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Reliable optimal controls for SEIR models in epidemiology

Simone Cacace, Alessio Oliviero

TL;DR

This work examines optimal control of SEIR epidemiological models using both Dynamic Programming (via the Hamilton-Jacobi-Bellman equation) and Pontryagin's maximum principle (via the Direct-Adjoint Looping method). It introduces a hybrid SL-DAL approach that uses a semi-Lagrangian solution to warm-start the forward-backward optimization, improving reliability and enabling validation via first- and second-order optimality conditions. Through numerical experiments across basic, waning-immunity, border-control, and ICU-constrained scenarios, the authors show that the combined method yields lower costs and more robust policies than either method alone. The results highlight the practical value of integrating global-optimal value-function information with local descent techniques for complex, high-dimensional epidemiological control problems.

Abstract

We present and compare two different optimal control approaches applied to SEIR models in epidemiology, which allow us to obtain some policies for controlling the spread of an epidemic. The first approach uses Dynamic Programming to characterise the value function of the problem as the solution of a partial differential equation, the Hamilton-Jacobi-Bellman equation, and derive the optimal policy in feedback form. The second is based on Pontryagin's maximum principle and directly gives open-loop controls, via the solution of an optimality system of ordinary differential equations. This method, however, may not converge to the optimal solution. We propose a combination of the two methods in order to obtain high-quality and reliable solutions. Several simulations are presented and discussed, also checking first and second order necessary optimality conditions for the corresponding numerical solutions.

Reliable optimal controls for SEIR models in epidemiology

TL;DR

This work examines optimal control of SEIR epidemiological models using both Dynamic Programming (via the Hamilton-Jacobi-Bellman equation) and Pontryagin's maximum principle (via the Direct-Adjoint Looping method). It introduces a hybrid SL-DAL approach that uses a semi-Lagrangian solution to warm-start the forward-backward optimization, improving reliability and enabling validation via first- and second-order optimality conditions. Through numerical experiments across basic, waning-immunity, border-control, and ICU-constrained scenarios, the authors show that the combined method yields lower costs and more robust policies than either method alone. The results highlight the practical value of integrating global-optimal value-function information with local descent techniques for complex, high-dimensional epidemiological control problems.

Abstract

We present and compare two different optimal control approaches applied to SEIR models in epidemiology, which allow us to obtain some policies for controlling the spread of an epidemic. The first approach uses Dynamic Programming to characterise the value function of the problem as the solution of a partial differential equation, the Hamilton-Jacobi-Bellman equation, and derive the optimal policy in feedback form. The second is based on Pontryagin's maximum principle and directly gives open-loop controls, via the solution of an optimality system of ordinary differential equations. This method, however, may not converge to the optimal solution. We propose a combination of the two methods in order to obtain high-quality and reliable solutions. Several simulations are presented and discussed, also checking first and second order necessary optimality conditions for the corresponding numerical solutions.
Paper Structure (20 sections, 51 equations, 17 figures, 1 table)

This paper contains 20 sections, 51 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. The controlled SEIR model
  3. Finite horizon problems and cost definition
  4. The Dynamic Programming approach and semi-Lagrangian schemes
  5. The variational approach and the Direct-Adjoint Looping method
  6. Comparing and combining semi-Lagrangian schemes and the Direct-Adjoint Looping method
  7. Feedback vs. open loop
  8. The role of initial data
  9. Convergence and error estimates
  10. State constraints
  11. Computational cost
  12. Combination of the two approaches
  13. Numerical simulations
  14. Test 1: basic model
  15. Test 2: temporary immunity
  16. ...and 5 more sections

Figures (17)

  • Figure 1: Diagram of the epidemic SEIR model
  • Figure 2: Trajectories of the normalised epidemic SEIR model. From left to right, top to bottom: susceptible, exposed, infective, recovered.
  • Figure 3: Diagram of the normalised epidemic SEIR model with restrictions $\lambda$ and vaccinations $\nu$.
  • Figure 4: Test 1. Optimal trajectories (dashed lines) and controls (full lines) for system \ref{['seir_controllato_ridotto']}. Solutions obtained with the semi-Lagrangian scheme on the left, with Direct-Adjoint Looping on the right.
  • Figure 5: Test 2. Optimal trajectories (dashed lines) and controls (full lines) for system \ref{['seir_immunità_temporanea']}. Solutions obtained with the semi-Lagrangian scheme on the left, with Direct-Adjoint Looping on the right.
  • ...and 12 more figures