XU Ming-xing,TENG Zhi-dong,ZHANG Long

(College of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China)

Abstract: A nonautonomous SIRS epidemic model with density dependence is proposed and studied.Threshold valuesfor the permanence and extinction of the disease are established.It is shown that the disease is extinct ifand the disease is permanent if As applications,the periodic and almost periodic models are discussed.The basic reproductive numbers R0 are obtained.It is shown that the disease is extinct if R0≤1 and the disease is permanent if R0>1.

Key words:Nonautonomous SIRS model;Threshold valuer;Basic reproductive number;Permanence;Extinction

0 Introduction

As is well known,the research on the epidemic dynamical models is a very important subject in the mathematical theory of epidemiology.Various epidemic dynamical models have been proposed and analyzed,and great progress has been achieved[1–5].Particularly,the autonomous epidemic models have been investigated extensively in recent years.Many important and interesting results are established.These results come down to the computation of threshold value and basic reproduction number,the local and global stability of the disease-free equilibrium and endemic equilibrium,the existence of periotic orbits,the persistence and extinction of the disease,the existence of backward bifurcation,etc.And many results have been applied successfully to solve the practical medicinal problems.

However,the nonautonomous phenomenon often occurs in many realistic epidemic models.It will be more identical with the sound background as we consider long-term dynamical behavior of an epidemic model,the parameters of the model usually will change with time.Therefore,the research on the nonautonomous epidemic models also plays very important role in epidemiology.We see that there have been some research works on the nonautonomous epidemic models[6,7].Particularly,the works in allusion with density dependent epidemic models can be found in[8–10]and the references cited therein.

In this paper we consider the following nonautonomous SIRS epidemic model,

Our purpose is to study the extinction and permanence of the disease for model(1).We will give some new threshold valuesWe will prove that the disease is extinct ifand permanent ifWhen model(1)degenerates into periodic or almost periodic case,the basic reproduction numberR0is obtained,andR0acts as a sharp threshold for the extinction and permanence of the disease.

This paper is organized as follows.Section 1 contains some basic preliminaries including hypotheses,some lammas,the definition of extinction,permanence of disease.The extinction of the disease is settled in Section 2.Section 3 deals with the permanence of the disease.In Section 4,we conclude some corollaries.

1 Preliminaries

In model(1),S(t),I(t)andR(t)denote the size of susceptible,infectious and recovered at timet,respectively,N(t)=S(t)+I(t)+R(t),β(t)denotes the disease transmission coefficient,b(t)andd(t)are the natural birth and death rates at timet,γ(t)and δ(t)are the instantaneous per capita rates of leaving the infection stage and removed stage at timet.f(t)=b(t)−d(t)is the intrinsic growth rate at timet,andK(t)is the carrying capacity of the population at timet,wherea∈[0,1].

For a bounded functionh(t)defined onR+=[0,+∞),we denoteThe initial condition for any solution of model(1)is given in the following form

InfectiveIis said to be extinct,if limt→∞I(t)=0 for any solution(S(t),I(t),R(t))of model(1)with initial value(2).InfectiveIis said to be permanent,if there are positive constants ν1,ν2such that

for any solution(S(t),I(t),R(t))of model(1)with initial value(2).

For model(1),we introduce the following assumptions.

(H1)Functionsb(t),K(t),β(t),d(t),γ(t),f(t),δ(t)andare continuous bounded and nonnegative onR+,andK(t)>0 for allt≥0.

(H2)There are positive constants ω1,ω2,ω3,ω4such that

Remark 1When model(1)degenerates into ω-periodic case,that is, β(t), γ(t),b(t),K(t),d(t),f(t),δ(t)are nonnegative continuous periodic functions with period ω>0,then(H2)degenerates into

Here,for any ω-periodic continuous function

When model(1)degenerates into almost periodic system,that is, β(t), γ(t),b(t),K(t),d(t),f(t)and δ(t)are nonnegative continuous almost periodic functions,then(H2)degenerates into

Here,for any almost periodic continuous function

Consider the well-known nonautonomous logistic equation

wherep(t)andq(t)are bounded continuous functions onR+and there is a constant ρ >0 such that lim0.We have the following results[11].

