Estimation of waning vaccine efficacy

Whether the protection afforded by a vaccine wanes over time and if so, by how much, are important public health questions with implications for mass immunization programs. However, the measurement of such effects is fraught with difficulties. Generally, clinical trials are too short to provide evidence on waning effects. Thus it is necessary to rely on epidemiologic data, which are subject to ascertainment bias. Furthermore, standard epidemiologic measures, such as hazard ratios, may not accurately reflect waning effects. We develop two models of waning vaccine efficacy, the selection and deterioration models. In the selection model, waning arises from heterogeneity in the duration of protection. In the deterioration model, the hazard ratio of infection in vaccinated individuals relative to unvaccinated individuals increases with time. The two models are the with-waning analogs of the previously defined "all or nothing" and "leaky" models. Likelihood functions based on these models are obtained for cohort and case-report data, as typically collected in surveillance studies of vaccine efficacy. These likelihood functions involve three groups of parameters, corresponding to the hazard of infection, vaccine efficacy (possibly including waning effects), and ascertainment of infections. Our strategy is to fit a range of models incorporating a variety of assumptions about the vaccine efficacy model, ascertainment bias, and age dependence of the hazard. If allowing for waning improves the fit of the model under a range of scenarios, then we infer that it is a genuine effect. We apply the methods to two sets of data on whooping cough vaccine efficacy in the United Kingdom