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We adapt a SEIRD differential model with asymptomatic population and Coviddeaths, which we call SEAIRD, to simulate the evolution of COVID-19, and add acontrol function affecting both the diffusion of the virus and GDP, featuringall direct and indirect containment policies; to model feasibility, the controlis assumed to be a piece-wise linear function satisfying additionalconstraints. We describe the joint dynamics of infection and the economy anddiscuss the trade-off between production and fatalities. In particular, wecarefully study the conditions for the existence of the optimal policy responseand its uniqueness. Uniqueness crucially depends on the marginal rate ofsubstitution between the statistical value of a human life and GDP; we show anexample with a phase transition: above a certain threshold, there is a uniqueoptimal containment policy; below the threshold, it is optimal to abstain fromany containment; and at the threshold itself there are two optimal policies. Wethen explore and evaluate various profiles of various control policiesdependent on a small number of parameters.

Mortality containment vs. economics opening: optimal policies in a SEIARD model