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This paper considers a stochastic SIR (susceptible$\to$infective$\to$removed)epidemic model in which individuals may make infectious contacts in two ways,both within `households' (which for ease of exposition are assumed to haveequal size) and along the edges of a random graph describing additional socialcontacts. Heuristically-motivated branching process approximations aredescribed, which lead to a threshold parameter for the model and methods forcalculating the probability of a major outbreak, given few initial infectives,and the expected proportion of the population who are ultimately infected bysuch a major outbreak. These approximate results are shown to be exact as thenumber of households tends to infinity by proving associated limit theorems.Moreover, simulation studies indicate that these asymptotic results providegood approximations for modestly sized finite populations. The extension tounequal sized households is discussed briefly.

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