  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.
