Long Time Behavior of an Two Diffusion Stochastic SIR Epidemic Model with Nonlinear Incidence and Treatment

In this paper we propose a stochastic SIR epidemic model to evaluate effect of the randomness on treatment and nonlinear incidence rate. More precisely, we perturb both nonlinear incidence and treatment rates in deterministic SIR model with Gaussian white noise and obtain two diffusion stochastic model. For the model, we theoretically prove that it's solution is positive and global, and then, we obtain the conditions under which we can claim the existence of the stationary distribution. Also, by constructing suitable Lyapunov functions, we establish sufficient conditions for p-th moment and almost sure exponential stability of disease-free equilibrium. Conditions for disease extinction are obtained, as well. We close the paper by presenting numerical simulations to verify our theoretical results. For that purpose we use real-life data for spread of cholera in the Department of Artibonite in Haiti, as well as for influenza A H1N1 in Guangdong Province, China.