A Fokker-Planck Framework for Control of Epidemics

TL;DR

The paper addresses robust control of epidemic spread when only the distribution of possible states is targeted, accounting for uncertainty in initial data. It formulates a PDE-constrained optimization problem for the Fokker-Planck equation associated with a stochastic SIR model and proves the existence of optimal controls along with first-order necessary conditions via a Pontryagin minimum principle. It introduces the Sequential Quadratic Hamiltonian (SQH) method to compute optimal controls and demonstrates the approach on a minimal stochastic SIR with three controls ($\alpha$, $v$, $\eta$) under different cost structures, showing that distribution-level control can slow, flatten, or preserve susceptible mass depending on policy priorities. The work offers a rigorous, implementable framework for policy design under uncertainty and points to scalable strategies for higher-dimensional, heterogeneous epidemic models using advanced PDE discretizations.

Abstract

We present a control framework for stochastic compartmental models in epidemiology. In this framework, rather than directly controlling the stochastic system, we perform optimal control of an associated Fokker-Planck equation, with the goal of steering the distribution of possible solutions of the stochastic system to some desirable state. In particular, this allows for robust control mechanism with uncertainty not only in the dynamics, but also in the initial data. We formulate and fully analyze a partial differential equation constrained optimization problem, including a proof of existence of optimal controls via analysis of the control-to-state map, and a characterization of optimal controls via the Pontryagin minimum principle. We describe the application of the sequential quadratic Hamiltonian method to our problem, which provides numerical approximations of optimal control maps. We demonstrate our method using a minimal stochastic susceptible-infected-recovered model with different choices of cost functionals that represent different policy-maker concerns.

Figures (5)

• Figure 1: The flow diagram for \ref{['eq:controlledSIR']} in the absence of noise. Colored lines represent our modifications to the basic SIR model; these are the terms where the control variables $\alpha,\eta,v$ appear.
• Figure 2: The dynamics corresponding to our choice of parameters in the uncontrolled case (i.e. $\alpha,\eta,v \equiv 0$). Top: the deterministic disease dynamics governed by \ref{['eq:controlledSIR']}. Bottom: snapshots of the solution $f(x,t)$ of the Fokker-Planck equation \ref{['eq:FPcontrolled']} in the form of contour plots with higher values in red and lower values in blue.
• Figure 3: Scenario 1: $G(x,t) = x_2 = I$, $K(x) \equiv 0$. In this scenario, the optimal control employs the maximal level of treatment efforts ($\eta(t)$) in the initial stage of infections, and as infections near their peak, these efforts are supplemented with modest use of NPIs ($\alpha(t)$) and vaccination ($v(t)$).
• Figure 4: Scenario 2: $G(x,t) = 1_{\{I\ge 0.15\}}(x)$, the indicator function of the set $\{I \ge 0.15\}$, $K(x) \equiv 0$. Here $0.15$ is imagined to be the infection threshold past which the healthcare system would be overburdened. In this scenario, the optimal control uses the maximal level of treatment efforts ($\eta(t)$) in the initial stage of infections, the maximum level of vaccination ($v(t)$) during the peak infections, and a substantially stronger employment of NPIs ($\alpha(t)$) than in Scenario 1. This has the effect of pushing the bulk of the mass for the distribution $f(x,t)$ below the line $I = 0.15$ (grey dotted line) much more quickly than in Scenario 1. With these controls, the dynamics (top left) exhibit a more prolonged but flattened epidemic.
• Figure 5: Scenario 3: $G(x,t) \equiv 0$, $K(x) = -\max(S-0.3,0)$, $v(t)\equiv 0$. Here we simulate a scenario where vaccination is unavailable, and the only goal is to maintain a sufficiently large susceptible population by the end time. In this case, the optimal strategy entails very strong use of NPIs ($\alpha(t)$), and little to no use of treatment efforts ($\eta(t)$). The result is that the mass of the distribution function $f(x,t)$ moves much more slowly toward states with lower $S$ value, and ends with a nontrivial portion of the mass above the threshold $S = 0.3$ (green dotted line). In this case, the particular dynamics land below this threshold at time $t = 10$ (top left).

Theorems & Definitions (16)

• Proposition 2.1
• Remark 2.1
• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Proposition 3.1
• Lemma 3.2
• proof
• Theorem 3.2