Stochastic dynamics of an SEIS epidemic model

In this paper, we investigate the stochastic disease dynamics of an SEIS epidemic model with latent patients and active patients. The two parameters R0s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}^{s}$\end{document} and R0∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}^{*}$\end{document} are identified as the disease-free and endemic dynamics of the model. More specifically, we give the almost surely exponential stability of the disease-free equilibrium in terms of R0s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}^{s}$\end{document}, and stochastic endemic dynamics in terms of R0∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}^{*}$\end{document}. The theoretical and numerical results may be useful for studying the dynamics of disease spreading in a randomly fluctuating environment.