Optimal Control of the SIR Model with Constrained Policy, with an Application to COVID-19

This article considers the optimal control of the SIR model with both transmission and treatment uncertainty. It follows the model presented in Gatto and Schellhorn (2021). We make four significant improvements on the latter paper. First, we prove the existence of a solution to the model. Second, our interpretation of the control is more realistic: while in Gatto and Schellhorn the control $\alpha$ is the proportion of the population that takes a basic dose of treatment, so that $\alpha >1$ occurs only if some patients take more than a basic dose, in our paper, $\alpha$ is constrained between zero and one, and represents thus the proportion of the population undergoing treatment. Third, we provide a complete solution for the moderate infection regime (with constant treatment). Finally, we give a thorough interpretation of the control in the moderate infection regime, while Gatto and Schellhorn focussed on the interpretation of the low infection regime. Finally, we compare the efficiency of our control to curb the COVID-19 epidemic to other types of control.