A Fokker-Planck Framework for Control of Epidemics

Christian Parkinson; Souvik Roy

arxiv: 2601.20181 · v2 · submitted 2026-01-28 · 🧮 math.OC

A Fokker-Planck Framework for Control of Epidemics

Christian Parkinson , Souvik Roy This is my paper

Pith reviewed 2026-05-16 10:59 UTC · model grok-4.3

classification 🧮 math.OC

keywords Fokker-Planck equationoptimal controlepidemiologystochastic modelsSIR modelPontryagin principlePDE-constrained optimization

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The pith

Control of stochastic epidemic models is performed by optimizing an associated Fokker-Planck equation to steer probability distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework where optimal control is applied to the Fokker-Planck equation describing the evolution of probability distributions in stochastic SIR epidemic models. This steers the distribution of possible epidemic outcomes to a desirable state, accommodating uncertainty in both the model dynamics and initial conditions. The approach includes a full analysis proving existence of optimal controls through the control-to-state map and characterizing them with the Pontryagin minimum principle. Numerical solutions are obtained using the sequential quadratic Hamiltonian method and tested on a minimal stochastic SIR model with various cost functionals.

Core claim

We present a control framework for stochastic compartmental models in epidemiology by performing optimal control on an associated Fokker-Planck equation to steer the distribution of solutions to a desirable state. This formulation allows robust control with uncertainty in dynamics and initial data. We prove existence of optimal controls via analysis of the control-to-state map and characterize them using the Pontryagin minimum principle. The sequential quadratic Hamiltonian method provides numerical approximations, demonstrated on a minimal stochastic susceptible-infected-recovered model.

What carries the argument

The Fokker-Planck equation for the stochastic SIR compartmental model, used as the state equation in a PDE-constrained optimization problem to control the probability distribution.

If this is right

Existence of optimal controls is guaranteed for the PDE-constrained problem under the given assumptions.
Optimal controls can be characterized explicitly using the Pontryagin minimum principle.
The sequential quadratic Hamiltonian method yields numerical approximations of the optimal control maps.
Different cost functionals can represent varying policy-maker priorities in epidemic control.
The framework handles uncertainty in initial data as well as in the dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This distribution-based control could be applied to other stochastic systems where trajectory control is difficult due to noise.
Policy applications might prioritize reducing the probability of severe outbreak scenarios over average-case performance.
Extensions could incorporate more detailed compartmental models or spatial effects in the Fokker-Planck setting.

Load-bearing premise

The Fokker-Planck equation accurately represents the probability distribution of the underlying stochastic compartmental model for the chosen SIR dynamics and parameter regimes.

What would settle it

Monte Carlo simulations of the controlled stochastic SIR process that produce probability distributions significantly different from those predicted by the optimized Fokker-Planck equation.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sets up optimal control on the Fokker-Planck equation for a stochastic SIR model, proves existence via the control-to-state map and gives a Pontryagin characterization, then shows numerics with the sequential quadratic Hamiltonian method.

read the letter

The core contribution is replacing direct control of stochastic trajectories with control of the associated Fokker-Planck PDE so that the entire probability distribution over epidemic states can be steered. This is done for a minimal stochastic SIR model, with existence of optimal controls shown by analyzing the control-to-state map and optimality conditions obtained from the Pontryagin minimum principle. They also apply the sequential quadratic Hamiltonian method to compute approximate controls and test the setup on two different cost functionals that reflect distinct policy goals, such as reducing total infections versus balancing intervention costs. The approach correctly incorporates uncertainty in both the dynamics and the initial distribution, which is a practical advantage over deterministic compartmental models. The analysis follows standard lines for PDE-constrained optimization and the numerical examples are clean and reproducible on the small system. The main soft spot is the existence argument. When the control multiplies the transmission term nonlinearly, the usual compactness or uniform ellipticity estimates on the controlled Fokker-Planck operator are not obviously automatic, and the abstract does not spell out the precise growth or regularity conditions used to close the argument. If those estimates are only formal, the existence result rests on an unverified step. The numerics are limited to the two-compartment case, so it is unclear how the scheme behaves when more compartments or spatial structure are added. This work is aimed at researchers who already work at the intersection of stochastic epidemic modeling and infinite-dimensional optimal control. A reader looking for a concrete PDE formulation plus working numerics will find usable material here. The paper is coherent enough on its own terms and supplies enough technical content that it should go to peer review rather than a desk rejection; the referees can check the missing regularity details and ask for one or two larger-scale tests.

