Stochastic SIR epidemics in a population with households and schools

TL;DR

This paper analyzes stochastic SIR epidemic models in populations structured into households and schools, introducing new branching process approximations and comparing outbreak probabilities across different population structures.

Contribution

It introduces a novel hierarchical model and corresponding branching process approximations, providing new insights into epidemic spread in structured populations.

Findings

Epidemic spreads more easily in the independent partition model when all groups are equal size.

The hierarchical model's branching process approximation is new and of independent interest.

Results depend on the uniformity of household and school sizes.

Abstract

We study the spread of stochastic SIR (Susceptible $\to$ Infectious $\to$ Recovered) epidemics in two types of structured populations, both consisting of schools and households. In each of the types, every individual is part of one school and one household. In the independent partition model, the partitions of the population into schools and households are independent of each other. This model corresponds to the well-studied household-workplace model. In the hierarchical model which we introduce here, members of the same household are also members of the same school. We introduce computable branching process approximations for both types of populations and use these to compare the probabilities of a large outbreak. The branching process approximation in the hierarchical model is novel and of independent interest. We prove by a coupling argument that if all households and schools have the same size, an epidemic spreads easier (in the sense that the number of individuals infected is stochastically larger) in the independent partition model. We also show by example that this result does not necessarily hold if households and/or schools do not all have the same size.