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Network-Based Epidemic Control Through Optimal Travel and Quarantine Management

Mahtab Talaei, Apostolos I. Rikos, Alex Olshevsky, Laura F. White, Ioannis Ch. Paschalidis

TL;DR

The paper develops a network-based framework for epidemic control with two complementary approaches: (i) optimal travel-rate reduction via minimizing the dominant eigenvalue $oldsymbol{ ho}_{ ext{max}}(M(t_0, au))$, implemented with projected gradient descent under an $ extell_1$ travel-change budget, and (ii) a quarantine-aware SIQR extension whose cost-minimization under a decay constraint is reducible to a matrix-balancing problem and solvable with Augmented Primal-Dual Gradient Dynamics (Aug-PDGD). The authors establish connections between the optimization constraints and the basic reproduction number $R_0$, and prove convergence properties (including semi-global exponential convergence to KKT points) for the quarantine algorithm. They validate the methods on a 14-node Massachusetts county network, demonstrating that increased travel-reduction budget and optimized quarantine rates substantially suppress infection spread while preserving socio-economic activity. Overall, the work provides scalable, structure-aware optimization tools for epidemic containment in connected populations, grounded in spectral analysis and matrix-balancing theory.

Abstract

Motivated by the swift global transmission of infectious diseases, we present a comprehensive framework for network-based epidemic control. Our aim is to curb epidemics using two different approaches. In the first approach, we introduce an optimization strategy that optimally reduces travel rates. We analyze the convergence of this strategy and show that it hinges on the network structure to minimize infection spread. In the second approach, we expand the classic SIR model by incorporating and optimizing quarantined states to strategically contain the epidemic. We show that this problem reduces to the problem of matrix balancing. We establish a link between optimization constraints and the epidemic's reproduction number, highlighting the relationship between network structure and disease dynamics. We demonstrate that applying augmented primal-dual gradient dynamics to the optimal quarantine problem ensures exponential convergence to the KKT point. We conclude by validating our approaches using simulation studies that leverage public data from counties in the state of Massachusetts.

Network-Based Epidemic Control Through Optimal Travel and Quarantine Management

TL;DR

The paper develops a network-based framework for epidemic control with two complementary approaches: (i) optimal travel-rate reduction via minimizing the dominant eigenvalue , implemented with projected gradient descent under an travel-change budget, and (ii) a quarantine-aware SIQR extension whose cost-minimization under a decay constraint is reducible to a matrix-balancing problem and solvable with Augmented Primal-Dual Gradient Dynamics (Aug-PDGD). The authors establish connections between the optimization constraints and the basic reproduction number , and prove convergence properties (including semi-global exponential convergence to KKT points) for the quarantine algorithm. They validate the methods on a 14-node Massachusetts county network, demonstrating that increased travel-reduction budget and optimized quarantine rates substantially suppress infection spread while preserving socio-economic activity. Overall, the work provides scalable, structure-aware optimization tools for epidemic containment in connected populations, grounded in spectral analysis and matrix-balancing theory.

Abstract

Motivated by the swift global transmission of infectious diseases, we present a comprehensive framework for network-based epidemic control. Our aim is to curb epidemics using two different approaches. In the first approach, we introduce an optimization strategy that optimally reduces travel rates. We analyze the convergence of this strategy and show that it hinges on the network structure to minimize infection spread. In the second approach, we expand the classic SIR model by incorporating and optimizing quarantined states to strategically contain the epidemic. We show that this problem reduces to the problem of matrix balancing. We establish a link between optimization constraints and the epidemic's reproduction number, highlighting the relationship between network structure and disease dynamics. We demonstrate that applying augmented primal-dual gradient dynamics to the optimal quarantine problem ensures exponential convergence to the KKT point. We conclude by validating our approaches using simulation studies that leverage public data from counties in the state of Massachusetts.
Paper Structure (22 sections, 9 theorems, 59 equations, 5 figures)

This paper contains 22 sections, 9 theorems, 59 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Network spread model of COVID-19
  4. Construction of Infection Flow and Travel Rates Matrices
  5. Stability and Splitting of Matrices
  6. Balancing of Nonnegative Matrices
  7. Problem Formulation
  8. Optimal Travel Rates for Epidemic Control
  9. Problem Structure
  10. Proposed Solution
  11. COVID-19 Model with Optimal Quarantine Rates
  12. Integrating Quarantining into a COVID-19 Model
  13. Problem Formulation for Minimizing Quarantine Rates
  14. Proposed Solution
  15. Simulation Results
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\lambda_0$ be a simple eigenvalue of a matrix $Z_0 \in \mathds{C}^{n \times n}$ and let $u_0$ be an associated eigenvector, so that $Z_0 u_0 = \lambda_0 u_0$. Then a (complex) function $\lambda$ and a (complex) vector function $u$ are defined for all $Z$ in some neighborhood $\mathcal{N}(Z_0) \ and Moreover, the functions $\lambda$ and $u$ are $\infty$ times differentiable on $\mathcal{N}(Z_

Figures (5)

  • Figure 1: Optimal values of $f(\tau)$ in problem \ref{['problem_formulation_P1']} for different budget parameters $b$ after changing the travel rates $\tau$ via \ref{['PGD_iteration']}.
  • Figure 2: Number of cumulative cases (infected, quarantined, and recovered) for different constraints over the travel rates for the state of Massachusetts.
  • Figure 3: Number of active (asymptomatic and symptomatic infected) cases for different constraints over the travel rates for the state of Massachusetts.
  • Figure 4: Number of cumulative cases (infected, quarantined, and recovered) assuming for different quarantine rates for the state of Massachusetts. For the optimal policy, $\alpha$ is set to $0.023$, which corresponds to halving the number of infected cases every $30$ days.
  • Figure 5: Number of active (asymptomatic and symptomatic infected) cases for different quarantine policies for the state of Massachusetts. For the optimal policy, $\alpha$ is set to $0.023$, which corresponds to halving the number of infected cases every $30$ days.

Theorems & Definitions (21)

  • Definition 1: Strongly Connected Matrix
  • Definition 2: Primitive Matrix
  • Definition 3: Matrix Stability and Function Convergence
  • Remark 1
  • Remark 2
  • Theorem 1: 1985:Magnus_191
  • proof
  • proof
  • Theorem 2
  • proof
  • ...and 11 more