Analysis of an SIRS epidemic model with nonlinear incidence and vaccination

Abstract

In this paper, we study an SIRS epidemic model with a nonlinear incidence rate and vaccination. We give the existence of positivity, and boundedness of the equilibrium of the model. We calculate the basic reproduction number of the proposed model by using the next generation matrix method. By constructing Lyapunov function, we show that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less or equal than one and that the endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are performed to investigate the effect of vaccinate on model behavior.