  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.
