An optimal feedback control that minimizes the epidemic peak in the SIR model under a budget constraint

We give the explicit solution of the optimal control problem which consists in minimizing the epidemic peak in the SIR model when the control is an attenuation factor of the infectious rate, subject to a L1 constraint on the control which represents a budget constraint. The optimal strategy is given as a feedback control which consists in a singular arc maintaining the infected population at a constant level until the immunity threshold is reached, and no intervention outside the singular arc. We discuss and compare this strategy with the one that minimizes the peak when fixing the duration of a single intervention, as already proposed in the literature. Numerical simulations illustrate the benefits of the proposed control.