The impact of household structure on disease-induced herd immunity

Abstract

The disease-induced herd immunity level h_D 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 h_D equals the classical herd immunity level h_C, 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 h_D<h_C, sometimes dramatically so. For an SEIR (susceptible rightarrow exposed rightarrow infective rightarrow 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 h_D>h_C 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 {tilde{h}}_D of h_D, supported by numerical studies using real-world household size distributions. For n=2, 3, we prove that {tilde{h}}_D>h_C when all households have size n, and conjecture that this inequality holds for any common household size n. We prove results comparing {tilde{h}}_D and h_C for epidemics which are highly infectious within households, and also for epidemics which are weakly infectious within households.