How much and when should we limit economic and social activity to ensure that the health-care system is not overwhelmed during an epidemic? We study a setting where ICU resources are constrained and suppression is costly. Providing a fully analytical solution we show that the common wisdom of “flattening the curve”, where suppression measures are continuously taken to hold down the spread throughout the epidemic, is suboptimal. Instead, the optimal suppression is discontinuous. The epidemic should be left unregulated in a first phase and when the ICU constraint is approaching society should quickly lock down (a discontinuity). After the lockdown, regulation should gradually be lifted, holding the rate of infected constant, thus respecting the ICU resources while not unnecessarily limiting economic activity. In a final phase, regulation is lifted.

We call this strategy “filling the box”. The cost under the optimal strategy is obtained in closed form as an explicit function of economic and medical fundamentals. We show that the policy is optimal also when there, in addition, is a small cost associated with the number of infected. The tighter the ICU constraint, the wider is the range of such cost for which the policy is still optimal. This suggests the primary focus of poor countries (with few ICU resources) should be to protect the health-care system, while richer countries (with extensive ICU resources) may strive to reduce the number of infected even more.