Epidemics Between Bailey and Reed-Frost: A Stochastic Version of the Kermack and Mckendrick Model

Abstract

A model is developed that describes evolution with respect to time of an infectious disease introduced into a population of susceptibles. The proposed model incorporates at one end Bailey's simple stochastic epidemic and at the other end the Reed-Frost chain-binomial models and is the natural stochastic analogue of Kermack and McKendrick's deterministic model. The epidemic process is characterized by the size of the population and by two infectivity functions. The first one relates to a time dependent outside source of infection. The second one describes the infectivity of an individual as a function of his age-of-infection, that is the time elapsed since his own infection. The proposed model consists of a set of partial differential equations which governs, steered by the given infectivity functions, the evolution with respect to time of a set of density functions. These density functions deliver a complete stochastic description of the infectious-age structure of the population at any moment of time. An expression for the size of the epidemic, that is the probability distribution of the number of infectives, as a function of time follows. Also expressions for the expected arrival times of infectives, useful for the inverse problem, are developed. By letting time tend to infinity earlier results for the final size of the epidemic are confirmed.