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The waning of immunity after recovery or vaccination is a major factoraccounting for the severity and prolonged duration of an array of epidemics,ranging from COVID-19 to diphtheria and pertussis. To study the effectivenessof different immunity level-based vaccination schemes in mitigating the impactof waning immunity, we construct epidemiological models that mimic the latter'seffect. The total susceptible population is divided into an arbitrarily largenumber of discrete compartments with varying levels of disease immunity. Wethen vaccinate various compartments within this framework, comparing the valueof $R_0$ and the equilibria locations for our systems to determine an optimalimmunization scheme under natural constraints. Relying on perturbativeanalysis, we establish a number of results concerning the location, existence,and uniqueness of the system's endemic equilibria, as well as results ondisease-free equilibria. In addition, we numerically simulate the dynamicsassociated with our model in the case of pertussis in Canada, fitting our modelto available time-series data. Our analytical results are applicable to a widerange of systems composed of arbitrarily many ODEs.

A Perturbative Approach to the Analysis of Many-Compartment Models Characterized by the Presence of Waning Immunity