A RANDOM GRAPH MODEL FOR THE FINAL‐SIZE DISTRIBUTION OF HOUSEHOLD INFECTIONS

Abstract

In epidemiological/disease control studies, one might be interested in estimating the parameters community probability infection (CPI) and the household secondary attack rate (SAR), as introduced by Longini and Koopman. The quasi-binomial distribution I (QBD I) with parameters n, p and θ, introduced by Consul, is proposed as a model for the final-size distribution of household infections, where p (CPI) is the probability of an individual being infected from the community and θ (SAR) is the rate of secondary transmission of infection within household. An individual can be infected either from within the household or from the community. Let X be the total number of infected members in a household of size n. Then the distribution of X is given by the QBD I with the probability mass function: \[ P(X=x;n,p,\theta )=\binom nxp(p+x\theta )⁁{x-1}(1-p-x\theta )⁁{n-x},\quad x=0,1,\ldots,n \] P(X=x; n, p, θ)=(n x)p(p+xθ)x-1(1−p−xθ)n-x, x=0,1,…,n with 0<p<1, θ⩾0 such that p+nθ<1. The epidemic model is derived from a directed random graph. Data from influenza epidemics in Asian and American households are used to test the model and a comparison is made with the Longini–Koopman model. It is shown empirically that the QBD I is as good as the L–K model in describing the household infectious disease data, and both models provide almost identical estimates for community and household transmission parameters although they are derived from different perspectives and conditions.