THE USE OF CHAIN-BINOMIALS WITH A VARIABLE CHANCE OF INFECTION FOR THE ANALYSIS OF INTRA-HOUSEHOLD EPIDEMICS

Abstract

An important characteristic of highly infectious diseases is that they tend to produce groups of cases occurring within households rather than separate cases scattered throughout the community. If all the cases in a household were due to some single source of infection such as typhoid-infected water, then we should expect the total number of cases to follow a binomial distribution. On the other hand, if the disease were introduced into the household by one of its members and then transmitted from person to person, a very different situation would arise. For a disease involving a short period of high infectivity and an approximately constant incubation period, we should expect to be able to distinguish different generations of the intra-household epidemic. There might be a binomial distribution of secondary cases resulting from contact with the primary case, to be followed later by another binomial distribution of tertiary cases amongst the susceptibles who had previously escaped, and so on. In his classic paper on this subject Greenwood (1931) introduced such a chainbinomial model for the investigation of measles epidemics, for which the assumptions made are thought to be approximately true. When the period of infectivity is more extended, as with scarlet fever or whooping cough for example, the stochastic model recently discussed by Bailey (1953) may be more appropriate. Greenwood examined data on the 1926 measles epidemic in St Pancras, and showed that, so far as the total number of cases in a household was concerned, the hypothesis of a simple binomial distribution was quite inadequate, while the distribution expected from the chain-binomial model gave a satisfactory fit to the observations. It is important to notice, however, that this material provided information only on the total size of the epidemic in a given household; no analysis of the individual links of the chain was possible. Now Wilson, Bennett, Allen & Worcester (1939), in their investigation of cases of measles occurring in Providence, Rhode Island, during 1929-34, were able to go a stage further and break down the data into its constituent parts. Of course it must be recognized that departures from a constant incubation period, or the possibility of multiple primary cases, give rise to certain difficulties in the so-called chaining of this kind of material. However, it seemed likely that in the present case this source of confusion would not be serious. Wilson et al. were able to show that in none of the groups of available data did the Greenwood model give an adequate description of the numbers of cases occurring in the separate generations through which the epidemic passed, though it did sometimes give a satisfactory fit to the distribution of the total number of cases in a household. In a later paper, Wilson (1947) considered a somewhat different approach. According