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An Age-Structured Vaccination Strategy for Epidemic Containment: A Model Predictive Control Approach

Candy Sonveaux, Morgane Dumont, Mirko Fiacchini, Mohamad Ajami

TL;DR

This work tackles COVID-19 containment by formulating an age-structured SIRD model with a vaccination input and embedding it within a Model Predictive Control (MPC) framework. The approach provides a rigorous theoretical foundation through recursive feasibility and asymptotic stability proofs, ensuring that the MPC cost upper-bounds the final death toll. Empirical results for Wallonia show the MPC vaccination strategy outperforms Belgium’s decreasing-age national policy in terms of faster disease eradication, fewer infections, and reduced deaths, while using vaccines more efficiently. The methodology offers a transferable, model-based framework for optimal vaccine allocation that can adapt to other epidemics and future data, with planned extensions to account for imperfect vaccine efficacy and ICU capacity constraints.

Abstract

This work presents a novel Model Predictive Control (MPC) approach to develop an optimal age-structured vaccination strategy for the containment of COVID-19 in Wallonia, Belgium. The proposed MPC framework is designed to minimize deaths, achieve early disease eradication, and adhere to operational constraints. By incorporating an age-structured Susceptible-Infected-Recovered-Deceased (SIRD) model with an additional term for vaccination, the MPC strategy dynamically adapts to the evolving epidemic state. A detailed proof of the asymptotic stability and recursive feasibility of the proposed MPC algorithm is provided. This ensures that the optimal cost at each step provides an upper bound on the minimal number obtainable of deaths at the end of the pandemic. Moreover, simulations demonstrate that the proposed MPC approach outperforms the decreasing age vaccination strategy adopted by the Belgian government during the first wave of vaccinations. The results highlight the potential of MPC-based vaccination strategies to reduce the total number of deaths, accelerate disease eradication, and optimize vaccine administration.

An Age-Structured Vaccination Strategy for Epidemic Containment: A Model Predictive Control Approach

TL;DR

This work tackles COVID-19 containment by formulating an age-structured SIRD model with a vaccination input and embedding it within a Model Predictive Control (MPC) framework. The approach provides a rigorous theoretical foundation through recursive feasibility and asymptotic stability proofs, ensuring that the MPC cost upper-bounds the final death toll. Empirical results for Wallonia show the MPC vaccination strategy outperforms Belgium’s decreasing-age national policy in terms of faster disease eradication, fewer infections, and reduced deaths, while using vaccines more efficiently. The methodology offers a transferable, model-based framework for optimal vaccine allocation that can adapt to other epidemics and future data, with planned extensions to account for imperfect vaccine efficacy and ICU capacity constraints.

Abstract

This work presents a novel Model Predictive Control (MPC) approach to develop an optimal age-structured vaccination strategy for the containment of COVID-19 in Wallonia, Belgium. The proposed MPC framework is designed to minimize deaths, achieve early disease eradication, and adhere to operational constraints. By incorporating an age-structured Susceptible-Infected-Recovered-Deceased (SIRD) model with an additional term for vaccination, the MPC strategy dynamically adapts to the evolving epidemic state. A detailed proof of the asymptotic stability and recursive feasibility of the proposed MPC algorithm is provided. This ensures that the optimal cost at each step provides an upper bound on the minimal number obtainable of deaths at the end of the pandemic. Moreover, simulations demonstrate that the proposed MPC approach outperforms the decreasing age vaccination strategy adopted by the Belgian government during the first wave of vaccinations. The results highlight the potential of MPC-based vaccination strategies to reduce the total number of deaths, accelerate disease eradication, and optimize vaccine administration.
Paper Structure (13 sections, 3 theorems, 14 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 13 sections, 3 theorems, 14 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Epidemiological Model
  3. Model Predictive Control
  4. Recursive feasibility and stability conditions for general MPC
  5. MPC-based vaccination for COVID-19 pandemic
  6. MPC-based vaccination implementation
  7. Results and Discussion
  8. Simulation parameters
  9. No vaccination strategy
  10. National vaccination strategy
  11. MPC vaccination strategy
  12. Discussion
  13. Conclusion

Key Result

Theorem 1

Given the dynamical system (eq:diff_eq), the optimization problem (eq:opt_prob) and the equilibrium set $X^* \subseteq \mathbb{X}$, suppose that: Then, the optimal control problem in (eq:opt_prob) is feasible $\forall x \in \mathcal{X}_N$, and there exist two $\mathcal{K}_\infty$ functions: $\alpha_d(\cdot)$ and $\alpha_u(\cdot)$, such that $\forall x \in \mathcal{X}_N$, and hence $V_N^0(\cdot)$

Figures (8)

  • Figure 1: Compartmental scheme of the age-structured SIRD model with an additional vaccination term for the $k^{th}$ age group
  • Figure 2: Illustration of the MPC scheme, inspired by seborg2016process
  • Figure 3: Populations evolution over time (in days) in the absence of vaccination
  • Figure 4: Populations evolution over time (in days) under the national vaccination strategy
  • Figure 5: Populations evolution over time (in days) under the MPC vaccination strategy
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 1: rawlings2017model
  • Corollary 1
  • Theorem 2
  • Proof 1