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Without widespread immunization, the road to recovery from the current COVID-19 lockdowns will optimally follow a path that finds the difficult balance between the social and economic benefits of liberty and the toll from the disease. We provide an approach that combines epidemiology and economic models, taking as given that the maximum capacity of the healthcare system imposes a constraint that must not be exceeded. Treating the transmission rate as a decreasing function of the severity of the lockdown, we first determine the minimal lockdown that satisfies this constraint using an epidemiology model with a homogeneous population to predict future demand for healthcare. Allowing for a heterogeneous population, we then derive the optimal lockdown policy under the assumption of homogeneous mixing and show that it is characterized by a bang–bang solution. Possibilities such as the capacity of the healthcare system increasing or a vaccine arriving at some point in the future do not substantively impact the dynamically optimal policy until such an event actually occurs.

We construct a general noncentral hypergeometric distribution, which models biased sampling without replacement. Our distribution is constructed from the combined order statistics of two samples; one of independent and identically distributed random variables with absolutely continuous distribution F and the other of independent and identically distributed random variables with absolutely continuous distribution G. The distribution depends on F and G only through F • G (−1) (F composed with the quantile function of G) and the standard hypergeometric distribution and Wallenius' noncentral hypergeometric distribution arise as special cases. We show in efficient economic markets the quantity traded has a general noncentral hypergeometric distribution.

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