Discrete-Time Household Epidemic Models

Abstract We present a general Markovian discrete-time SIR household epidemic model based on time units of a day. The model is flexible in how within-household infection depends upon the number of infectives at a given time and the interactions between global (between-household) and local (within-household) infections over the course of a day. Consequently, the temporal behaviour of the epidemic is important in studying final outcomes of the epidemic such as the final size. A branching process approximation is derived for the early stages of the epidemic initiated by a single infective. We also obtain a functional central limit theorem for the temporal evolution of the epidemic starting from a strictly positive fraction of the population infected in the limit as the population size tends to infinity. By combining the branching process approximation and functional central limit theorem we provide insight into the final size of the epidemic model. This enables us to provide fresh understanding of special cases of the generic model such as a time-of-day model, where individuals alternate between infecting in the community and within their household, and a household version of the Greenwood model.