Abstract A stochastic model for an SIR epidemic among a population of households that contains J types of individuals is considered. Infectives make two kinds of contacts: local contacts with individuals in their own household and global contacts with individuals from the entire population. Global mixing is proportionate . The behaviour of the model as the population size tends to infinity is analysed. An approximating branching process for the early stages of an epidemic is used to determine several different reproduction numbers and the early exponential growth rate. The means of certain final outcome quantities of an epidemic which takes off are determined, together with an associated multivariate central limit theorem. The assumption of proportionate global mixing leads to considerable simplification in both the calculation and proof of asymptotic properties, since key underlying processes are one-dimensional rather than J -dimensional.
Ball et al. (Sun,) studied this question.