Bifurcation Analysis of a COVID-19 Dynamical Model in the Presence of Holling Type-II Saturated Treatment with Reinfection

In this study, we have proposed a mathematical model to determine the dynamical behavior of COVID-19 transmission incorporating bilinear incidence rate and saturated treatment. One of our important assumptions is the occurrence of reinfection of COVID-19, which is studied to be significant. The governing model yields up to multiple equilibrium points depending on different parameter sets. Local and global stability analysis with the help of suitable Lyapunov coefficients has been established for both disease-free and endemic steady states. It is found from analysis that the disease may still persist in the population even 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}$$\mathcal {R}_0 <1$$\end{document}, if there is a limitation in the treatment facilities because of the state of saturation. Consequently, the phenomenon of backward bifurcation is detected. The model dynamics show the existence of transcritical and backward bifurcation under certain parametric conditions. Analytical demonstration of backward bifurcation in the system indicates that diminishing the basic reproductive number below unity is insufficient to prevent the spread of the disease. The model system is also examined for saddle-node bifurcation due to its nature of bistability in the equilibrium points. Additionally, sensitivity analyses for the variables in the basic reproduction number have been carried out to identify the variables that have the greatest impact on the course of the disease. Numerical simulations are utilized to clearly validate the theoretical results so as to show how the suggested mathematical model may be employed.