  We formulate an optimal control problem to determine the lockdown policy to
curb an epidemic where other control measures are not available yet. We present
a unified framework to model the epidemic and economy that allows us to study
the effect of lockdown on both of them together. The objective function
considers cost of deaths and infections during the epidemic, as well as
economic losses due to reduced interactions due to lockdown. We tune the
parameters of our model for Covid-19 epidemic and the economies of Burundi,
India, and the United States (the low, medium and high income countries). We
study the optimal lockdown policies and effect of system parameters for all of
these countries. Our framework and results are useful for policymakers to
design optimal lockdown strategies that account for both epidemic related
infections and deaths, and economic losses due to lockdown.
