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Regional Control Strategies for a Spatiotemporal SQEIAR Epidemic Model: Application to COVID-19

Elghandouri Mohammed, Ezzinbi Khalil, Youness Mezzan

TL;DR

This work develops a spatiotemporal SQEIAR PDE model for epidemics, introducing a quarantined population $Q$ and regional controls $u(t,x)$ (treatment) and $v(t,x)$ (regional quarantine) over subregions to minimize susceptibility, exposure, and infection while accounting for reinfection. Existence and regularity of solutions are established via operator-theoretic methods and a truncation approach to handle nonlinearities, and an optimal control problem with a convex cost functional $\\mathcal{J}$ is solved, yielding explicit adjoint-based optimality conditions for $u^*$ and $v^*$. A numerical COVID-19 illustration using finite differences demonstrates substantial reductions in peak infections and increased quarantined individuals under the proposed controls, validating the framework and its potential to guide regional policy decisions. The analysis further discusses how distributing quarantine across more regions (increasing $n$) affects control efficacy and cost, offering direction for future optimization of regional partitioning in spatial epidemic management.

Abstract

In this work, we develop a spatial SEIAR-type epidemic model considering a quarantined population (denoted as Q), which we call the SQEIAR model. The dynamics of the SQEIAR model are described by six Partial Differential Equations (PDEs) that represent the changes in the susceptible, quarantined, exposed, asymptomatic, infected, and recovered populations. Our goal is to reduce the number of susceptible, exposed, asymptomatic, and infected individuals while accounting for the environment, which plays a critical role in the spread of epidemics. We then propose a novel strategy for epidemic control, incorporating two key control measures: regional quarantine for the susceptible population and treatment for the infected. This approach serves as an alternative to widespread quarantine, minimizing the economic, social, and other potential impacts. Additionally, we consider the possibility of reinfection among recovered individuals, a common occurrence in many diseases. To demonstrate the practical utility of our results, a numerical example centered on COVID-19 is presented.

Regional Control Strategies for a Spatiotemporal SQEIAR Epidemic Model: Application to COVID-19

TL;DR

This work develops a spatiotemporal SQEIAR PDE model for epidemics, introducing a quarantined population and regional controls (treatment) and (regional quarantine) over subregions to minimize susceptibility, exposure, and infection while accounting for reinfection. Existence and regularity of solutions are established via operator-theoretic methods and a truncation approach to handle nonlinearities, and an optimal control problem with a convex cost functional is solved, yielding explicit adjoint-based optimality conditions for and . A numerical COVID-19 illustration using finite differences demonstrates substantial reductions in peak infections and increased quarantined individuals under the proposed controls, validating the framework and its potential to guide regional policy decisions. The analysis further discusses how distributing quarantine across more regions (increasing ) affects control efficacy and cost, offering direction for future optimization of regional partitioning in spatial epidemic management.

Abstract

In this work, we develop a spatial SEIAR-type epidemic model considering a quarantined population (denoted as Q), which we call the SQEIAR model. The dynamics of the SQEIAR model are described by six Partial Differential Equations (PDEs) that represent the changes in the susceptible, quarantined, exposed, asymptomatic, infected, and recovered populations. Our goal is to reduce the number of susceptible, exposed, asymptomatic, and infected individuals while accounting for the environment, which plays a critical role in the spread of epidemics. We then propose a novel strategy for epidemic control, incorporating two key control measures: regional quarantine for the susceptible population and treatment for the infected. This approach serves as an alternative to widespread quarantine, minimizing the economic, social, and other potential impacts. Additionally, we consider the possibility of reinfection among recovered individuals, a common occurrence in many diseases. To demonstrate the practical utility of our results, a numerical example centered on COVID-19 is presented.
Paper Structure (8 sections, 6 theorems, 21 equations, 17 figures, 1 table)

This paper contains 8 sections, 6 theorems, 21 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. SQEIAR Epidemic Model
  3. Existence of solutions
  4. Existence of the optimal solutions
  5. Necessary Optimality Conditions
  6. Numerical simulation and some discussions
  7. Discussion and Implications
  8. Conclusion

Key Result

Theorem 1.1

pazy Let $F:\mathbb{R}^{+}\times X\to X$ be locally integrable w.r.t the first argument and locally Lipschitz continuous w.r.t the second argument. If $B$ is the infinitesimal generator of a $C_0$-semigroup $(T(t))_{t\geq 0}$ on $X$, then for every $y_0\in X$, there exists a $t_{max}\leq +\infty$ su has a unique (mild) solution $y$ on $[0,t_{max})$. Moreover, if $t_{max}<+\infty$, then $\limsup\li

Figures (17)

  • Figure 1: Quarantine control must be applied in the highlighted sub-regions $\omega_i$ of $\Omega$, where key factors influencing disease spread are present.
  • Figure 2: Biological Processes of the SQEIAR Model with our Proposed Controllers.
  • Figure 3: Changes in the Susceptible Group Without Control Actions.
  • Figure 4: Changes in the Susceptible Group in the Presence of Controls.
  • Figure 5: Changes in the Quarantined Group Without Control Actions.
  • ...and 12 more figures

Theorems & Definitions (8)

  • Theorem 1.1
  • Proposition 3.1
  • Theorem 3.2
  • Definition 4.1
  • Definition 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Theorem 5.2