Dynamics and approximation of positive solution of the stochastic SIS model affected by air pollutants

In this paper, we develop a stochastic susceptible-infective-susceptible (SIS) model, in which the transmission coefficient is a function of air quality index (AQI). By using Markov semigroup theory, the existence of kernel operator is obtained. Then, the sufficient conditions that guarantee the stationary distribution and extinction are given by Foguel alternative, Khasminsk$\check{\rm l}$ function and Itô formula. Next, a positivity-preserving numerical method is used to approximate the stochastic SIS model, meanwhile for all $ p > 0 $, we show that the algorithm has the $ p $th-moment convergence rate. Finally, numerical simulations are carried out to illustrate the corresponding theoretical results.