The Early and Final States of an Epidemic in a Large Heterogeneous Population by a Small Initial Number of Infectives

Abstract

This paper describes the early and final properties of a general S–I–R epidemic process in which the infectives behave independently, each infective has a random number of contacts with the others in the population, and individuals vary in their susceptibility to infection. For the case of a large initial number of susceptibles and a small (finite) initial number of infectives, we derive the threshold behavior and the limiting distribution for the final state of the epidemic. Also, we show strong convergence of the epidemic process over any finite time interval to a birth and death process, extending the results of Ball (1983). These complement some results due to Butler (1994), who considers the case of a large initial number of infectives.