Application of a reaction-diffusion model with different incidence rates: COVID-19 strains evolution

COVID-19 is continuously evolving, and the JN.1 strain has become the primary prevalent strain in the world. This paper focuses on the JN.1 strain and establishes a two-strain reaction-diffusion model with different incidence rates. In this paper, the saturation incidence rate is used to describe the individual's fear of the JN.1 strain. First, we discuss the well-posedness with positivity and boundedness, obtain the reproduction number R0\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}, and study the long-term behavior of the solution, including extinction, uniform persistence, local stability and global asymptotical stability. Second, we discretize the continuous system using the NSFD method and explore the well-posedness of the discrete system. Third, we use the method of sensitivity analysis to reveal the effect of parameters on the basic reproduction number. Susceptible individuals are more easily infected as the spatial diffusion coefficient increases. Besides, it is beneficial to form herd immunity. Moreover, this research reveals the evolution trend of COVID-19: the virus infection rate continuously increases while the mortality rate caused by the virus decreases after increases, tending to 0 since the beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} strain. The final result of the SARS-CoV-2 continued evolution is a higher infection rate and lower mortality rate, consistent with the long-term evolution trend of the virus.