Contributions to the mathematical theory of epidemics. II. —The problem of endemicity

Abstract

In a previous communication an attempt was made to investigate mathe­matically the course of an epidemic in a closed population of susceptible individuals. In order to simplify the problem certain definite assumptions were made, namely, that all individuals were equally susceptible, and that death resulted, or complete immunity was conferred, as the result of an attack. The infectivity of the individual and his chances of death or recovery were represented by arbitrary functions, and the chance of a new infection occurring was assumed to be proportional to the product of the infected and susceptible members of the population. In spite of the introduction of the arbitrary functions, it was shown that in general a critical density of population existed, such that if the actual density was less than this, no epidemic could occur, but if it exceeded this by n an epidemic would appear on the introduction of a focus of infection, and further that if n was small relative to the population density, the size of the epidemic would be 2 n per unit area. It was shown that these conclusions could be readily extended to the case of a metaxenous disease, that is, one in which transmission takes place through an intermediate host. It is the purpose of the present paper to consider the effect of the continuous introduction of fresh susceptible individuals into the population. It appeared desirable to investigate this point, since it might make it possible to interpret certain aspects of the incidence of disease not only in human communities where there is usually an influx of fresh susceptible individuals either by immigration or by birth, but also in the animal experiments carried out by Topley and others—where fresh animals were introduced at a constant rate into the cages in which cases of disease were already present—from which certain definite results were obtained.