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State Feedback as a Strategy for Control and Analysis of COVID-19

Leonardo R. S. Rodrigues, Felipe Gabrielli

TL;DR

This work analyzes a four‑compartment SIQR framework for COVID‑19 with vaccination as a control to mitigate spread. It derives and analyzes equilibria, establishes disease‑free and endemic stability via Lyapunov and Routh–Hurwitz criteria, and introduces controllability and a Riccati‑based optimal control framework for vaccination strategy, solved numerically with Runge–Kutta methods in MATLAB. The key result is the reproduction number $R_0=\frac{\Delta}{\mu+v}\cdot\frac{\alpha}{\gamma+\mu+\eta}$, which governs the transition between disease elimination and endemic persistence, with vaccination ($v>0$) reducing $R_0$ and thus promoting global stability. The study demonstrates that vaccination can outperform isolation alone in reducing transmission, and that an optimal vaccination policy can achieve faster stabilization at lower control costs. The framework provides a rigorous, testable approach for evaluating vaccination strategies in the presence of quarantine and recovery dynamics.

Abstract

This paper presents a study on a compartmental epidemic model for COVID-19, examining the stability of its equilibrium points upon the introduction of vaccination as a strategy to mitigate the spread of the disease. Initially, the SIQR (Susceptible-Infectious-Quarantine-Recovered) mathematical model and its technical aspects are introduced. Subsequently, vaccination is incorporated as a control measure within the model scope. Equilibrium points and the basic reproductive number are determined, followed by an analysis of their stability. Furthermore, controllability characteristics and Optimal Control strategies for the system are investigated, supplemented by numerical simulations.

State Feedback as a Strategy for Control and Analysis of COVID-19

TL;DR

This work analyzes a four‑compartment SIQR framework for COVID‑19 with vaccination as a control to mitigate spread. It derives and analyzes equilibria, establishes disease‑free and endemic stability via Lyapunov and Routh–Hurwitz criteria, and introduces controllability and a Riccati‑based optimal control framework for vaccination strategy, solved numerically with Runge–Kutta methods in MATLAB. The key result is the reproduction number , which governs the transition between disease elimination and endemic persistence, with vaccination () reducing and thus promoting global stability. The study demonstrates that vaccination can outperform isolation alone in reducing transmission, and that an optimal vaccination policy can achieve faster stabilization at lower control costs. The framework provides a rigorous, testable approach for evaluating vaccination strategies in the presence of quarantine and recovery dynamics.

Abstract

This paper presents a study on a compartmental epidemic model for COVID-19, examining the stability of its equilibrium points upon the introduction of vaccination as a strategy to mitigate the spread of the disease. Initially, the SIQR (Susceptible-Infectious-Quarantine-Recovered) mathematical model and its technical aspects are introduced. Subsequently, vaccination is incorporated as a control measure within the model scope. Equilibrium points and the basic reproductive number are determined, followed by an analysis of their stability. Furthermore, controllability characteristics and Optimal Control strategies for the system are investigated, supplemented by numerical simulations.
Paper Structure (12 sections, 11 theorems, 91 equations, 10 figures, 1 table)

This paper contains 12 sections, 11 theorems, 91 equations, 10 figures, 1 table.

Table of Contents

  1. . Introdution
  2. SIQR Mathematical Model
  3. . Metodology
  4. Equilibrium Points and Basic Reproductive Number
  5. Positivity and Boundedness
  6. Stability of Equilibrium Points
  7. Control of finite dimensional linear systems
  8. Kalman's controllability condition
  9. Optimal Control Model
  10. Numerical Simulations
  11. . Results
  12. . Conclusion

Key Result

Theorem 4.1

The closed region $\Omega =\left\{ (S,I,Q,R) \in \mathbb{R}^{4}_{+}: N(t)\leq \frac{\Delta}{\mu}\right\}$ is positive invariant ste for the model (sist_vac).

Figures (10)

  • Figure 1: SIQR's dynamic flow.
  • Figure 2: Controllabillity Matrix
  • Figure 3: Evolution of the P norm
  • Figure 4: Variational curves of $S$, $I$, $Q$, and $R$
  • Figure 5: Variational curves of $S$, $I$, $Q$, and $R$
  • ...and 5 more figures

Theorems & Definitions (25)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof : Proof
  • Definition 4.5
  • Theorem 5.1
  • ...and 15 more