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Optimal Control of an Epidemic with Intervention Design

Behrooz Moosavi Ramezanzadeh

TL;DR

This work tackles the challenge of optimally controlling an epidemic described by a delayed SEIR model under a hard healthcare capacity constraint over an infinite horizon. It develops a rigorous variational framework that connects hard state constraints with penalized objectives via Moreau--Yosida regularization and uses $\Gamma$-convergence to justify finite-horizon approximations; PMP-based optimality conditions reveal a rich structure, including singular boundary-maintenance regimes. The combination of a well-posed delayed SEIR, a principled hard-soft constraint bridge, and convergence results yields insights into how vaccination and NPIs should be timed and scaled, with shadow prices clarifying the economic value of capacity and speed. Practically, the results emphasize rapid, large-scale vaccination and timely NPIs to prevent capacity breaches, quantify the marginal value of increasing capacity, and provide a framework for policy design under real-world delays.

Abstract

This paper investigates the optimal control of an epidemic governed by a SEIR model with operational delays in vaccination and non pharmaceutical interventions. We address the mathematical challenge of imposing hard healthcare capacity constraints (e.g., ICU limits) over an infinite time horizon. To rigorously bridge the gap between theoretical constraints and numerical tractability, we employ a variational framework based on Moreau--Yosida regularization and establish the connection between finite- and infinite-horizon solutions via $Γ$-convergence. The necessary conditions for optimality are derived using the Pontryagin Maximum Principle, allowing for the characterization of singular regimes where the optimal strategy maintains the infection level precisely at the capacity boundary. Numerical simulations illustrate these theoretical findings, quantifying the shadow prices of infection and costs associated with intervention delays.

Optimal Control of an Epidemic with Intervention Design

TL;DR

This work tackles the challenge of optimally controlling an epidemic described by a delayed SEIR model under a hard healthcare capacity constraint over an infinite horizon. It develops a rigorous variational framework that connects hard state constraints with penalized objectives via Moreau--Yosida regularization and uses -convergence to justify finite-horizon approximations; PMP-based optimality conditions reveal a rich structure, including singular boundary-maintenance regimes. The combination of a well-posed delayed SEIR, a principled hard-soft constraint bridge, and convergence results yields insights into how vaccination and NPIs should be timed and scaled, with shadow prices clarifying the economic value of capacity and speed. Practically, the results emphasize rapid, large-scale vaccination and timely NPIs to prevent capacity breaches, quantify the marginal value of increasing capacity, and provide a framework for policy design under real-world delays.

Abstract

This paper investigates the optimal control of an epidemic governed by a SEIR model with operational delays in vaccination and non pharmaceutical interventions. We address the mathematical challenge of imposing hard healthcare capacity constraints (e.g., ICU limits) over an infinite time horizon. To rigorously bridge the gap between theoretical constraints and numerical tractability, we employ a variational framework based on Moreau--Yosida regularization and establish the connection between finite- and infinite-horizon solutions via -convergence. The necessary conditions for optimality are derived using the Pontryagin Maximum Principle, allowing for the characterization of singular regimes where the optimal strategy maintains the infection level precisely at the capacity boundary. Numerical simulations illustrate these theoretical findings, quantifying the shadow prices of infection and costs associated with intervention delays.

Paper Structure

This paper contains 23 sections, 10 theorems, 42 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Analysis of Controlled System
  3. Dynamic Optimization
  4. Contribution and Framework
  5. Problem Formulations
  6. Admissible controls and Dynamics
  7. Running cost
  8. Hard-constrained Formulation
  9. Penalized Formulation
  10. Existence and Variational Convergence
  11. Topology and Convergence Framework
  12. Moreau--Yosida regularization: linking soft and hard constraints
  13. Variational convergence: finite to infinite horizon
  14. Optimality conditions and economic interpretation
  15. Adjoint system and Shadow Prices
  16. ...and 8 more sections

Key Result

Theorem 2.2

Let $I=[0,+\infty)$. Under Assumption assump:bounds, there exists a unique global solution $(s,e,i,r)\in W^{1,\infty}_{\mathrm{loc}}(I;\mathbb{R}^4)$ to sdeq:1. Moreover, the trajectory remains in the compact simplex $\{(s,e,i,r)\in[0,1]^4:\ s+e+i+r=1\}$, so the vector field is uniformly bounded; in and $i(t)>0$ for every finite $t\ge 0$ whenever $i_0>0$.

Figures (6)

  • Figure 1: Circular diagram of the SEIR epidemic model. Nodes represent compartments, arrows represent transitions, and colored boxes indicate control interventions.
  • Figure 2: Epidemic dynamics under fixed intervention strategies (left) and corresponding phase-plane trajectories (right).
  • Figure 3: Optimal control trajectories: Infected fraction relative to $I_{max}$ (top), optimal vaccination $u(t)$ (middle), and optimal suppression $h(t)$ (bottom).
  • Figure 4: Time path of shadow prices $\lambda_s(t)$, $\lambda_e(t)$, and $\lambda_i(t)$ under optimal control.
  • Figure 5: Correlation heatmap of parameters ($\beta, u_{max}, h_{max}$) and outcomes ($J_T, peak_i$).
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem 2.2: Existence, Uniqueness, and Non-negativity
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Corollary 2.5: Basic Asymptotic Properties
  • proof
  • Proposition 2.6: Final-size bound under maximal suppression
  • proof
  • Corollary 2.7
  • ...and 11 more