The impact of household structure on disease-induced herd immunity

The disease-induced herd immunity level [Formula: see text] is the fraction of the population that must be infected by an epidemic to ensure that a new epidemic among the remaining susceptible population is not supercritical. For a homogeneously mixing population [Formula: see text] equals the classical herd immunity level [Formula: see text], which is the fraction of the population that must be vaccinated in advance of an epidemic so that the epidemic is not supercritical. For most forms of heterogeneous mixing [Formula: see text], sometimes dramatically so. For an SEIR (susceptible [Formula: see text] exposed [Formula: see text] infective [Formula: see text] recovered) model of an epidemic among a population that is partitioned into households, in which individuals mix uniformly within households and, in addition, uniformly at a much lower rate in the population at large, we show that [Formula: see text] unless variability in the household size distribution is sufficiently large. Thus, introducing household structure into a model typically has the opposite effect on disease-induced herd immunity than most other forms of population heterogeneity. We reach this conclusion by considering an approximation [Formula: see text] of [Formula: see text], supported by numerical studies using real-world household size distributions. For [Formula: see text], we prove that [Formula: see text] when all households have size n, and conjecture that this inequality holds for any common household size n. We prove results comparing [Formula: see text] and [Formula: see text] for epidemics which are highly infectious within households, and also for epidemics which are weakly infectious within households.