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Optimal Control of Covid-19 Interventions in Public Health Management

Isabella Kemajou-Brown, Romario Gildas Foko Tiomela, Olawale Nasiru Lawal, Samson Adekola Alagbe, Serges Love Teutu Talla

TL;DR

The paper tackles the problem of optimally deploying COVID-19 interventions by formulating a nine-control, compartmental dynamic system with states $S,E,I_a,I_s,H,R,D$. It applies Pontryagin's Maximum Principle to derive three independent single-criterion objective functionals $J_1$, $J_2$, and $J_3$ corresponding to cost, effectiveness, and feasibility, each accompanied by a Hamiltonian, adjoint system, and explicit optimality conditions. Existence of optimal controls is established under affine dynamics and convex Lagrangians, with section-specific results yielding control laws such as $u_i^*= ext{projection of }\omega_i^*\text{ into }[0,u_{max}]$ (and variants) where $\omega_i^*$ depend on state- and adjoint-variables. The study illuminates how each criterion shapes intervention patterns and highlights the need for multi-objective extensions to capture trade-offs in practice. This framework provides a principled basis for policy planning under cost, health impact, and implementation feasibility, and sets the stage for numerical simulations and eventual multi-criteria optimization.

Abstract

This study explores the application of Pontryagin's Maximum Principle to derive optimal strategies for controlling the spread of COVID-19, leveraging a novel compartmental model to capture the disease dynamics. We prioritize three key criteria: cost, effectiveness, and feasibility, each examined independently to evaluate their unique contributions to pandemic management. By addressing these criteria, this study aims to design intervention strategies that are scientifically robust, practical, and economically sustainable. Furthermore, the focus on cost, effectiveness and feasibility seeks to provide policymakers with actionable insights for implementing interventions that maximize public health benefits while remaining feasible under real-world conditions.

Optimal Control of Covid-19 Interventions in Public Health Management

TL;DR

The paper tackles the problem of optimally deploying COVID-19 interventions by formulating a nine-control, compartmental dynamic system with states . It applies Pontryagin's Maximum Principle to derive three independent single-criterion objective functionals , , and corresponding to cost, effectiveness, and feasibility, each accompanied by a Hamiltonian, adjoint system, and explicit optimality conditions. Existence of optimal controls is established under affine dynamics and convex Lagrangians, with section-specific results yielding control laws such as (and variants) where depend on state- and adjoint-variables. The study illuminates how each criterion shapes intervention patterns and highlights the need for multi-objective extensions to capture trade-offs in practice. This framework provides a principled basis for policy planning under cost, health impact, and implementation feasibility, and sets the stage for numerical simulations and eventual multi-criteria optimization.

Abstract

This study explores the application of Pontryagin's Maximum Principle to derive optimal strategies for controlling the spread of COVID-19, leveraging a novel compartmental model to capture the disease dynamics. We prioritize three key criteria: cost, effectiveness, and feasibility, each examined independently to evaluate their unique contributions to pandemic management. By addressing these criteria, this study aims to design intervention strategies that are scientifically robust, practical, and economically sustainable. Furthermore, the focus on cost, effectiveness and feasibility seeks to provide policymakers with actionable insights for implementing interventions that maximize public health benefits while remaining feasible under real-world conditions.

Paper Structure

This paper contains 8 sections, 7 theorems, 90 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal control of a COVID-19 model
  4. Model Description
  5. Cost Objective Function
  6. Effectiveness Objective Function
  7. Feasibility Objective Function
  8. Concluding remarks

Key Result

Proposition 2.1

If a pair $(x(\cdot), u(\cdot))$ with $x\in PC^1([a,b];\mathbb{R}^n)$ and $u\in PC([a,b];{\mathcal{U}})$ is a solution to problem PontMax, then there exists $\alpha\in PC^1([a,b];\mathbb{R}^n)$ such that the following conditions hold: where the Hamiltonian $H$ is defined by and $\alpha\in\mathbb{R}^n$ is the adjoint variable (or costate) associated with the state $x$.

Figures (1)

  • Figure 1: Dynamics of COVID-19 spread

Theorems & Definitions (13)

  • Definition 2.1: Control System
  • Definition 2.2: Control-affine system
  • Proposition 2.1: Pontryagin's Maximum Principle
  • Remark 2.1
  • Definition 2.3: Lagrange function
  • Proposition 2.2: Existence of an optimal control
  • Remark 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.2
  • ...and 3 more