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 with a compartmental model for the dynamics of an epidemic that extends the standard SIR model by assuming that individuals’ immunity gradually declines across several different immunity levels. In addition to the usual dynamics of an infectious disease, we also consider effects such as waning immunity and vaccination. The maximum immunity level can be reached either through vaccination or through recovery from infection, and it is associated with the smallest R-number among all possible levels. By considering relevant values of the model parameters, we discuss the conditions and relationships among them under which an epidemic of this type can eventually be eradicated by an appropriate vaccination strategy within realistic constraints. Within the model under consideration, this is linked to a stability condition and to the possible uniqueness of the trivial equilibrium solution. We also perform several numerical simulations that illustrate these theoretical considerations.