Models for diseases with exposed periods

A general model for a disease without immunity against reinfection having arbitrary distributions of exposed and infective periods was formulated by Hethcote, Stech and van den Driessche [5]. They showed that for contact numbers exceeding 1, the endemic equilibrium is asymptotically stable if either the exposed period of the infective period is exponentially distributed or if both exposed and infective period have fixed length, and they conjectured that the endemic equilibrium is always asymptotically stable. We show that the endemic equilibrium is asymptotically stable if the mean exposed period is less than the mean infective period, or if the contact number is sufficiently large, or if the exposed period distribution function is convex. However, we also show that for a more general type of model in which the infective period distribution can depend on the length of the exposed period it is possible to have instability of the endemic equilibrium and a Hopf bifurcation.