Equilibrium Solutions of a Modified SIR Model with Vaccination and Several Levels of Immunity

Abstract

We consider a system of ordinary differential equations which extends the well-known SIR model for the dynamics of an epidemic. The main feature is that the population is divided in several subgroups according to their immunity level, which has as a consequence different infection rates. The maximum level of immunity can be achieved either by recovering from an infection, or by possible vaccination. We consider the cases that the vaccination rate is independent on the size of infected population, or that it depends also on this value by a power law. In addition, we assume that the immunity level can decay in time. The goal of this paper is to analyze the existence and uniqueness of equilibrium solutions, which can be either a trivial (disease-free) equilibrium, with no infections, or an endemic equilibrium, with a certain amount of infected individuals. Moreover, we give conditions for the local asymptotic stability of the unique trivial equilibrium solution. It will turn out that, if this is the case, then there exists no endemic equilibrium, which means that the epidemic can be eradicated, by arriving at herd immunity. On the other hand, if the trivial equilibrium is unstable, then we prove the existence of an endemic equilibrium which, under natural conditions, turns out to be unique. The stability of the endemic equilibrium remains still an open problem.