Urn models and vaccine efficacy estimation.

We derive the distribution of the number of infections among unvaccinated and vaccinated individuals for model 1 (leaky) and model 2 (all/nothing) vaccines, assuming random mixing of a homogeneous population. For all/nothing vaccines, we show that the distribution of the number of infected vaccinated individuals conditioning on n observed infections follows a hypergeometric distribution, and the vaccine efficacy estimate (VE) can be derived from the usual estimate of the total population size in a capture-recapture sampling program. For leaky vaccines, we show that the number of vaccinated infected follows a distribution that was first derived by Wallenius. We found that the current point estimates of VE for each model perform very well, but the urn model construction presented here provides a strong framework for estimation and hypothesis testing on the parameters, and can be applied when the available data are a sample of the population. Since the method does not require an underlying transmission model, it can be applied to estimate the VE for non-contagious diseases.