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Endemic Models with Arbitrarily Distributed Periods of Infection

Endemic Models with Arbitrarily Distributed Periods of Infection

Department of Mathematics

Arizona State University

thieme@math.la.asu.edu

A model is developed and analyzed for the spread of an infectious disease in a population with constant recruitment of new susceptibles. The model allows for arbitrarily many stages of infection all of which have general length distributions and disease mortalities. A basic reproduction ratio is derived and related to the existence of an endemic equilibrium, to the stability of the disease free-equilibrium, and to weak and strong endemicity (persistence) of the disease. A characteristic equation is found the zeros of which determine the local stability of the endemic equilibrium, and sufficient stability conditions are given for the case that infected individuals do not return into the susceptible class. If, in addition, the disease dynamics are much faster than the demographic dynamics, the different time scales make it possible to find explicit formulas for the inter-epidemic 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 a quarantine period to destabilize the endemic equilibrium.

1998-1999 Mathematics in Biology