  The recent COVID-19 pandemic highlighted the need of non-pharmaceutical
interventions in the first stages of a pandemic. Among these, lockdown policies
proved unavoidable yet extremely costly from an economic perspective. To better
understand the tradeoffs between economic and epidemic costs of lockdown
interventions, we here focus on a simple SIR epidemic model and study lockdowns
as solutions to an optimal control problem. We first show numerically that the
optimal lockdown policy exhibits a phase transition from suppression to
mitigation as the time horizon grows, i.e., if the horizon is short the optimal
strategy is to impose severe lockdown to avoid diffusion of the infection,
whereas if the horizon is long the optimal control steers the system to herd
immunity to reduce economic loss. We then consider two alternative policies,
motivated by government responses to the COVID-19 pandemic, where lockdown
levels are selected to either stabilize the reproduction number (i.e., "flatten
the curve") or the fraction of infected (i.e., containing the number of
hospitalizations). We compute analytically the performance of these two
feedback policies and compare them to the optimal control. Interestingly, we
show that in the limit of infinite horizon stabilizing the number of infected
is preferable to controlling the reproduction number, and in fact yields close
to optimal performance.
