The dynamics of sexually transmitted diseases with men who have sex with men

In this paper, we give a rather complete analysis for a susceptible-infective sexually transmitted disease (STD) model, where the males are divided into two different groups based on their different sexual orientation. The threshold 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} of STD model is obtained. If R0<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{0}<1$$\end{document}, the disease-free equilibrium is globally asymptotically stable. Further, we investigate the existence and stability of the boundary equilibria that characterize the males of the different sexual orientation. We also investigate the existence and stability of the positive equilibrium, which characterizes the possibility of coexistence of male heterosexual and male homosexual. We obtain sufficient and necessary conditions for the existence and global stability of these equilibria. We see that the proportion of heterosexuality in MSM affects the stability of the system. The theoretical results are verified by numerical simulation.