Turning Points And Relaxation Oscillation Cycles in Simple Epidemic Models

We study the interplay between effects of disease burden on the host population and the effects of population growth on the disease incidence, in an epidemic model of SIR type with demography and disease-caused death. We revisit the classical problem of periodicity in incidences of certain autonomous diseases. Under the assumption that the host population has a small intrinsic growth rate, using singular perturbation techniques and the phenomenon of the delay of stability loss due to turning points, we prove that large-amplitude relaxation oscillation cycles exist for an open set of model parameters. Simulations are provided to support our theoretical results. Our results offer new insight into the classical periodicity problem in epidemiology. Our approach relies on analysis far away from the endemic equilibrium and contrasts sharply with the method of Hopf bifurcations.