼ lim inf t!1 t 0 t 0 t;s!1

ðbz Þ0 ¼ lim inf

1 t

Z

ðbz Þ0 : ðl þ cÞ0

t

z ðtÞ be some fixed solution of equation (2.2) with initial value z ð0Þ > 0. Corollary 4.4. When system (3.1) is x-periodic and assumptions (H1 ), (H2 ) hold, then the infective I are permanent provided that

R03 ¼

ðbz Þ ðl þ cÞ

> 1;

and the disease is extinct if R03 6 1. Here, z ðtÞ is the globally uniformly attractive nonnegative x-periodic solution of equation (2.2). Corollary 4.5. When system (3.1) is almost periodic and assumptions (H1 ), (H2 ) hold, then the infective I are permanent provided that

R04 ¼

mðbz Þ > 1; mðl þ cÞ

and the disease is extinct if R04 6 1. Here, z ðtÞ is the globally uniformly attractive nonnegative almost periodic solution of equation (2.2). When system (3.1) reduces to x-periodic system, i.e., rðtÞ, KðtÞ > 0, bðtÞ, lðtÞ, rðtÞ, cðtÞ, and dðtÞ are all nonnegative continuous periodic function with period x > 0, we also have the global stability results. Theorem 4.1. For x-periodic system (3.1), we assume that (H1 ), (H2 ) hold. If the following conditions ðiÞ and ðiiÞ hold, then there is one positive x-periodic solution, which is globally asymptotically stable. 

(i) R03 ¼ lbzþc > 1, (ii) if there are positive constants qi , K i ði ¼ 1; 2Þ and di ði ¼ 1; 2; 3; 4Þ and nonnegative integrable functions ai ðtÞ ði ¼ 1; 2Þ Rt defined on ½0; 1Þ, satisfying d1 6 d4 , d3 6 d2 and s ai ðsÞds P K i þ qi ðt  sÞ for all t P s P 0: And also

d2 cðtÞ  d1 bðtÞ < a1 ðtÞ; d4 bðtÞ  d3 ðlðtÞ þ dðtÞÞ < a2 ðtÞ; hold for all t P 0. Here, z is the globally uniformly attractive nonnegative x-periodic solution of equation (2.2).

ð4:1Þ

874

J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

Proof 3. From ðiÞ, we see infective are permanent, then by (3.1) and Lemma 2.1, we conclude RðtÞ and NðtÞ are also permanent. Hence, from [26,27], we know system (3.1) has one positive x-periodic solution. Suppose ðI ðtÞ; R ðtÞ; N  ðtÞÞ is the xperiodic solution. Let IðtÞ; RðtÞ; NðtÞ be a positive solution of (3.1). From Lemma 2.1, we obtain NðtÞ ! N  ðtÞ as t ! 1, and there are positive constants r1 and r 2 such that

r1 6 IðtÞ; RðtÞ 6 r 2

for all t P 0:

ð4:2Þ

Choose the Liapunov function as follows:

VðtÞ ¼ d1 j ln IðtÞ  ln I ðtÞj þ d2 jRðtÞ  R ðtÞj: Calculating the upper right derivative of VðtÞ yields

Dþ VðtÞ 6 d1 bðtÞjN  N  j þ ðd2 cðtÞ  d1 bðtÞÞjI  I j þ ðd1 bðtÞ  d2 ðlðtÞ þ dðtÞÞÞjR  R j 6 d1 bðtÞjN  N  j  a1 ðtÞjI  I j  a2 ðtÞjR  R j; 6 d1 bðtÞjN  N j  kðtÞVðtÞ; 1 a2 ðtÞ where kðtÞ ¼ minfa1 dðtÞr ; d2 g P 0. Applying the differential inequality theorem and the variation of constants formula of 1

solutions of first-order linear differential equation, we have 

VðtÞ 6 e

Rt 0

kðsÞds

Z

t

d1 bðsÞjNðsÞ  N ðsÞje

Rs 0

kðqÞdq

ds þ Vð0Þ

 ð4:3Þ

0

for all t P 0: Since NðtÞ ! N  ðtÞ as t ! 1; by the properties of function ai ðtÞ, it is not hard to prove that VðtÞ ! 0 as t ! 1. That shows IðtÞ ! I ðtÞ, RðtÞ ! R ðtÞ as t ! 1; i.e., ðIðtÞ; RðtÞ; NðtÞÞ is globally asymptotically stable. The proof is complete. h 5. Numerical simulations In this section, we will give some examples to verify our theoretical results. We will consider a simplified nonautonomous SIRS model given by

8 dS rN ¼ bðtÞSI  lðtÞS þ ðbðtÞ  KðtÞ ÞN þ dR; > dt > > > < dI ¼ bðtÞSI  lðtÞI  cðtÞI; dt > dR > ¼ cðtÞI  ðlðtÞ þ dÞR: > > : dt NðtÞ ¼ SðtÞ þ IðtÞ þ RðtÞ:

