We embed a lockdown choice in a simplified epidemiological model and derive formulas for the optimal lockdown intensity and duration. The optimal policy reflects the rate of time preference, epidemiological factors, the hazard rate of vaccine discovery, learning effects in the health care sector, and the severity of output losses due to a lockdown. In our baseline specification a Covid-19 shock as currently experienced by the US optimally triggers a reduction in economic activity by two thirds, for about 50 days, or approximately 9.5 percent of annual GDP.