The impact of household structure on disease-induced herd immunity

Abstract

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