Analysis of a temperature-dependent model for water-borne disease transmission dynamics

It is well known that water-borne diseases occur seasonally and are strongly associated with the temperature. In this paper, a non-autonomous mathematical model for water-borne disease is proposed and analyzed. Three temperature-dependent parameters are included in the proposed model, related to the growth rate and death rate of pathogen in aquatic environment, and disease transmission rate from pathogen to human. In this paper, a threshold condition in terms of RC(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R_C(t)$$\end{document} is obtained to account for the extinction or the persistence of the disease. The impact of temperature variability on the spread of disease over the study region is also shown from the analysis of the temperature-dependent threshold RC(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_C(t)$$\end{document}. It is also shown that the proposed non-autonomous system has a non-trivial disease-free periodic state which is globally asymptotically stable whenever RC(t)<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_C(t) < 1$$\end{document}. The autonomous version of the proposed model is also analyzed. Results show that the autonomous system is locally and globally asymptotically stable for the disease-free state. The endemic equilibrium of the autonomous model 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} is also derived, and it is shown that the endemic solution for the autonomous system is globally asymptotically stable. Numerical simulation has been carried out to illustrate our theoretical results and also predict the effectiveness of the control strategies.