ð5:1Þ

Corresponding auxiliary system is

  dz z ¼ rz 1  : dt KðtÞ

ð5:2Þ

In system (5.1), let bðtÞ ¼ 1 þ 0:5 sinð2ptÞ, bðtÞ ¼ 0:6 þ 0:5 sinðptÞ, cðtÞ ¼ 0:5 þ 0:3 sinðptÞ, d ¼ 0:2, r ¼ 0:4, 10 lðtÞ ¼ 0:6 þ 0:5 sinð2ptÞ and KðtÞ ¼ 2þsinð ptÞ. It is easy to verify that assumptions (H1 ), (H2 ) hold. Thus from Lemma 2.1, we see that system (5.2) has a globally asymptotically stable positive periodic solution z ðtÞ with period 2, given by

e



z ðtÞ ¼ 1 z0

þ

Rt 0

rðxÞdx

Rt

Rx

rðxÞ e 0 KðxÞ

0

rðsÞds

; dx

R2 rðsÞds where z0 ¼ R e 0 R x 1 . By (3.2), we obtain R0 ¼ 3:792 > 0. Then by Theorem 3.1, the disease of system (5.1) is permanent. 2 rðxÞ rðsÞds e

0 KðxÞ

0

dx

Direct computation gives R03 ¼ 2:72 > 1, if we choose d1 ¼ d3 ¼ d4 ¼ 1, d2 ¼ 53, qi ¼ K i ¼ 1; i ¼ 1; 2, and ai ðtÞ ¼ 1 for all t P 0, Rt i ¼ 1; 2, then s ai ðsÞds P K i þ qi ðt  sÞ for all t P s P 0: It is easy to verify that the first equation in (4.1) does not hold true. Using the program ode45 of Matlab for solving, the numerical simulation shows that system (5.1) has a unique 2-periodic solution which is globally asymptotically stable (see Fig. 1), although the condition (4.1) is not true. This may imply that sometimes the condition ðiiÞ in Theorem 4.1 is redundant, and R03 > 1 alone is sufficient for the globally asymptotically stability of the periodic solution. But if we let bðtÞ ¼ 0:2 þ 0:1 sinðptÞ, and fix the other parameter values as those in Fig. 1, also the assumptions (H1 ), (H2 ) hold true, then from (3.16), we have R1 ¼ 0:2 < 0; and from Corollary 4.1 we obtain that R03 ¼ 10=11 < 1, thus from Theorem 3.2 and Corollary 4.4, the disease of system (5.1) is extinct, corresponding simulations are depicted in Fig.2. For childhood diseases, a child becomes infected after contact with another infective. This contact rate fluctuates with the seasons and can be approximated in several ways. We choose the transmission rate, bðtÞ ¼ b0 ð1 þ g cosð2ptÞÞ; where b0 is the baseline transmission parameter, 0 6 g < 1 measures the amplitude of the seasonal variation in transmission. We take the parameters in system (1.2) as follows:

875

J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

b0 ¼ 44; lðtÞ ¼ rðtÞ ¼ 0:00013; bðtÞ ¼ 0:00026; KðtÞ ¼ 10; dðtÞ ¼ 1:8; cðtÞ ¼ 36: Clearly, for any g 2 ½0; 1Þ, the assumptions (H1 ), (H2 ) hold for system (1.2), and note that we can choose the constants xi ¼ 1 ði ¼ 1; 2; 3; 4Þ in (H2 ). Again we use the program ode45 of Matlab for obtaining numerical simulations that can be seen in Fig. 3 ðg ¼ 0:1Þ and 4 ðg ¼ 0:36Þ. Although system (1.2) is 1-periodic, 2-periodic solution can be found (see Fig. 4). S(t) 2.5 1.5 2

I(t) R(t)

1

1.5

0.5 1 0 R(t)

0.5

2 1.5

0

1

I(t) 0

20

40

0.5

60

1

1.5

2.5

2 S(t)

t Fig. 1. The left figure gives the graphes of S, I, R as functions of time for system (5.1). The right figure gives the ðS; I; RÞ phase-space plot of system (5.1). Here, R0 ¼ 3:792 > 0, and the disease is permanent. System (5.1) has a periodic solution.

5 S(t)

4.5 4

0.25 0.2

3

0.15

R(t)

3.5

2.5

0.1

2

0.05

1.5

0

1

0.8

0.5 0

I(t) 0

R(t) 20

40

0.6

0.4

I(t)

60

0.2 1

2

4

3

5

S(t)

t Fig. 2. The left figure gives the graphes of S, I, R as functions of time for system (5.1). The right figure gives the ðS; I; RÞ phase-space plot of system (5.1). Here, R1 ¼ 0:2 < 0, and the disease is extinct.

