  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.
