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, respectively. 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 result generalizes one available in the literature for the equivalent problem formulated for the classical SIR model, which can be recovered as a special case of our model when the infection rate is constant. Our contribution enhances the understanding of epidemic management with adaptive human behavior, offering insights for robust containment strategies.

💡 Research Summary

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The paper addresses the optimal control of an epidemic model that incorporates endogenous behavioral responses through a state‑dependent infection rate. In the classical SIR framework the infection rate β is constant, which ignores how individuals adapt their contacts when the perceived risk changes. The authors therefore consider a behavioral‑feedback SIR model (CBF‑SIR) in which β = β(x, y) depends on the current fractions of susceptible (x) and infected (y) individuals. The function β is assumed to be continuously differentiable and to satisfy two monotonicity conditions: it increases with the susceptible fraction (∂β/∂x > 0) and does not increase with the infected fraction (∂β/∂y ≤ 0). These assumptions capture the empirical facts that a larger pool of susceptibles tends to raise the effective contact rate, while a higher number of active cases induces voluntary distancing or mask‑wearing, thereby reducing transmission.

An external controller can reduce the effective infection rate by applying a control signal u(t)∈