Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate

Abstract

We study a class of discrete SIRS epidemic models with nonlinear incidence rate $F(S)G(I)$ and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction number , then the disease-free equilibrium is globally asymptotically stable, and if ${ℛ}_0>1$, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only when ${ℛ}_0>1$, the disease in the model is permanent. Some special cases of $F(S)G(I)$ are discussed. Particularly, when $F(S)G(I)=\beta SI/(1+\lambda I)$, it is obtained that the endemic equilibrium is globally asymptotically stable if and only if ${ℛ}_0>1$. Furthermore, the numerical simulations show that for general incidence rate $F(S)G(I)$ the endemic equilibrium may be globally asymptotically stable only as ${ℛ}_0>1$.