Lemma 1(a).There is a constantM0>0 such that for any positive solutionz(t)of equation(3),limsupt→∞z(t)≤M0.

(b).If there is a constant γ >0 such that limthen there are positive constantsm0andM0such that for any positive solutionz(t)of equation(3),

(c).If there is a constant η >0 such that limthen for any positive solutionz(t)of equation

(d).Each fixed positive solutionz(t)of equation(3)is globally uniformly attractive.

(e).When equation(3)is ω-periodic,then equation(3)has a unique globally asymptotically stable positive ω-periodic solution.

(f).When equation(3)is ω-periodic,then for any positive solutionz(t)of equation(3),limt→∞z(t)=0.

(g).When equation(3)is almost periodic,m(p)>0 andm(q)>0,then equation(3)has a unique globally asymptotically stable positive almost periodic solution.

(h).When equation(3)is almost periodic,m(p)≤0 andm(q)>0,then for any positive solutionz(t)of equation(3),limt→∞z(t)=0.

Next,we consider the following nonautonomous linear equation

We have the following result.

Lemma 2If there are constants η >0 and γ >0 such that

then any solutions of equation(4)with positive initial value are ultimately bounded.

Lemma 3Supposea(t)andb(t)are continuous bounded functions defined onR+,there is a constant ω5>0 such thatThen for any constant ε1>0 there exist ε2>0,T>0 and η >0 such that

for allt≥T,and for any constantH>0,there is aP>0 such that for any α≥Pandt≥0

LetN(t)=S(t)+I(t)+R(t),then model(1)becomes into the following form

Lemma 4Suppose that(H1),(H2)hold.Then any solution(N(t),I(t),R(t))of model(5)with initial condition(2)is nonnegative and bounded onR+.

The proofs of Lemmas 1-4 are not hard,here we hence omit them.

2 Extinction

In model(1),total populationN(t)satisfies the following equation

From Lemma 1,there are positive constantsmandMsuch that for any positive solutionN(t)of equation(6),m≤liminft→∞N(t)

whereN∗(t)be some fixed solution of equation(6)with initial valueN∗(0)>0.We first give the following extinction theorem.

Theorem 1Suppose that(H1)and(H2)hold,if there is a constant λ>0 such that

then infectiveIin model(1)is extinct.

ProofFirstly,we prove thatis independent of the choice ofN∗(t).In fact,Lemma 1(d)implies that for any ε>0 small enough and any solutionN(t)of equation(6)with initial valueN(0)>0 there exists aT≥0 such that ast≥T,N∗(t)−ε≤N(t)≤N∗(t)+ε.Hence,for allt≥T,

and

Thus,we obtainBy the arbitrariness of ε,we finally obtain

Iffrom the first equation of model(5),we have

From(7),by Lemma 1(c),using the comparison theorem we can obtain that limt→∞I(t)=0.

From the first equation of model(5),we have

Integrating from 0 tot,we have

From(8)and(9)we finally have limt→∞I(t)=0.This completes the proof.

3 Permanence

Theorem 2Suppose that(H1)and(H2)hold,if there is a constant λ>0 such that

then infectiveIin model(1)is permanent.

ProofFirstly,we can provefor any solutionN(t)of equation(6)withN(0)>0.LetM1andM2are I(t)and R(t)’s limsup.

