One of the principal ways nations are responding to the COVID-19 pandemic is by locking down portions of their economies to reduce infectious spread . This is expensive in terms of lost jobs, lost economic productivity, and lost freedoms . So it is of interest to ask: What is the optimal intensity with which to lockdown, and how should that intensity vary dynamically over the course of an epidemic? This paper explores such questions with an optimal control model that recognizes the particular risks when infection rates surge beyond the healthcare system's capacity to deliver appropriate care . The analysis shows that four broad strategies emerge, ranging from brief lockdowns that only``smooth the curve"to sustained lockdowns that prevent infections from spiking beyond the healthcare system's capacity . Within this model, it can be optimal to have two separate periods of locking down, so returning to a lockdown after initial restrictions have been lifted is not necessarily a sign of failure . Relatively small changes in judgments about how to balance health and economic harms can alter dramatically which strategy prevails . Indeed, there are constellations of parameters for which two or even three of these distinct strategies can all perform equally well for the same set of initial conditions; these correspond to so-called triple Skiba points . The performance of trajectories can be highly nonlinear in the state variables, such that for various times t, the optimal unemployment rate could be low, medium, or high, but not anywhere in between . These complex dynamics emerge naturally from modeling the COVID-19 epidemic and suggest a degree of humility in policy debates . Even people who share a common understanding of the problem's economics and epidemiology can prefer dramatically different policies . Conversely, favoring very different policies is not evidence that there are fundamental disagreements.