Optimal Control of Behavioral-Feedback SIR Epidemic Model

TL;DR

This work extends classical SIR epidemic modeling by incorporating behavioral feedback into the infection rate and studies its optimal control under an infinite-horizon, threshold-constrained objective. It proves that, under mild regularity and monotonicity conditions on the feedback, the infection curve is unimodal and the optimal policy is a threshold-based filling-the-box strategy: let the outbreak unfold to a prescribed infection threshold, then apply the minimum intervention needed to hold the infected fraction at the threshold until the reproduction number drops below one and the outbreak fades. The analysis combines geometric phase-space methods for the uncontrolled system with a carefully constructed candidate value function to establish optimality and uniqueness, even in the absence of a global invariant. A key contribution is the demonstration that the filling-the-box policy, previously known for the classical SIR model, extends to a broad class of behavioral-feedback SIR models, thereby guiding robust containment strategies in the presence of adaptive human behavior.

Abstract

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 \emph{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.

Figures (3)

• Figure 1: Numerical simulation of the CBF-SIR epidemic model \ref{['control-system']} with $\beta(x,y)= 0.35 x(1-y)$ and $\gamma=0.05$. The top plot shows in blue the uncontrolled solution and in green the solution corresponding to the optimal control signal $u^*(t)$\ref{['feedback']}-\ref{['eq:mu']} with threshold $\bar{y} = 0.2$. The optimal control signal is reported in the bottom plot.
• Figure 2: Phase portrait of the uncontrolled BF-SIR model \ref{['uncontrol-system']} on the state space $\mathcal{S}$, with recovery rate $\gamma=0.05$ and infection rate \ref{['eq:example2']} with $b(x)=(x+2)/10$ and $a =0.5$.
• Figure 3: Numerical counterexample illustrating the filling-the-box strategy for two different thresholds: (a) $\bar{y} = 0.11$ and (b) $\bar{y} = 0.154$. In each case, the top plot shows the infection dynamics: the uncontrolled epidemic is represented by the blue dashed curve, while the controlled trajectory under the filling-the-box strategy is plotted in green. The bottom plot reports the corresponding control function applied over time.

Theorems & Definitions (31)

• Proposition 1
• proof
• Remark 1
• Example 1
• Definition 1
• Theorem 1
• proof
• Proposition 2
• proof
• Lemma 1