  Motivated by the issue of COVID-19 mitigation, in this work we tackle the
general problem of optimally controlling an epidemic outbreak of a communicable
disease structured by time since exposure, by the aid of two types of control
instruments namely, social distancing and vaccination by a vaccine at least
partly effective in protecting from infection. Effective vaccines are assumed
to be made available only in a subsequent period of the epidemic so that - in
the first period - epidemic control only relies on social distancing, as it
happened for the COVID-19 pandemic. By our analyses, we could prove the
existence of (at least) one optimal control pair, we derived first-order
necessary conditions for optimality, and proved some useful properties of such
optimal solutions. A worked example provides a number of further insights on
the relationships between key control and epidemic parameters.