From(H1),(H2)and Lemma 3,choose constants ε1>0,ε2>0,T1>0 and η2>0 such that

for all

Firstly,we claim that

If it is not true,then there existsT2>T1such thatI(t)< ε1for allt≥T2.Suppose thatR(t)≥ ε2fort≥T2,then from the second equation of model(5),we have

for allt≥T2.From(10)it follows limt→∞R(t)= −∞,which is a contradiction.Hence,there is a τ1≥T2such thatR(τ1)< ε2.Now,we prove

fort≥ τ1.If it is not true,then there is a τ2> τ1satisfyingR(τ2)> ε2+γΛω2ε1.Hence,there is a τ3∈ (τ1,τ2)such thatR(τ3)= ε2andR(t)> ε2fort∈ (τ3,τ2).Choose an integerp≥ 0 such that τ2∈ [τ3+pω2,τ3+(p+1)ω2).Integrating the second equation of model(5)from τ3to τ2,and using(10),we obtain

This is a contradiction.Hence,(14)is true.

From the first equation of model(5),we have

for allt≥ τ1.Integrating from τ1tot,we have

From(11),it follows limt→∞I(t)=+∞.This is a contradiction.Hence,(13)is true.

Secondly,we prove that there is a constant ν1>0 such that

From(10)-(12),Lemma 3 and(H2),there existP>0 and η3>0 such that

for any α ≥Pandt≥ 0.Choose an integerC>0 large enough such that

for anyt≥ 0 and ξ0∈ [0,P).From(13),there are only two possibilities as follows

(1)I(t)≥ ε1for all larget;(2)I(t)oscillates about ε1for all larget.

We will show thatastis large sufficiently,whereEvidently,we only need to consider case(2).Lett1andt2be sufficiently large such that

Ift2−t1≤(C+2)P,then from the first equation of model(5)and using Lemma 4,we obtain

which impliesI(t)≥ε1exp(−L(C+2)P)for allt∈[t1,t2].

Ift2−t1>(C+2)P,then from(19),we haveI(t)≥ ε1exp(−L(C+2)P)for allt∈ [t1,t1+(C+2)P].IfR(t)≥ ε2for allt∈[t1,t1+P],then integrating the second equation in model(5)fromt1tot1+P,and using(16),we obtain

This is a contradiction.Hence,there is at∗∈ [t1,t1+P]such thatR(t∗)< ε2.From(14),we can obtain

for allt∈ [t∗,t2].Obviously,ast∈ [t1,t1+2P],by(19),we obtain

We claim thatI(t)≥ ν1for allt∈ [t1+(C+2)P,t2].If it is not true,then there is at=t1+(C+2)P+T0,such atI(t)= ν1andI(t)≥ν1on[t1,t],whereT0=nP+ξ≥0,n≥0 is an integer,ξ∈[0,P).

From the first equation of model(5)and(20),we have

for allt∈ [t1+2P,t2].Integrating fromt1+2Ptotand using(17),(18),we further have

This is a contradiction.So,I(t)≥ ν1for allt∈ [t1,t2].Hence,liminft→∞I(t)≥ ν1.From Lemma 4,we finally have that infectiveIis permanent.This completes the proof.

4 Some corollaries

As applications of Theorems 1 and 2,we have the following corollaries.

Corollary 1Suppose that(H1)and(H2)hold and there is a constant λ>0 such that

andN∗(t)is some fixed solution of equation(6)withN∗(0)>0.Then infectiveIis extinct.

Corollary 2Suppose that(H1)and(H2)hold and there is a constant λ>0 such that

andN∗(t)is some fixed solution of equation(6)withN∗(0)>0.Then infectiveIis permanent.

Corollary 3When model(1)is ω-periodic,(H1)and(H2)hold,then infectiveIis extinct provided thatwhereN∗(t)is the globally attractive negative ω-periodic solution of equation(6).

Corollary 4Suppose that model(1)is periodic,(H1)and(H2)hold.Then infectiveIis permanent if and only ifR0>1.

Corollary 5When model(1.1)is almost periodic,(H1)and(H2)hold,then infectiveIis extinct provided thatwhereN∗(t)is the globally uniformly attractive nonnegative almost periodic solution of equation(6).

Corollary 6Suppose that model(1)is almost periodic,(H1)and(H2)hold.Then infectiveIis permanent if and only ifR0>1.