Global stability for a multi-group SIRS epidemic model with varying population sizes
Introduction
Multi-group epidemic models have been studied in the literature of mathematical epidemiology to describe the transmission dynamics of various infectious diseases such as measles, mumps, gonorrhea, West-Nile virus and HIV/AIDS. A heterogeneous host population can be divided into several homogeneous groups according to modes of transmission, contact patterns, or geographic distributions, so that within-group and inter-group interactions could be modeled separately.
Certainly, there are several group models for example, as patch models, see Wang and Xiao [1] and Arino [2] and as transport-related models, see Liu and Zhou [3], Liu and Takeuchi [4] and Nakata [5] and references therein. In 2006, Guo et al. [6] have first succeeded to establish the complete global dynamics for a multi-group SIR model, by making use of the theory of non-negative matrices, Lyapunov functions and a subtle grouping technique in estimating the derivatives of Lyapunov functions guided by graph theory.
However, many researchers on multi-group SIR models, commonly follow this research approach to analyze the global stability of various multi-group SIR epidemic models (see for example, [7], [8], [9], [10], [11], [12], [13], [14]).
On the other hand, recently, Nakata et al. [15] and Enatsu et al. [16] proposed a simple idea to extend the Lyapunov functional techniques in McCluskey [17] for SIR epidemic models to SIRS epidemic models (see Lemma 4.3), and Muroya et al. [18] succeeded to prove the global stability for a class of multi-group SIR epidemic models without the grouping technique by graph theory in Guo et al. [6].
Motivated by these facts, in this paper, we are interested in the global stability of the following multi-group SIRS epidemic model which has cross patch infection between different groups: , and , denote the numbers of susceptible, infected and recovered individuals in city at time , respectively. , is the recruitment rate of the population, , , is the natural death rates of susceptible, infected and recovered individuals in city , , and denotes the natural recovery rate of the infected individuals in city , , respectively. Functions describing the dynamics within city of each population of individuals, might involve all populations of individuals that are present in the city, and we suppose that there are no between-city interactions, though. We assume that two cities are connected by the direct transport such as airplanes or trains, etc. Therefore, for the model (1.1), the only input is the recruitment. Moreover, not only for infective individuals in city , the disease is transmitted to the susceptible individuals by the incidence rate with a transmission rate , but also we consider cross patch infection between different groups such that for each who travel from other city into city , the disease is transmitted by the incidence rate with a transmission rate . We assume that parameters are positive constants, and are nonnegative constants.
The initial conditions of system (1.1) take the form
By the biological meanings of natural death rates of susceptible, infected and recovered individuals, we may assume that Moreover, for simplicity in this paper, we assume that that is, an infected individual in the first group can cause infection to a susceptible individual in the second group through an infection path.
Put where the positive -column vector and denotes the spectral radius of the matrix defined by
Observe that if , then the variables do not appear in (1.1) and hence, in this case, we may consider only the reduced system for and as follows.
For system (1.7), the result of Guo et al. [6] is as follows.
Theorem A For (1.7), assume that and (1.4) holds. Then, for , the disease-free equilibrium of system (1.7) is globally asymptotically stable in , and for , there exists an endemic equilibrium of system (1.1) (see [19]) which is globally asymptotically stable in , where is the interior of the feasible region defined byand .
Motivated by the above results, in this paper, applying both well-known techniques in Guo et al. [6] (see Lemma 4.1, Lemma 4.2) and some special techniques in Nakata et al. [15] and Enatsu et al. [16] with McCluskey [17] and Muroya et al. [18] (see Lemma 4.3, Lemma 4.4, Lemma 4.5), we establish sufficient conditions for the global asymptotic stability of the multi-group SIRS epidemic model (1.1) (see (1.10) in Theorem 1.1).
Note that it is sufficient to use Lemma 4.1, Lemma 4.2 for a multi-group SIR model in Guo et al. [6] which is the special case of (1.1), and our proof no longer needs such a grouping technique by graph theory in Guo et al. [6], but for the case of (1.1), we need more lemmas (see Lemma 4.3, Lemma 4.4, Lemma 4.5) which seem to be complicated but these techniques just come from Nakata et al. [15] and Enatsu et al. [16].
The main theorem in this paper is as follows.
Theorem 1.1 For system (1.1), assume that (1.3), (1.4) hold. Then, for , the disease-free equilibrium is globally asymptotically stable in , and for , system (1.1) is uniformly persistent in (which means that there exists a compact set in the interior of such that all the solutions of system (1.1) with the initial conditions (1.2) ultimately enter ) and there exists at least one endemic equilibrium in , where is the interior of the feasible region defined byMoreover, for , ifthen is globally asymptotically stable in .
The organization of this paper is as follows. To prove Theorem 1.1, we consider the reduced system (1.1). In Section 2, we offer positiveness and eventual boundedness of solutions for system (1.1). In Section 3, following the proof techniques in Guo et al. [6], we similarly prove the global asymptotic stability of the disease-free equilibrium for and the uniform persistence of system (1.1) and the existence of the endemic equilibrium of system (1.1) for (see Proposition 3.1 and Corollary 3.1). In Section 4, for , using Lyapunov function techniques to the system (1.1) (see Lemma 4.1, Lemma 4.2, Lemma 4.3, Lemma 4.4, Lemma 4.5), under the condition (1.10), we prove the global asymptotic stability for the endemic equilibrium of (1.1).
Section snippets
Positiveness and eventual boundedness of solutions of (1.1)
In this section, we consider positiveness and eventual boundedness of solutions of (1.1). Let be a positive -column vector and be the total population in city .
Then, we have the following lemma on positiveness and eventual boundedness of solutions of (1.1).
Lemma 2.1 and under the condition (1.3), it holds that
Global stability of the disease-free equilibrium for
We can obtain the following proposition, whose proof is similar to that of Guo et al. [6, Proposition 3.1] (see the proof of Proposition 3.1 in Appendix).
Proposition 3.1 If , then the disease-free equilibrium of system(1.1) is the unique equilibrium of(1.1) and it is globally asymptotically stable in . If , then is unstable and system(1.1) is uniformly persistent in .
Proof of the first part of Theorem 1.1 for If , then by Proposition 3.1, we can obtain the first part of Theorem 1.1. □
Uniform
Global stability of the endemic equilibrium for
In this section, we assume , and we prove that an endemic equilibrium of (1.1) is globally asymptotically stable in . By Corollary 3.1, there exists an endemic equilibrium such that We rewrite (1.1) as
Now, for some positive constants , let us consider
Acknowledgments
The authors wish to express their gratitude to an anonymous referee and also the editor for their helpful comments which improve the quality of the paper in the present style.
The first author’s research was supported by Scientific Research (c), No. 24540219 of Japan Society for the Promotion of Science. The second author’s research was supported by JSPS Fellows, No. 237213 of Japan Society for the Promotion of Science. The third author’s research was supported by JSPS Fellows, No. 222176 of
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