Reaction-Diffusion-Integral System Modeling SARS-CoV-2 Infection-Induced versusVaccine-Induced Immunity: Analytical Solutions and Stability Analysis

Abstract

Understanding protection from SARS-CoV-2 infection and severe COVID-19 induced by natural SARS-CoV-2 infection versus vaccination is essential for informed vaccine mandate decisions. In this article, we construct a system of reaction-diffusion-integral equations to describe the development of vaccinated population, not previously infected, and pre-infected population, vaccinated or not, subject to continued exposure to coronavirus leading to possible re-infection. The model accounts for the differences in induced immunity in the two populations and the spread of infection due to movements of various populations in space. To realistically describe the nature of immunity, which has been found to decline with time following vaccination or infection, the rate of infection is expressed here as an integral of a function of the specific rate of infection that increases exponentially with time, depending on how long it is after the subjects have been infected with, or vaccinate against, the virus. The model is analyzed for its stability, and the contour plot is presented. The analytical solutions of the model system are derived in the form of traveling waves, using the modified extended hyperbolic tangent method. Inspection and interpretation of the different shapes of these plots yield valuable insights.