Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate

In this paper, we investigate a disease transmission model of SIRS type with latent period τ⩾0 and the specific nonmonotone incidence rate, namely, kexp(−dτ)S(t)I(t−τ)1+αI2(t−τ). For the basic reproduction number R0>1, applying monotone iterative techniques, we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in [Hai-Feng Huo, Zhan-Ping Ma, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 459-468]. Moreover, combining both monotone iterative techniques and the Lyapunov functional techniques to an SIR model by perturbation, we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium.