  We adapt a SEIRD differential model with asymptomatic population and Covid
deaths, which we call SEAIRD, to simulate the evolution of COVID-19, and add a
control function affecting both the diffusion of the virus and GDP, featuring
all direct and indirect containment policies; to model feasibility, the control
is assumed to be a piece-wise linear function satisfying additional
constraints. We describe the joint dynamics of infection and the economy and
discuss the trade-off between production and fatalities. In particular, we
carefully study the conditions for the existence of the optimal policy response
and its uniqueness. Uniqueness crucially depends on the marginal rate of
substitution between the statistical value of a human life and GDP; we show an
example with a phase transition: above a certain threshold, there is a unique
optimal containment policy; below the threshold, it is optimal to abstain from
any containment; and at the threshold itself there are two optimal policies. We
then explore and evaluate various profiles of various control policies
dependent on a small number of parameters.
