First order strong convergence and extinction of positivity preserving logarithmic truncated Euler-Maruyama method for the stochastic SIS epidemic model

The well-known stochastic SIS model characterized by highly nonlinear in epidemiology has a unique positive solution taking values in a bounded domain with a series of dynamical behaviors. However, the approximation methods to maintain the positivity and long-time behaviors for the stochastic SIS model, while very important, are also lacking. In this paper, based on a logarithmic transformation, we propose a novel explicit numerical method for a stochastic SIS epidemic model whose coefficients violate the global monotonicity condition, which can preserve the positivity of the original stochastic SIS model. And we show the strong convergence of the numerical method and derive that the rate of convergence is of order one. Moreover, the extinction of the exact solution of stochastic SIS model is reproduced. Some numerical experiments are given to illustrate the theoretical results and testify the efficiency of our algorithm.