Estimating vaccine efficacy from small outbreaks

Let C_V and C₀ denote the number of cases among vaccinated and unvaccinated individuals, respectively, and let v be the proportion of individuals vaccinated. The quantity ê = 1 -(1-v)C _V/(vC₀) = 1 -(relative attack rate) is the most used estimator of the effectiveness of a vaccine to protect against infection. For a wide class of vaccine responses, a family of transmission models and three types of community settings, this paper investigates what ê actually estimates. It does so under the assumption that the community is large and the vaccination coverage is adequate to prevent major outbreaks of the infectious disease, so that only data on minor outbreaks are available. For a community of homogeneous individuals who mix uniformly, it is found that ê estimates a quantity with the interpretation of 1 -(mean susceptibility, per contact, of vaccinees relative to unvaccinated individuals). We provide a standard error for ê in this setting. For a community with some heterogeneity ê can be a very misleading estimator of the effectiveness of the vaccine. When individuals have inherent differences, ê estimates a quantity that depends also on the inherent susceptibilities of different types of individual and on the vaccination coverage for different types. For a community of households, ê estimates a quantity that depends on the rate of transmission within households and on the reduction in infectivity induced by the vaccine. In communities that are structured, into households or age-groups, it is possible that ê estimates a value that is negative even when the vaccine reduces both susceptibility and infectivity.