[Paper Review] Epidemic control via stochastic optimal control

Andrew Lesniewski|arXiv (Cornell University)|Apr 14, 2020

COVID-19 epidemiological studies22 references4 citations

TL;DR

This paper formulates optimal epidemic control using a stochastic SIR model with vaccination and isolation as control variables, applying a stochastic minimum principle to derive forward-backward stochastic differential equations (FBSDEs) solvable via Monte Carlo simulation. The key contribution is a numerically tractable framework that identifies cost-optimal mitigation strategies under uncertainty in infection rates, showing that high-cost policies trigger aggressive early interventions, while low-cost policies enable sustained, less intensive control.

ABSTRACT

We study the problem of optimal control of the stochastic SIR model. Models of this type are used in mathematical epidemiology to capture the time evolution of highly infectious diseases such as COVID-19. Our approach relies on reformulating the Hamilton-Jacobi-Bellman equation as a stochastic minimum principle. This results in a system of forward backward stochastic differential equations, which is amenable to numerical solution via Monte Carlo simulations. We present a number of numerical solutions of the system under a variety of scenarios.

Motivation & Objective

• To develop a mathematically rigorous framework for optimal control of epidemic spread under stochastic infection rates.
• To address the challenge of limited and unreliable early epidemic data in informing public health policy.
• To model the impact of vaccination and isolation as control mechanisms within a stochastic SIR framework.
• To derive a numerically solvable system of equations that captures optimal control policies under varying cost regimes.
• To evaluate the effectiveness of combined isolation and vaccination strategies under different economic and health cost constraints.

Proposed method

• Formulates a stochastic SIR model where the infection rate β is subject to random shocks via a Brownian motion, introducing stochasticity into the dynamics.
• Reformulates the Hamilton-Jacobi-Bellman equation as a stochastic minimum principle to derive a system of forward-backward stochastic differential equations (FBSDEs).
• Uses Monte Carlo simulations to numerically solve the FBSDE system and compute optimal control policies for vaccination and isolation.
• Defines a cost function with parameters L, M, N to represent running costs of interventions, enabling analysis under high- and low-cost regimes.
• Applies the FBSDE solution to simulate optimal control trajectories for susceptible (S), infected (I), and control variables (vaccination and isolation rates).
• Validates the approach through numerical experiments across multiple cost function configurations, comparing raw and optimal epidemic trajectories.

Experimental results

Research questions

• RQ1 How can optimal control be applied to a stochastic SIR model with uncertain infection rates to minimize epidemic impact?
• RQ2 What is the structure of the optimal control policy for isolation and vaccination under different cost functions?
• RQ3 How do high-cost versus low-cost intervention regimes affect the timing, intensity, and effectiveness of mitigation strategies?
• RQ4 To what extent can stochastic optimal control reduce peak infection levels and total infections compared to uncontrolled spread?
• RQ5 How do the optimal control policies for isolation and vaccination interact when applied simultaneously?

Key findings

• Under high-cost isolation policies, the optimal strategy involves aggressive early isolation, reducing peak infections and flattening the epidemic curve significantly.
• For high-cost vaccination, the optimal policy triggers a rapid, large-scale vaccination campaign that drastically reduces the susceptible population and suppresses transmission.
• In low-cost scenarios, the optimal control policy applies sustained, moderate levels of isolation and vaccination, avoiding sharp interventions while still containing the epidemic.
• The combined isolation and vaccination policy under high-cost conditions leads to a sharp decline in both susceptible and infected fractions, with optimal control intensifying early and tapering off as the epidemic wanes.
• Under low-cost conditions, the optimal policies for isolation and vaccination are more evenly distributed over time, with both interventions active throughout the epidemic duration.
• Numerical results show that the optimal control framework consistently reduces the total number of infections and delays the peak compared to uncontrolled dynamics, especially under high-cost regimes.