Genuine and spurious bistability in a simple epidemic model with waning immunity

Abstract

We study an infection-age structured epidemic model in which both the infectivity and the rate of loss of immunity depend on the time-since-infection. The model can be equivalently viewed as a nonlinear renewal equation for the incidence of infection or as a partial differential equation for the density of infected individuals. We explicitly consider gamma, rather than Erlang, distributed durations of infection using a combination of ODE approximations and numerical bifurcation methods. We show that the shape of this distribution strongly influences stability of the endemic equilibrium, even when the basic reproduction number $R_0$ and the mean duration of infectiousness are fixed. Moreover, we establish the existence of regions of bistability, where a stable endemic equilibrium coexists with a stable periodic orbit. To our knowledge, this provides the first example of bistability in infection-age structured models with waning immunity alone. Finally, we show how common compartmental modelling approaches, which impose implicit assumptions on the distribution of the duration of infection, can lead to spurious dynamical outcomes. Taken together, our analysis underscores the crucial role of distributional structure in epidemic modelling and provides new insights into the rich dynamics of infection-age structured SIS/SIRS models.