Endemic Models with Arbitrarily Distributed Periods of Infection II: Fast Disease Dynamics and Permanent Recovery

Abstract

A model for the spread of an infectious disease in a population with constant recruitment of new susceptibles, developed in previous work, is further analyzed in the case that disease survivors are permanently immune and that the disease dynamics are much faster than the demographic dynamics. Though the model allows for arbitrarily many stages of infection, all of which have general length distributions and disease survival functions, the different time scales make it possible to find explicit formulas for the interepidemic period (distance between peaks or valleys of disease incidence) and the local stability or instability of the endemic equilibrium. It turns out that the familiar formula for the length of the interepidemic period of childhood diseases has to be reinterpreted when the exponential length distribution of the infectious period is replaced by a general distribution. Using scarlet fever in England and Wales (1897--1978) as an example, we illustrate how different assumptions for the length distributions of the exposed and infectious periods (under identical average lengths) lead to quite different values for the minimum length of the quarantine period to destabilize the endemic equilibrium.