Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model with saturated incidence rate

In this paper, we present an SIRS (Susceptible, Infective, Recovered, Susceptible) epidemic model with a saturated incidence rate and disease causing death in a population of varying size. We define a parameter R-0* from which a study of stability is made. In the deterministic case, we discuss the existence of an endemic equilibria and prove the global stability of the disease-free equilibrium. By introducing a perturbation in the contact rate through a white noise, we consider a stochastic version. We prove the existence of a global positive solution which lives in a certain domain. Thereafter, we prove the global stability nth moment of the system as soon as the intensity of the white noise is below a certain threshold. Finally, we perform some numerical simulations to compare the dynamic behaviors of the deterministic system and the stochastic system.