This paper presents a mathematically formalized approach which points out the relation between efficacy and effectiveness of vaccines. The first term denotes the relative degree of protection in clinical trials or under ideal conditions, while the latter is based on observed real-life data. We define the efficacy by a similar formula to the effectiveness, but the probabilities involved in the relative risk are conditional with respect to the exposure to the virus. If exposure and vaccination status are independent, the two quantities are equal. Otherwise, the observed value of the effectiveness is a biased one, as it could be seen for example in the context of COVID-19 vaccination. This phenomenon is explained by the main result of this paper, which shows a relation between effectiveness and efficacy involving the degree of discrimination, a factor which quantifies the asymmetric exposure to the virus of the vaccinated compared to the unvaccinated group.



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