Turning Points And Relaxation Oscillation Cycles in Simple Epidemic Models

Abstract

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.

A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model

“…The existence of periodic orbits of (8) was proved by Li et. al [17]. In Section 4, we demonstrate that a prescribed number of relaxation oscillations for system (8) can be obtained by varying the perturbation term f pN q.…”

“…Note that the line Z 0 " tpS, I, N q : I " 0, S " D D`p N u is a set of equilibria of (8) in the invariant plane tI " 0u. It is known [8,17] that Z 0 consists of the endpoints of a family of heteroclinic orbits (see Figure 2). The existence of periodic orbits of (8) was proved by Li et.…”

“…Remark 9. The period T of the limit cycle is referred to as interepidemic period (IEP) in [17], where it was shown numerically that T is proportional to 1{ . Their observation is consistence with the asymptotic formula (74).…”

“…It was proved in [17] that, for some parameters, system (8) exhibits a stable periodic orbit while the positive equilibrium is asymptotically stable, which implies that there must be at least one unstable periodic orbit. It was commented in [17,Section 4.3] that, in general, the unstable periodic orbit is not necessarily a relaxation oscillation but a small periodic orbit through a subcritical Hopf bifurcation. Using Theorem 4.1, we are able to confirm that there is an isolated unstable periodic orbit for small ą 0 that forms a relaxation oscillation.…”

“…which was proposed and investigated by Graef et al [8] and Li et al [17]. Here N " S`I`R and bpN q " DN` f pN q, with f pN q " rNˆ1´N N max˙, (7) and B N gpS, N q ą 0 and B N gpS, N q ą 0 @ S ě 0, N ě 0.…”