Threshold behaviour and final outcome of an epidemic on a random network with household structure

TLDR

The study introduces a stochastic SIR epidemic model where transmission occurs within equal‑size households and along random‑graph social contacts. Branching‑process approximations yield a threshold parameter, formulas for the probability of a major outbreak and the expected final epidemic size, and the model is extended to unequal‑size households. The approximations are proven exact as household number grows and simulations show they accurately predict outcomes in moderately sized populations.

Abstract

In this paper we consider a stochastic SIR (susceptible→infective→removed) epidemic model in which individuals may make infectious contacts in two ways, both within ‘households’ (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly sized finite populations. The extension to unequal-sized households is discussed briefly.