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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 $\mathscr{R}_0>1$ then the disease in the model is uniform persistent. Furthermore, when $\mathscr{R}_0>1$, by constructing new Lyapunov functions we establish the sufficient conditions of the global asymptotic stability for the endemic equilibrium.

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