Dynamical Behavior of the Heroin Epidemic Model on a Finite Weighted Network

We are concerned with a heroin epidemic model with relapse on a finite weighted network. The well-posedness of solutions to this problem is investigated. By means of the next-generation matrix method, we define the basic reproduction number on network and give a certain estimate as a threshold parameter by R~0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{R}_0 $$\end{document} due to the effect of population mobility. Applying Lyapunov functions method, we show that the drug-free equilibrium is globally asymptotically stable if R~0<R0<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{R}_0<R_0<1 $$\end{document}, where 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} is the basic reproduction number corresponding to the model without population mobility, while the endemic equilibrium is globally asymptotically stable if R~0>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{R}_0>1 $$\end{document}. Based on this, we deduce that the transmission on network brings more difficulty for the control and prohibition of heroin epidemic. The numerical simulations further illustrate the effect of population mobility on network. The final sensitivity analysis of R~0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widetilde{R}_0 $$\end{document} to the effective contact rate and ratio of seeking treatment reveal that prevention is more effective than treatment for the control of heroin epidemic.