An estimation procedure for household disease data

Abstract

SUMMARY A procedure is proposed for estimating a parameter of an epidemic model from household data. The procedure is essentially a method of moments for martingales. It is computationally convenient and is less restrictive in its model assumptions than the maximum likelihood methods used so far. The approach is illustrated with measles data and is also shown to be useful when households contain susceptibles of different types. This paper is concerned with data generated by the spread of an infectious disease within households. Each of the households initially has one or more infectives, and susceptibles may become infected by adequately close contacts with infectives from the same household. A newly infected individual passes through a stage during which he is latent and then through a stage during which he is infectious, before being removed by isolation, by death or by naturally losing his infectiousness and becoming immune for the remaining duration of the epidemic. Such data are usually analysed by maximum likelihood methods using one of two types of epidemic models to describe the spread of the disease within a household. Either a chain binomial model, like those initially introduced by Greenwood and Reed-Frost, is used or the Kermack-McKendrick model, as formulated in its stochastic form by Bartlett (1949), is used. An extensive discussion of these models and methods is given in the book by Bailey (1975). The method proposed in this paper is based on an approach suggested for counting processes in a Berkeley Ph.D. thesis by 0. 0. Aalen and is essentially a method of moments for martingales constructed from the epidemic model. There are two advantages of the resulting estimator and its asymptotic normality. One advantage is its computational convenience and another is that it applies for an epidemic model with more realistic assump- tions about the durations of the latent and infectious periods than those made in the chain binomial and Kermack-McKendrick models. We take for our epidemic model a continuous time model. The duration of the latent period and the duration of the infectious period are taken to be independent random variables which, apart from in ? 4, are taken to have arbitrary distributions. We let v denote the mean dura- tion of the infectious period. Infection occurs, within households, according to an orderly point process with intensity at any time being jointly proportional to the number of suscep- tibles and the number of infectious individuals in the household at that time. The constant of proportionality ,B is the within-household infection rate.