Qualitative analysis of a mathematical model with presymptomatic individuals and two SARS-CoV-2 variants

Abstract

The SARS-CoV-2 continues to spread across the world. During this COVID-19 pandemic, several variants of the SARS-CoV-2 have been found. Some of these new variants like the VOC-202012/01 of lineage B.1.1.7 or the most recently B.1.617 emerging in India have a higher infectiousness than those previously prevalent. We propose a mathematical model based on ordinary differential equations to investigate potential consequences of the appearance of a new more transmissible SARS-CoV-2 strain in a given region. The proposed mathematical model incorporates the presymptomatic and asymptomatic subpopulations in addition to the more usual susceptible, exposed, infected, and recovered subpopulations. This is important from a realistic point of view since it has been found recently that presymptomatic and asymptomatic individuals are relevant spreaders of the SARS-CoV-2. Using the next-generation matrix method, we find the basic reproduction number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0}$$\end{document} , an important threshold parameter that provides insight regarding the evolution and outcome of a certain instance of the COVID-19 pandemic. The local and global stability of system equilibria are also presented. In particular, for the global stability we construct a Lyapunov functional and use the LaSalle invariant principle to prove that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. On the other hand, if \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} the endemic equilibrium is globally asymptotically stable. Finally, we present numerical simulations to numerically support the analytic results and to show the impact of the introduction of a new more contagious SARS-CoV-2 variant in a population.