Permanence and extinction for a nonautonomous SEIRS epidemic model

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Abstract

In this paper, we study the long-time behavior of a nonautonomous SEIRS epidemic model. We obtain new sufficient conditions for the permanence (uniform persistence) and extinction of infectious population of the model. By numerical examples we show that there are cases such that our results improve the previous results obtained in [T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Bio. 69 (2007) 2537–2559]. We discuss a relation between our results and open questions proposed in the paper.

Introduction

In this paper, we consider the following nonautonomous SEIRS epidemic modeldS(t)dt=Λ(t)-β(t)S(t)I(t)-μ(t)S(t)+δ(t)R(t),dE(t)dt=β(t)S(t)I(t)-μ(t)+ε(t)E(t),dI(t)dt=ε(t)E(t)-μ(t)+γ(t)I(t),dR(t)dt=γ(t)I(t)-μ(t)+δ(t)R(t)with initial valueS(0)>0,E(0)0,I(0)>0,R(0)0.Here S(t),E(t),I(t) and R(t) denote the size of susceptible, exposed (not infectious but infected), infectious and recovered population at time t0, respectively. Λ(t) denotes the birth rate, β(t) denotes the disease transmission coefficient, μ(t) denotes the mortality, ε(t) denotes the rate of developing infectivity, γ(t) denotes the recovery rate and δ(t) denotes the rate of losing immunity at time t.

In the field of mathematical epidemiology, the qualitative analysis of mathematical epidemic models has been carried out by many authors (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] and references therein). One of the main streams of the field is the analysis of autonomous models (see for instance [8], [9], [13], [14], [18], [19] and references therein). For instance, in case where system (1.1) is autonomous (that is, all parameters are given by time-independent functions Λ(t)=Λ,β(t)=β,μ(t)=μ,ε(t)=ε,γ(t)=γ and δ(t)=δ), we obtain the basic reproduction number (see e.g. [5]) asR0=εβ(μ+ε)(μ+γ)Λμ.It is well known that the infectious disease dies out if R01 and the disease persists if R0>1 (see [10], [14]).

On the other hand, in the real world, quite a few infectious diseases spread seasonally (one of the reasons of such a phenomenon is, for instance, the seasonal change of the number of infectious vectors [3]). Therefore, the study of periodic epidemic models has recently been carried out enthusiastically (see e.g. [2], [3], [4], [11], [12], [15], [16], [20], [21], [22], [23] and references therein). The definition of the basic reproduction number R0 for periodic epidemic models was firstly given by Bacaër and Guernaoui [3]. For system (1.1) with periodic parameters, Nakata and Kuniya [12] proved that R0 plays the role as a threshold parameter for determining the global dynamics of solutions, that is, the disease-free periodic solution is globally asymptotically stable if R0<1 and the disease persists if R0>1.

The nonautonomous case is an extension of the periodic case. The study of the basic reproduction number R0 for general time-heterogeneous epidemic models has recently been carried out by Inaba [7] and Thieme [17].

Zhang and Teng [22] analyzed the dynamics of nonautonomous SEIRS epidemic model (1.1) and obtained some sufficient conditions for the permanence and extinction of the infectious population. One can notice that results obtained in Theorems 4.1 and 5.1 in their paper do not determine the disease dynamics completely, since those conditions do not give a threshold-type condition even in the autonomous case.

In this paper we obtain new sufficient conditions for the permanence and extinction of system (1.1). We prove that our conditions gives the threshold-type result by the basic reproduction number given as in (1.3) when every parameter is given as a constant parameter. Thus our result is an extension result of the threshold-type result in the autonomous system. Our results may contribute to predict the disease dynamics, such as permanence and extinction of the infectious population, when the phenomena is modeled as a nonautonomous system.

This paper is organized as follows. In Section 2 we present preliminary setting and propositions, which we use to analyze the long-time behavior of system (1.1) in the following sections. In Sections 3 Extinction of infectious population, 4 Permanence of infectious population we prove our main theorems on the extinction and permanence of infectious population of system (1.1). In Section 5, we derive explicit conditions for the existence and permanence of infectious population of system (1.1) for some special cases. We prove that when every parameter is given as a constant parameter our conditions for the permanence and extinction becomes the threshold condition by the basic reproduction number. In Section 6 we provide numerical examples to illustrate the validity of our results. Moreover, those examples illustrate the cases where our theoretical result can determine the dynamics even conditions proposed in [22] are not satisfied.

