Optimal Control of Behavioral-Feedback SIR Epidemic Model

We consider a behavioral-feedback SIR epidemic model, in which the infection rate depends in feedback on the fractions of susceptible and infected agents. Contrary to the classic SIR dynamics, the considered model allows one to account for endogenous adaptation mechanisms of the agents in response to the epidemics, such as voluntary social distancing, or the adoption of face masks. For this model, we formulate an optimal control problem for a social planner that has the ability to reduce the infection rate to keep the infection curve below a certain threshold within an infinite time horizon, while minimizing the intervention cost. Based on the dynamic properties of the model, we prove that, under quite general conditions on the infection rate, the filling-the-box strategy is the optimal control. This strategy consists in letting the epidemics spread without intervention until the threshold is reached, then applying the minimum control that leaves the fraction of infected individuals constantly at the threshold until the reproduction number becomes less than one and the infection naturally fades out. Our contribution enhances the understanding of epidemic management while accounting for adaptive human behavior, offering novel insights for robust containment strategies.