## Taiwanese Journal of Mathematics

### GLOBAL BEHAVIORS FOR A CLASS OF MULTI-GROUP SIRS EPIDEMIC MODELS WITH NONLINEAR INCIDENCE RATE

#### Abstract

In this paper, we study a class of multi-group SIRS epidemic models with nonlinear incidence rate which have cross patch infection between different groups. The basic reproduction number $\mathscr{R}_0$ is calculated. By using the method of Lyapunov functions, LaSalle's invariance principle, the theory of the nonnegative matrices and the theory of the persistence of dynamical systems, it is proved that if $\mathscr{R}_0\leq 1$ then the disease-free equilibrium is globally asymptotically stable, and if $\mathscr{R}_0\gt 1$ then the disease in the model is uniform persistent. Furthermore, when $\mathscr{R}_0\gt 1$, by constructing new Lyapunov functions we establish the sufficient conditions of the global asymptotic stability for the endemic equilibrium.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 5 (2015), 1509-1532.

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133721

Digital Object Identifier
doi:10.11650/tjm.19.2015.4205

Mathematical Reviews number (MathSciNet)
MR3412018

Zentralblatt MATH identifier
1357.92076

Subjects
Primary: 34D23: Global stability 92D30: Epidemiology

#### Citation

Tang, Qian; Teng, Zhidong; Jiang, Haijun. GLOBAL BEHAVIORS FOR A CLASS OF MULTI-GROUP SIRS EPIDEMIC MODELS WITH NONLINEAR INCIDENCE RATE. Taiwanese J. Math. 19 (2015), no. 5, 1509--1532. doi:10.11650/tjm.19.2015.4205. https://projecteuclid.org/euclid.twjm/1499133721

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