Household epidemic models revisited

TL;DR

This paper analyzes a stochastic household epidemic model with global and local contacts, deriving large population properties, and investigates how modifications affect outbreak probabilities and final size, with results on monotonicity under certain conditions.

Contribution

It introduces a generalized household epidemic model with a new analysis framework, including a central limit theorem for the epidemic's final size and monotonicity results for outbreak probability and size.

Findings

Large population properties derived, including a CLT for final size.

Outbreak probability increases with household size and contact probability.

Final size increases with household size and contact probability under log-convexity.

Abstract

We analyse a generalized stochastic household epidemic model defined by a bivariate random variable $(X_G, X_L)$, representing the number of global and local infectious contacts that an infectious individual makes during their infectious period. Each global contact is selected uniformly among all individuals and each local contact is selected uniformly among all other household members. The main focus is when all households have the same size $h \geq 2$, and the number of households is large. Large population properties of the model are derived including a central limit theorem for the final size of a major epidemic, the proof of which utilises an enhanced embedding argument. A modification of the epidemic model is considered where local contacts are replaced by global contacts independently with probability $p$. We then prove monotonicity results for the probability of the major outbreak and the limiting final fraction infected $z$ (conditioned on a major outbreak). a) The probability of a major outbreak is shown to be increasing in both $h$ and $p$ for any distribution of $X_L$. b) The final size $z$ increases monotonically with both $h$ and $p$ if the probability generating function (pgf) of $X_L$ is log-convex, which is satisfied by traditional household epidemic models where $X_L$ has a mixed-Poisson distribution. Additionally, we provide counter examples to b) when the pgf of $X_L$ is not log-convex.