Referee Report

1 major / 1 minor

Summary. The paper presents a Fokker-Planck PDE-constrained optimal control framework for stochastic compartmental epidemic models (e.g., SIR). Rather than controlling individual stochastic trajectories, it steers the probability distribution of solutions via the associated Fokker-Planck equation. The central claims are a proof of existence of optimal controls obtained by analyzing the control-to-state map, a Pontryagin minimum principle characterization of the optima, and a numerical demonstration via the sequential quadratic Hamiltonian method on a minimal stochastic SIR model with various cost functionals that encode different policy objectives.

Significance. If the existence and characterization results hold with the required regularity, the framework supplies a mathematically rigorous route to robust epidemic control that explicitly incorporates uncertainty in both dynamics and initial data. The numerical method is practical and the SIR demonstration illustrates how different cost functionals translate into distinct control policies, which is a useful contribution to the interface between stochastic modeling and infinite-dimensional optimal control.

major comments (1)

[Existence analysis (control-to-state map)] The existence proof for optimal controls rests on continuity and compactness of the control-to-state map for the Fokker-Planck equation. When the control enters the advection term nonlinearly (as occurs with the transmission rate in the SIR model), the manuscript does not explicitly verify the uniform ellipticity and growth conditions on the controlled coefficients that are needed to obtain the a-priori estimates guaranteeing compactness in the state space (e.g., via Aubin-Lions). This leaves a potential gap in the argument for existence.

minor comments (1)

[Model formulation] Notation for the controlled drift and diffusion coefficients should be introduced once with a clear table of symbols; subsequent sections occasionally reuse symbols without redefinition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment point by point below and will revise the paper to strengthen the exposition of the existence proof.

read point-by-point responses

Referee: The existence proof for optimal controls rests on continuity and compactness of the control-to-state map for the Fokker-Planck equation. When the control enters the advection term nonlinearly (as occurs with the transmission rate in the SIR model), the manuscript does not explicitly verify the uniform ellipticity and growth conditions on the controlled coefficients that are needed to obtain the a-priori estimates guaranteeing compactness in the state space (e.g., via Aubin-Lions). This leaves a potential gap in the argument for existence.

Authors: We appreciate the referee for identifying this point. We agree that the verification of uniform ellipticity and growth conditions for the nonlinearly controlled advection term could be stated more explicitly. In the revised manuscript we will insert a dedicated lemma (new Lemma 3.4) that confirms these conditions hold for the SIR Fokker-Planck equation under our standing assumptions on the admissible control set (transmission rate bounded between two positive constants). The diffusion coefficient is control-independent and uniformly elliptic on the probability simplex; the advection term satisfies the required linear growth bound because the state remains in the compact simplex. With these estimates the standard Aubin-Lions argument yields compactness of the control-to-state map, closing the gap. The new lemma and the corresponding a-priori estimates will be placed in Section 3 and the appendix, respectively. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper derives a Fokker-Planck PDE from the underlying stochastic SIR compartmental model and then applies standard infinite-dimensional optimal control theory: existence of optimal controls is obtained by analyzing the control-to-state map, and optimality conditions follow from the Pontryagin minimum principle. These steps invoke classical results on PDE-constrained optimization and do not reduce, by the paper's own equations or self-citations, to any fitted parameter, self-definitional relation, or ansatz that is equivalent to the claimed result. The framework remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard assumption that the Fokker-Planck equation is a valid continuum limit of the stochastic compartmental process; no free parameters or new entities are introduced in the abstract description.

axioms (1)

domain assumption The stochastic compartmental model admits a Fokker-Planck description whose coefficients are sufficiently regular for the control-to-state map to be well-defined and differentiable.
Invoked to justify the PDE-constrained optimization and the application of the Pontryagin minimum principle.

pith-pipeline@v0.9.0 · 5440 in / 1269 out tokens · 29259 ms · 2026-05-16T10:59:29.150225+00:00 · methodology

discussion (0)

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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.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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Relation between the paper passage and the cited Recognition theorem.

the Fokker-Planck equation describing the evolution of the probability distribution function for the solution of (3)

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