Global stability for a multi-group SIRS epidemic model with varying population sizes

Abstract

In this paper, by extending well-known Lyapunov function techniques to SIRS epidemic models, we establish sufficient conditions for the global stability of an endemic equilibrium of a multi-group SIRS epidemic model with varying population sizes which has cross patch infection between different groups. Our proof no longer needs such a grouping technique by graph theory commonly used to analyze the multi-group SIR models.

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:

$$\{\begin{array}{c}\frac{d{S}_k}{dt}=b_k-{\mu }_{k1}{S}_k-S_k\left(\sum _{j=1}^n{\beta }_{kj}{I}_j\right)+{\delta }_k{R}_k\text{,}\\ \frac{d{I}_k}{dt}=S_k\left(\sum _{j=1}^n{\beta }_{kj}{I}_j\right)-({\mu }_{k2}+{\gamma }_k)I_k\text{,}\\ \frac{d{R}_k}{dt}={\gamma }_k{I}_k-({\mu }_{k3}+{\delta }_k)R_k\text{,}& k=1,2,\dots ,n\text{.}\end{array}$$

$S_k\left(t\right)$, $I_k\left(t\right)$ and $R_k\left(t\right)$, $k=1,2,\dots ,n$ denote the numbers of susceptible, infected and recovered individuals in city $k$ at time $t$, respectively. $b_k$, $k=1,2,\dots ,n$ is the recruitment rate of the population, ${\mu }_{ki}$, $i=1,2,3$, is the natural death rates of susceptible, infected and recovered individuals in city $k$, $k=1,2,\dots ,n$, and ${\gamma }_k$ denotes the natural recovery rate of the infected individuals in city $k$, $k=1,2,\dots ,n$, respectively. Functions describing the dynamics within city $k$ 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 $I_k$ in city $k$, the disease is transmitted to the susceptible individuals $S_k$ by the incidence rate ${\beta }_{kk}{S}_k{I}_k$ with a transmission rate ${\beta }_{kk}$, but also we consider cross patch infection between different groups such that for each $I_j,j\ne k$ who travel from other city $j$ into city $k$, the disease is transmitted by the incidence rate ${\beta }_{kj}{S}_k{I}_j$ with a transmission rate ${\beta }_{kj}$. We assume that parameters $b_k,{\mu }_{k1},{\mu }_{k2},{\mu }_{k3},{\gamma }_k$ are positive constants, and ${\beta }_{kj},{\delta }_k$ are nonnegative constants.

The initial conditions of system (1.1) take the form

$$S_k\left(0\right)={\varphi }_1^k>0\text{,}{I}_k\left(0\right)={\varphi }_2^k>0\text{,}{R}_k\left(0\right)={\varphi }_3^k>0\text{,}k=1,2,\dots ,n\text{.}$$

By the biological meanings of natural death rates ${\mu }_{k1},{\mu }_{k2},{\mu }_{k3}$ of susceptible, infected and recovered individuals, we may assume that

$${\mu }_{k1}\le min({\mu }_{k2},{\mu }_{k3})\text{,}k=1,2,\dots ,n\text{.}$$

Moreover, for simplicity in this paper, we assume that

$$\text{the }n\times n\text{ matrix }B={\left({\beta }_{kj}\right)}_{n\times n}\text{ is irreducible ,}$$

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

$${\tilde{R}}_0=\rho \left(\tilde{M}\left(S^0\right)\right)\text{,}$$

where the positive $n$-column vector $S^0={(S_1^0,S_2^0,\dots ,S_n^0)}^T={(b_1/{\mu }_{11},b_2/{\mu }_{21},\dots ,b_n/{\mu }_{n1})}^T$ and $\rho \left(\tilde{M}\left(S^0\right)\right)$ denotes the spectral radius of the matrix $\tilde{M}\left(S^0\right)$ defined by

$$\tilde{M}\left(S^0\right)={\left(\frac{{\beta }_{kj}{S}_k^0}{{\mu }_{k2}+{\gamma }_k}\right)}_{n\times n}\text{.}$$

