The stationary distribution in a class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion

Abstract

A class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion is investigated. By using the Lyapunov function method, the existence of global positive solutions and the ultimate boundedness with probability one are obtained. By using the Markov semigroups theory, Fokker–Planck equation and Khasminskiǐ functions, the existence of unique stationary distribution for the model is established. That is, when the stochastic basic reproduction number R0S>1 and some extra conditions are satisfied then the distribution density of any positive solutions of the model converges to a unique invariant density as t→+∞. Finally, the main conclusions and open problems are illustrated and verified by the numerical simulations.