  A large number of recent studies consider a compartmental SIR model to study
optimal control policies aimed at containing the diffusion of COVID-19 while
minimizing the economic costs of preventive measures. Such problems are
non-convex and standard results need not to hold. We use a Dynamic Programming
approach and prove some continuity properties of the value function of the
associated optimization problem. We study the corresponding
Hamilton-Jacobi-Bellman equation and show that the value function solves it in
the viscosity sense. Finally, we discuss some optimality conditions. Our paper
represents a first contribution towards a complete analysis of non-convex
dynamic optimization problems, within a Dynamic Programming approach.
