Optimal control for a SIR epidemic model with limited quarantine

Social distance, quarantines and total lock-downs are non-pharmaceutical interventions that policymakers have used to mitigate the spread of the COVID-19 virus. However, these measures could be harmful to societies in terms of social and economic costs, and they can be maintained only for a short period of time. Here we investigate the optimal strategies that minimize the impact of an epidemic, by studying the conditions for an optimal control of a Susceptible-Infected-Recovered model with a limitation on the total duration of the quarantine. The control is done by means of the reproduction number [Formula: see text], i.e., the number of secondary infections produced by a primary infection, which can be arbitrarily varied in time over a quarantine period T to account for external interventions. We also assume that the most strict quarantine (lower bound of [Formula: see text]) cannot last for a period longer than a value [Formula: see text]. The aim is to minimize the cumulative number of ever-infected individuals (recovered) and the socioeconomic cost of interventions in the long term, by finding the optimal way to vary [Formula: see text]. We show that the optimal solution is a single bang-bang, i.e., the strict quarantine is turned on only once, and is turned off after the maximum allowed time [Formula: see text]. Besides, we calculate the optimal time to begin and end the strict quarantine, which depends on T, [Formula: see text] and the initial conditions. We provide rigorous proofs of these results and check that are in perfect agreement with numerical computations.