Effects of Nonmonotonic Functional Responses on a Disease Transmission Model: Modeling and Simulation

Abstract

In this article, a novel susceptible–infected–recovered epidemic model with nonmonotonic incidence and treatment rates is proposed and analyzed mathematically. The Monod–Haldane functional response is considered for nonmonotonic behavior of both incidence rate and treatment rate. The model analysis shows that the model has two equilibria which are named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The stability analysis has been performed for the local and global behavior of the DFE and EE. With the help of the basic reproduction number

\left(R_0\right)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \left( {R_{0} } \right) $$\end{document}

, we investigate that DFE is locally asymptotically stable when

\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}

and unstable when

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}$$ R_{0} > 1 $$\end{document}

. The local stability of DFE at

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}$$ R_{0} = 1 $$\end{document}

has been analyzed, and it is obtained that DFE exhibits a forward transcritical bifurcation. Further, we identify conditions for the existence of EE and show the local stability of EE under certain conditions. Moreover, the global stability behavior of DFE and EE has been investigated. Lastly, numerical simulations have been done in the support of our theoretical findings.

Keywords

Monod–Haldane functional / Basic reproduction number / Local and global stability / Bifurcation

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Funding

Delhi Technological University

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature