Comparison of deterministic and stochastic SIRS epidemic model with saturating incidence and immigration

The purpose of this work is to compare the stochastic and deterministic versions of an SIRS epidemic model. The SIRS models studied here include constant inflows of new susceptibles, infectives and removeds. These models also incorporate saturation incidence rate and disease-related death. First, we study the global stability of deterministic model with and without the presence of a positive flow of infectives into the population. Next, we extend the deterministic model system to a stochastic differential system by incorporating white noise. We show there is a unique positive solution to the system, and the long-time behavior of solution is studied. Mainly, we show how the solution goes around the endemic equilibrium and the disease-free equilibrium of deterministic system under different conditions on the intensities of noises and the parameters of the model. Finally, we introduce some numerical simulation graphics to illustrate our main results.