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

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.

Analysis of a stochastic SIR epidemic on a random network incorporating household structure

“…We can view this as a branching process if we consider the households infected by a local epidemic initiated by a single infective within a typical household to be the children of that household. It is proved in Ball et al [11] that (when the household size is fixed) this approximation becomes exact as m → ∞, in that the size of the epidemic process converges in distribution to the total progeny of the approximating branching process.…”

“…In addition, the paper [11] assumes that all households are of the same size, but we conjecture that the results still hold when household size varies under very mild conditions on the household size distribution. The paper [11] focuses on establishing rigorous limit theorems, based on couplings between processes associated with the epidemic and certain branching processes, which show that many quantities of interest in the model can be expressed, asymptotically, in terms of the criticality and extinction probabilities of these branching processes. In this paper we focus on the more applied aspects of analysing this model, using these more theoretical results to derive methods for finding properties of the model.…”

“…In a recent paper, Ball et al [11], the authors formulate and analyse a new SIR (susceptible → infective → removed) epidemic model featuring mixing on two levels: local contacts with individuals in the same household and global contacts with an individual's neighbours in a random network with specified degree distribution. In that paper, rigorous branching process approximations are developed, which lead to a threshold theorem determining whether a major outbreak is possible and the probability of such an outbreak, as well as results which allow one to determine the expected proportion of the population that will be infected by such a major outbreak.…”

“…Though the proofs in [11] are valid for general infectious period distributions, explicit calculations for evaluating the probability of a major outbreak are only done for two special cases of infectious period distribution. In addition, the paper [11] assumes that all households are of the same size, but we conjecture that the results still hold when household size varies under very mild conditions on the household size distribution.…”

Solvability of implicit final size equations for SIR epidemic models

“…Our aim is to investigate the final epidemic size given by this heterogeneous mean-field approximation. We note that networks with heterogeneous degree distribution can also be modeled by the edge based compartmental model [17,23] and the household structure can also be taken into account [10].…”