Global stability for SIRS epidemic models with general incidence rate and transfer from infectious to susceptible

Abstract

We study a class of SIRS epidemic dynamical models with a general nonlinear incidence rate and transfer from infectious to susceptible. The incidence rate includes a wide range of monotonic, concave incidence rates and some non-monotonic or concave cases. We apply LaSalle’s invariance principle and Lyapunov’s direct method to prove that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number $$R_0\le 1$$ , and the endemic equilibrium is globally asymptotically stable if $$R_0>1$$ , under some conditions imposed on the incidence function f(S, I).