On the problem of minimizing the epidemic final size for SIR model by social distancing

We revisit the problem of minimizing the epidemic final size in the SIR model through social distancing of a bounded intensity. In the existing literature, this problem was considered imposing a priori interval structure on the time period when interventions are enforced. We show that the support of the optimal control is still a single time interval when considering the more general class of controls with an $ L^1 $ constraint on the confinement effort that reduces the infection rate. There is thus no benefit in splitting interventions on several disjoint time periods. However, if the infection rate is known beforehand to change with time once from one value to another one, then we show that the optimal solution may consist in splitting the interventions in at most two disjoint time periods.