Backward bifurcation underlies rich dynamics in simple disease models

Abstract

In this paper, dynamical systems theory and bifurcation theory are applied to investi- gate the rich dynamical behaviours observed in three simple disease models. The 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiol- ogy, in-host disease, and autoimmunity. These closely related models display interesting dynamical behaviors including bistability, recurrence, and regular oscillations, each of which has possible clinical or public health implications. In this contribution we elucidate the key role of backward bifurcation in the parameter regimes leading to the behaviors of interest. We demonstrate that backward bifurcation facilitates the appearance of Hopf bifurcations, and the varied dynamical behaviors are then determined by the properties of the Hopf bifurcation(s), including their location and direction. A Maple program devel- oped earlier is implemented to determine the stability of limit cycles bifurcating from the Hopf bifurcation. Numerical simulations are presented to illustrate phenomena of interest such as bistability, recurrence and oscillation. We also discuss the physical motivations for the models and the clinical implications of the resulting dynamics.