1−Periodic Solution

1

S(t)

0.9 0.8 0.7 0.6 0.5 0.4 0.3

R(t)

0.2 0.1 0

I(t) 0

5

10 t

15

20

Fig. 3. The existence of a 1-periodic solution of system (1.2) with initial condition (0.56, 0.04, 0.4).

876

J. Liu, T. Zhang / Applied Mathematical Modelling 34 (2010) 866–877

2−Periodic Solution

1 0.9

S(t)

0.8 0.7 0.6 0.5 0.4

R(t)

0.3 0.2 0.1 0

I(t) 0

2

4

t

6

8

10

Fig. 4. The existence of a 2-periodic solution of system (1.2) with initial condition (0.56, 0.04, 0.4).

6. Conclusions Classical epidemic models with density dependent birth rate are studied in the situation where all coefficients depend on time. For these nonautonomous systems threshold conditions for the permanence of the infection and for the extinction of the disease are established. The condition for permanence has the form of a lim inf condition for some time-dependent net infection rate while the condition for extinction assumes the form of a lim sup condition. Hence, in the general case the main result are not threshold criteria in a strict sense. However, in the periodic and almost periodic cases, the conditions merge into a sharp threshold criterion and a reproduction number can be defined. Also for the periodic system, global stability of periodic solution is obtained. We construct two examples to indicate our main results. In the first example which consider the case that the disease is permanent, and there is a unique periodic solution which is globally asymptotically stable. We have also shown that when the contact rate changes, the disease dies out. In the second example, we suppose that the rate of contact bðtÞ is a seasonally forced function, we have shown that as the amplitude of the seasonal variation in transmission g varies, both 1-periodic solution and 2-periodic solution can occur, even though system (1.2) is 1-periodic. It will be interesting to consider the global stability of the model, this seems to be a difficult problem since the model is nonautonomous. We leave this for future investigations. Acknowledgement J. Liu’s research was partially supported by the National Natural Sciences Foundation of People’s Republic of China (Grant No. 10701062). T. Zhang’s research was supported by the Scientific Research Programmes of Colleges in Xinjiang (XJEDU2008S10). We would like to thank the editor and referees for their careful reading and valuable comments which led to an improvement of our original manuscript. References [1] K. Cooke, P. van den Driessche, X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999) 332–352. [2] H.W. Hethcote, P. van den Driessche, Two SIS epidemiologic models with delays, J. Math. Biol. 40 (2000) 3–26. [3] Z. Ma, J. Liu, J. Li, Stability analysis for differential infectivity epidemic models, Nonlinear Anal.: Real World Appl. 4 (2003) 841–856. [4] J. Liu, T. Zhang, Bifurcation analysis of an SIS epidemic model with nonlinear birth rate, Chaos, Solitons & Fractals 40 (2009) 1091–1099. [5] Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. Math. Appl. 56 (2008) 3118–3129. [6] N. Yoshida, T. Hara, Global stability of a delayed SIR epidemic model with density dependent birth and death rates, J. Comput. Appl. Math. 201 (2007) 339–347. [7] H.R. Thieme, Uniform weak implies uniform strong persistence also for non-autonomous semiflows, Proc. Am. Math. Soc. 127 (1999) 2395–2403. [8] T. Zhang, Z. Teng, Permanence and extinction for a nonautonomous SIRS epidemic model with time delay, Appl. Math. Model. 33 (2009) 1058–1071. [9] C. Egami, Positive periodic solutions of nonautonomous delay competitive systems with weak Allee effect, Nonlinear Anal.: Real World Appl. 10 (2009) 494–505. [10] H.T. Banks, V.A. Bokil, S. Hu, Monotone approximation for a nonlinear size and class age structured epidemic model, Nonlinear Anal.: Real World Appl. 8 (2007) 834–852. [11] Z. Teng, Z. Li, H. Jiang, Permanence criteria in nonautonomous predator-prey Kolmogorov systems and its applications, Dyn. Syst. 19 (2004) 171–194. [12] G. Herzog, R. Redheffer, Nonautonomous SEIRS and Thron models for epidemiology and cell biology, Nonlinear Anal.: Real World Appl. 5 (2004) 33–44. [13] S.F. Dowell, Seasonal variation in host susceptibility and cycles of certain infectious diseases, Emerg. Infect. Dis. 7 (2001) 369–374. [14] W. Wang, X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat. 20 (2008) 699–717. [15] F. Zhang, X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl. 325 (2007) 496–516. [16] H.R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci. 166 (2000) 173–201. [17] J. Zhang, J. Lou, Z. Ma, J. Wu, A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China, Appl. Math. Comput. 162 (2005) 909–924. [18] M.Y. Li, J.R. Graef, L. Wang, J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci. 160 (1999) 191–213.

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