Analyzing the Impact of the Parameter Values on the Qualitative Behaviour of the Solutions of a Modified SIR Model with Vaccination and Several Levels of Immunity

We consider a system of ordinary differential equations associated to a compartmental model for the dynamics of an epidemic which extends the standard SIR model by assuming that the immunity of individuals is gradually decaying along several different immunity levels. Besides the usual dynamics of an infectious disease, we consider also effects like waning of immunity and vaccination. The maximum immunity level can be reached either by vaccination or by recovery from an infection and it has associated the smallest R-number among all possible levels. By considering relevant values of the model parameters, we discuss conditions and relations between them under which an epidemic of this type can be eventually eradicated by a proper vaccination strategy within realistic possibilities. Within the considered model, this is connected to a stability condition and with the possible uniqueness of the trivial equilibrium solution. We also perform several numerical simulations which illustrate these theoretical considerations.