Stability analysis and asymptotic behavior of a stochastic delayed SIRS epidemic model with a class of nonlinear incidence rates

Abstract

This paper deals with a stochastic version of a deterministic delayed SIRS (Susceptible-InfectiveRemoved-Suceptible) epidemic model with nonlinear incidence rate. The stochastic model is obtained by taking into account random perturbations in the contact rate β due to environmental variations. We assume that the stochastic perturbation intensity is proportional to the number of infectious. Firstly, the existence of a unique global positive solution of the stochastic differential equations with delay describing the model is proved. Then, the stability of the disease-free equilibrium point is established under suitable conditions on the parameters of the model. We also study the asymptotic behaviour of the solution when the disease-free equilibrium (DFE) is unstable. Finally, numerical simulations are introduced to support our results.