Stability of epidemic models with waning immunity

Abstract

SIRS type epidemiological model has a fundamental form to study the role of temporal immunity of recovered individuals in disease transmission dynamics and several variant models have been considered in the last century, but up to now dynamical aspects of the model are not fully elucidated. We here look over previous studies concerning qualitative analysis for a family of SIRS type epidemiological models. To this aim we construct a general model in a form of delay equation, a coupled system of a renewal equation and delay differential equations, structuring infected population by infection-age (time elapsed since infection). We re-examine the structure of equilibria and stability of the disease free equilibrium. We then introduce slightly improved stability conditions for an endemic equilibrium. Specifying modelling ingredients we derive two special cases where the model can be represented as a system of ordinary and delay differential equations that have appeared in the literature. For those models we have a powerful tool, namely Lyapunov function, to study global stability of the endemic equilibrium. Relating epidemic models are also discussed.