Contribution to the Theory of Epidemics

Abstract

In the present paper we shall discuss the following model of epidemics proposed and studied by Neyman and Scott [1]. We assume that an individual, infected at a given moment, becomes infectious after a certain fixed period of incubation and that the interval of time during which he may infect others is of length 0. (These assumptions about the incubation period and period of infectiousness are not essential for our model, since we shall be interested only in sizes of “generations” of epidemics, which can be defined without using time coordinates.) Between the time the individuals get infected and become infectious they may travel over the habitat, denoted in this paper by X. Thus, our model will depend upon

• 1. The family of probability measures µ _u (•), u ∈ X, governing the movements of individuals between the time they get infected and the time they become infectious. [We assume that the set X is a measure space and that all µ _u (•) are defined on the same Borel σ-field of subsets of X.] We interpret µ _u (X) as the probability that an individual infected at u shall become infectious at some point in the set X.

• 2. The family of probability distributions p (k | x), x ∈ X, k = 0, 1, 2,…, where p (k | x) is interpreted as the probability that an infectious at x will infect exactly k individuals. We assume that the functions p (k | x) are measurable for every k. We also assume that all individuals travel and infect independently of each other.

This investigation originated from a seminar on problems of health held at the Statistical Laboratory, University of California, Berkeley, and was supported (in part) by a research grant (No. GM-10525) from the National Institutes of Health, Public Health Service.