Stability and oscillations of multistage SIS models depend on the number of stages

In this paper we consider multistage SIS models of infectious diseases, where infected individuals are passing through infectious stages I1,I2,⋯In and then return to the susceptible compartment. First we calculate the basic reproduction number R0, and prove that the disease dies out for R0≤1, while a unique endemic equilibrium exists for R0>1. Our main result is that the stability of the endemic equilibrium depends on the number of stages: the endemic equilibrium is always stable when n ≤ 3, while for any n > 3 it can be either stable or unstable, depending on the particular choice of the parameters. We generalize previous stability results for SIRS models as well and point out a mistake in the literature for multistage SEIRS models. Our results have important implications on the discretization of infectious periods with varying infectivity.