  This article considers the minimization of the total number of infected
individuals over the course of an epidemic in which the rate of infectious
contacts can be reduced by time-dependent nonpharmaceutical interventions. The
societal and economic costs of interventions are taken into account using a
linear budget constraint which imposes a trade-off between short-term heavy
interventions and long-term light interventions. We search for an optimal
intervention strategy in an infinite-dimensional space of controls containing
multiple consecutive lockdowns, gradually imposed and lifted restrictions, and
various heuristic controls based for example on tracking the effective
reproduction number. Mathematical analysis shows that among all such
strategies, the global optimum is achieved by a single constant-level lockdown
of maximum possible magnitude. Numerical simulations highlight the need of
careful timing of such interventions, and illustrate their benefits and
disadvantages compared to strategies designed for minimizing peak prevalence.
Rather counterintuitively, adding restrictions prior to the start of a
well-planned intervention strategy may even increase the total incidence.
