Random periodic solution for a stochastic SIS epidemic model with constant population size

In this paper, a stochastic susceptible-infected-susceptible (SIS) epidemic model with periodic coefficients is formulated. Under the assumption that the total population is fixed by N, an analogue of the threshold R0T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${R_{0}^{T}}$\end{document} is identified. If R0T>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${R_{0}^{T} > 1}$\end{document}, the model is proved to admit at least one random periodic solution which is nontrivial and located in (0,N)×(0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0,N)\times(0,N)$\end{document}. Further, the conditions for persistence and extinction of the disease are also established, where a threshold is given in the case that the noise is small. Comparing with the threshold of the autonomous SIS model, it is generalized to its averaged value in one period. The random periodic solution is illuminated by computer simulations.