---
title: "Fokker-Planck Optimal Control Framework for Stochastic Epidemic Models"
id: "pubmed-42747468"
canonical_url: "https://medichelpline.com/clinical-feed/pubmed-42747468"
content_type: "clinical_feed_article"
specialty: "Infectious Disease"
source_name: "PubMed / NCBI"
source_url: "https://pubmed.ncbi.nlm.nih.gov/42747468/"
doi: "10.1007/s00285-026-02462-7"
published_at: "2026-09-16T00:00:00.000Z"
evidence_level: "Journal Article"
license: "CC-BY-NC-4.0 / Informational Use"
---
# Fokker-Planck Optimal Control Framework for Stochastic Epidemic Models
## Provenance & Clinical Metadata
- **Canonical URL:** https://medichelpline.com/clinical-feed/pubmed-42747468
- **Specialty:** [Infectious Disease](https://medichelpline.com/clinical-feed/infectious-disease.md)
- **Primary Source:** PubMed / NCBI
- **Source URL:** [Original Journal Publication](https://pubmed.ncbi.nlm.nih.gov/42747468/)
- **DOI:** [10.1007/s00285-026-02462-7](https://doi.org/10.1007%2Fs00285-026-02462-7)
- **Published At:** 2026-09-16T00:00:00.000Z
- **Evidence Rating:** Journal Article
## Executive GIST (TL;DR)
- The paper develops a control framework for **stochastic compartmental models** in epidemiology that controls an associated **Fokker-Planck** equation rather than the stochastic trajectories directly. - Controlling the Fokker-Planck PDE steers the probability distribution of possible epidemic realizations toward a desired state, allowing robustness to uncertainty in both dynamics and initial conditions. - The authors formulate a PDE-constrained optimization problem and provide a full mathematical analysis: they prove existence of optimal controls by studying the control-to-state map and characterize optimal controls via the **Pontryagin minimum principle**. - For numerical approximation, the study describes application of the **sequential quadratic Hamiltonian method** to compute optimal control maps for the PDE-constrained problem. - The methodology is illustrated using a minimal stochastic **susceptible-infected-recovered (SIR)** model and explores different cost functionals that reflect alternative policy-maker objectives. - Keywords emphasized in the source include Epidemiology, Fokker-Planck equation, Optimal control, PDE-constrained optimization, and the sequential quadratic Hamiltonian method. - The work is authored by Christian Parkinson and Souvik Roy, published in J Math Biol (2026) with PMID 42747468 and DOI 10.1007/s00285-026-02462-7. The authors declare no conflicts of interest. - Funding acknowledgements listed include grants from the Division of Mathematical Sciences (2309491 and 2230790), indicating support for the applied mathematical development behind the framework.
## Clinical Analysis & Structured Key Points
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Affiliations Expand ### Affiliations * 1 Department of Mathematics & Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA. chparkin@msu.edu. * 2 Department of Mathematics, University of Texas at Arlington, Arlington, TX, USA. * PMID: **42747468** * DOI: [ 10.1007/s00285-026-02462-7 ](https://doi.org/10.1007/s00285-026-02462-7) Item in Clipboard # A Fokker-Planck framework for control of epidemics Christian Parkinson et al. J Math Biol. 2026. Show details Display options Display options Format Abstract PubMed PMID J Math Biol Actions * [ Search in PubMed ](https://pubmed.ncbi.nlm.nih.gov/?term=%22J+Math+Biol%22%5Bjour%5D&sort=date&sort_order=desc) * [ Search in NLM Catalog ](https://www.ncbi.nlm.nih.gov/nlmcatalog?term=%22J+Math+Biol%22%5BTitle+Abbreviation%5D) * [ Add to Search ](https://pubmed.ncbi.nlm.nih.gov/42747468/) . 2026 Sep 16;93(4):47. doi: 10.1007/s00285-026-02462-7. ### Authors [Christian Parkinson](https://pubmed.ncbi.nlm.nih.gov/?term=Parkinson+C&cauthor_id=42747468)[ 1 ](https://pubmed.ncbi.nlm.nih.gov/42747468/#short-view-affiliation-1 "Department of Mathematics & Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA. chparkin@msu.edu."), [Souvik Roy](https://pubmed.ncbi.nlm.nih.gov/?term=Roy+S&cauthor_id=42747468)[ 2 ](https://pubmed.ncbi.nlm.nih.gov/42747468/#short-view-affiliation-2 "Department of Mathematics, University of Texas at Arlington, Arlington, TX, USA.") ### Affiliations * 1 Department of Mathematics & Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI, USA. chparkin@msu.edu. * 2 Department of Mathematics, University of Texas at Arlington, Arlington, TX, USA. * PMID: **42747468** * DOI: [ 10.1007/s00285-026-02462-7 ](https://doi.org/10.1007/s00285-026-02462-7) Item in Clipboard Cite Display options Display options Format Abstract PubMed PMID ## 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. **Keywords:** Epidemiology; Fokker-Planck equation; Optimal control; PDE constrained optimization; Sequential quadratic Hamiltonian method. © 2026. The Author(s). [PubMed Disclaimer](https://pubmed.ncbi.nlm.nih.gov/disclaimer/) ## Conflict of interest statement Declarations. Conflict of interest: The authors declare no conflict of interest. ## References 1. 1. Annunziato M, Borzì A (2018) A Fokker-Planck control framework for stochastic systems. EMS Surveys Math Sci 5(1):65–98 - [DOI](https://doi.org/10.4171/emss/27) 2. 1. Antil H, Dondl P, Striet L (2021) Approximation of integral fractional laplacian and fractional PDEs via sinc-basis. SIAM J Sci Comput 43(4):A2897–A2922 - [DOI](https://doi.org/10.1137/20m1374122) 3. 1. Antil H, Leykekhman D (2018) A brief introduction to PDE-constrained optimization. Front in PDE-constrained optim 163:434 4. 1. Barbu T, Moroşanu C, Pavăl S-D (2025) A Fokker-Planck model for optical flow estimation and image registration. Mathematics 13(17):2807 - [DOI](https://doi.org/10.3390/math13172807) 5. 1. Bertozzi AL, Franco E, Mohler G, Short MB, Sledge D (2020) The challenges of modeling and forecasting the spread of COVID-19. 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