Section snippets

Preliminaries

As in [22] we put the following assumptions for system (1.1).

Assumption 2.1

  • (i)

    Functions Λ,β,μ,δ,ε and γ are positive, bounded and continuous on 0,+ and β(0)>0.

  • (ii)

    There exist constants ωi>0 i=1,2,3 such thatliminft+tt+ω1β(s)ds>0,liminft+tt+ω2μ(s)ds>0,liminft+tt+ω3Λ(s)ds>0.

In what follows, we denote by N(t) the solution ofdN(t)dt=Λ(t)-μ(t)N(t)with initial value N(0)=S(0)+E(0)+I(0)+R(0)>0. By adding equations of (1.1), we easily see that N(t)=S(t)+E(t)+I(t)+R(t) means the size of total population at

Extinction of infectious population

In this section, we obtain sufficient conditions for the extinction of infectious population of system (1.1). The definition of the extinction is as follows:

Definition 3.1

We say that the infectious population of system (1.1) is extinct iflimt+I(t)=0.

We give one of the main results of this paper.

Theorem 3.2

If there exist positive constants λ>0,p>0 and T1>0 such thatR1(λ,p)limsupt+tt+λβ(s)N(s)p-(μ(s)+ε(s))ds<0,R1(λ,p)limsupt+tt+λε(s)1p-(μ(s)+γ(s))ds<0and G(p,t)<0 for all tT1, then the infectious population

Permanence of infectious population

In this section, we obtain sufficient conditions for the permanence of infectious population of system (1.1). The definition of the permanence is as follows:

Definition 4.1

We say that the infectious population of system (1.1) is permanent if there exist positive constants I1>0 and I2>0, which are independent from the choice of initial value satisfying (1.2), such that0<I1liminft+I(t)limsupt+I(t)I2<+.

We give one of the main results of this paper.

Theorem 4.2

If there exist positive constants λ>0,p>0 and T1>0 such

Applications

In this section, we consider some special cases of system (1.1). Applying Theorem 3.2, Theorem 4.2, we derive explicit conditions for the extinction and permanence of infectious population of system (1.1).

First, we assume that all coefficients of system (1.1) are given by identically constant functions. Then, (1.1) becomes an autonomous system. We show that, in this case, our results obtained in Sections 3 Extinction of infectious population, 4 Permanence of infectious population become a

Numerical examples

In this section we perform numerical simulations in order to verify the validity of Theorem 3.2, Theorem 4.2 and to show that in some special cases, our results can improve the previous results for the permanence and extinction of system (1.1) obtained by Zhang and Teng [22].

FixΛ(t)1,μ(t)1,ε(t)=0.31+0.5cos(2πt),γ(t)=0.51+0.5cos(2πt)and δ(t)0.1. Then, from (2.1), we have limt+N(t)=1. Here we assume N(0)=1 and thus N(t)1.

Let β(t)=6.491+0.5cos(2πt). Then, system (1.1) becomes periodic

Discussion

In this paper, we have investigated the global dynamics of a nonautonomous SEIRS epidemic model (1.1). We obtain new sufficient conditions for the extinction and permanence of infectious population of system (1.1) in Theorem 3.2, Theorem 4.2, respectively. We analyze the dynamics of system (1.1) via considering the behavior of a function defined as in (2.3), see Lemma 2.3, Lemma 4.3.

In Section 5, we prove that when every parameter of system (1.1) is given as a constant parameter, our conditions

Acknowledgements

The authors would like to thank Prof. Yoshiaki Muroya and the reviewers very much for their valuable comments and suggestions on an earlier version of this paper, and Prof. Nicolas Bacaër for having a discussion with us at the conference “R0 and related concepts: methods and illustrations” held at Paris on 29-31, October, 2008. TK was supported by Japan Society for the Promotion of Science (JSPS), No. 222176. YN was supported by Spanish Ministry of Science and Innovation (MICINN), MTM2010-18318.

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