Numerical analysis of split-step [formula omitted] methods with truncated Wiener process for a stochastic SIS epidemic model

Abstract

In this paper, the split-step θ methods for the stochastic SIS model are considered, and then the numerical positivity, convergence and dynamical behaviors are analyzed. Obviously, the numerical positivity makes sense biologically and is achieved by the truncated Wiener processes. Because the global Lipschitz condition is not satisfied to diffusion terms of the stochastic SIS model, an alternative approach is implemented for the convergence by the local Lipschitz condition and the numerical boundedness in the almost sure sense. Moreover, numerical extinction and persistence are presented by introducing a localized stochastic stability function, which is different from traditional methods. By using the exponential estimation of the localized stability function and the large number law for martingales, numerical extinction and persistence are obtained. Finally, numerical examples are given to illustrate our results.