Observe that if ${\delta }_k=0,k=1,2,\dots ,n$, then the variables $R_k,k=1,2,\dots ,n$ do not appear in (1.1) and hence, in this case, we may consider only the reduced system for $S_k$ and $I_k,k=1,2,\dots ,n$ as follows.

$$\{\begin{array}{c}\frac{d{S}_k}{dt}=b_k-{\mu }_{k1}{S}_k-S_k\left(\sum _{j=1}^n{\beta }_{kj}{I}_j\right)\text{,}\\ \frac{d{I}_k}{dt}=S_k\left(\sum _{j=1}^n{\beta }_{kj}{I}_j\right)-({\mu }_{k2}+{\gamma }_k)I_k\text{.}\end{array}$$

For system (1.7), the result of Guo et al. [6] is as follows.

Theorem A

For (1.7), assume that ${\mu }_{k1}\le {\mu }_{k2},k=1,2,\dots ,n$ and (1.4) holds. Then, for ${\hat{R}}_0\le 1$, the disease-free equilibrium ${\hat{E}}^0=({\hat{S}}_1^0,0,{\hat{S}}_2^0,0,\dots ,{\hat{S}}_n^0,0)$ of system (1.7) is globally asymptotically stable in $\hat{\Gamma }$, and for ${\hat{R}}_0>1$, there exists an endemic equilibrium ${\hat{E}}^{\ast }=({\hat{S}}_1^{\ast },{\hat{I}}^{\ast },{\hat{S}}_2^{\ast },{\hat{I}}_2^{\ast },\dots ,{\hat{S}}_n^{\ast },{\hat{I}}_n^{\ast })$ of system (1.1) (see [19]) which is globally asymptotically stable in ${\hat{\Gamma }}^0$, where ${\hat{\Gamma }}^0$ is the interior of the feasible region $\hat{\Gamma }$ defined by

$$\hat{\Gamma }=\{(S_1,I_1,S_2,I_2,\dots ,S_n,I_n)\in R_{+}^{2n}\mid S_k\le \frac{b_k}{{\mu }_{k1}},S_k+I_k\le \frac{b_k}{{\mu }_{k1}},k=1,2,\dots ,n\}\text{,}$$

and $R_{+}^m=\{(x_1,x_2,\dots ,x_m):x_k\ge 0,k=1,2,\dots ,m\}$.

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 $\delta =0$ 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 $\delta >0$ 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 ${\tilde{R}}_0\le 1$, the disease-free equilibrium $E^0=(S_1^0,0,0,S_2^0,0,0,\dots ,S_n^0,0,0)$ is globally asymptotically stable in $\Gamma$, and for ${\tilde{R}}_0>1$, system (1.1) is uniformly persistent in ${\Gamma }^0$ (which means that there exists a compact set $K$ in the interior of ${\Gamma }^0$ such that all the solutions $(S_1,I_1,R_1,S_2,I_2,R_2,\dots ,S_n,I_n,R_n)$ of system (1.1) with the initial conditions (1.2) ultimately enter $K$ ) and there exists at least one endemic equilibrium $E^{\ast }=(S_1^{\ast },I_1^{\ast },R_1^{\ast },S_2^{\ast },I_2^{\ast },R_2^{\ast },\dots ,S_n^{\ast },I_n^{\ast },R_n^{\ast })$ in ${\Gamma }^0$, where ${\Gamma }^0$ is the interior of the feasible region $\Gamma$ defined by

$$\Gamma =\{(S_1,I_1,R_1,S_2,I_2,R_2,\dots ,S_n,I_n,R_n)\in R_{+}^{3n}\mid S_k\le S_k^0,S_k+I_k+R_k\le \frac{b_k}{{\mu }_{k1}},k=1,2,\dots ,n\}\text{.}$$

Moreover, for ${\tilde{R}}_0>1$, if

then $E^{\ast }$ is globally asymptotically stable in ${\Gamma }^0$.

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 ${\tilde{R}}_0\le 1$ and the uniform persistence of system (1.1) and the existence of the endemic equilibrium $E^{\ast }$ of system (1.1) for ${\tilde{R}}_0>1$ (see Proposition 3.1 and Corollary 3.1). In Section 4, for ${\tilde{R}}_0>1$, 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).