Stochastic Optimal Control of an Epidemic Under Partial Information

TL;DR

The paper tackles a stochastic optimal control problem for an epidemic under partial information, modeling with an $SI^{\pm}R^{\pm}H$ compartmental structure and incorporating social distancing, testing, and vaccination. It converts the partially observed problem into a fully observed Markov decision process via diffusion approximation and an extended Kalman filter, then solves the resulting Bellman equation with two scalable numerical schemes: (i) state discretization with optimal quantization and linear interpolation, and (ii) quantization with value-function regression using educated basis functions. Numerical experiments show that the optimal policy typically combines lockdown, testing, and vaccination to curb infections while respecting hospital capacity, with comparable results between the two methods and clear insights into how undetected infections drive interventions. The work integrates filtering, dynamic programming, and quantization-based techniques to provide practical decision-support for epidemic management under uncertainty.**

Abstract

In this paper, we address a social planner's optimal control problem for a partially observable stochastic epidemic model. The control measures include social distancing, testing, and vaccination. Using a diffusion approximation for the state dynamics of the epidemic, we apply filtering arguments to transform the partially observable stochastic optimal control problem into an optimal control problem with complete information. This transformed problem is treated as a Markov decision process. The associated Bellman equation is solved numerically using optimal quantization methods for approximating the expectations involved to mitigate the curse of dimensionality. We implement two approaches, the first involves state discretization coupled with linear interpolation of the value function at non-grid points. The second utilizes a parametrization of the value function with educated ansatz functions. Extensive numerical experiments are presented to demonstrate the efficacy of both methods.

Figures (20)

• Figure 1: Flowchart of the $SI^{\pm}R^{\pm}H$ compartmental model with control variables $u^L,u^T$ and $u^V$. Hidden compartments are in red and observable compartments in blue.
• Figure 2: Typical curve of running costs functions featuring the fixed cost jump for $x>0$, linear cost in the interval $(0,\overline{x})$ and linear plus quadratic cost for $x> \overline{x}$.
• Figure 3: Sample paths of state components and correlation coefficient in the absence of control measures
• Figure 4: Sample paths of state components and correlation coefficient moderate control measures, $(u^L,u^T,u^V)=(0.2,0.03,0.015)$
• Figure 5: Linear interpolation method: value function and optimal decision rule at time $118$ for small values of $( m^2, q^1,q^2, \rho, z^2, z^3)$

Theorems & Definitions (12)

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