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

TL;DR

This paper analyzes a stochastic SIR epidemic model on a network with household structure, deriving threshold conditions and outbreak probabilities, validated by limit theorems and simulations for finite populations.

Contribution

It introduces a branching process approximation for epidemic spread on household-structured networks, providing threshold criteria and outbreak probability calculations with rigorous limit theorems.

Findings

Threshold parameter determines epidemic outbreak potential.

Approximate formulas are accurate for large and moderate populations.

Simulation results support theoretical predictions.

Abstract

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 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.