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Hierarchical Analysis and Control of Epidemic Spreading over Networks using Dissipativity and Mesh Stability

Shirantha Welikala, Hai Lin, Panos J. Antsaklis

TL;DR

The paper develops a hierarchical, dissipativity-based framework for analyzing and controlling epidemic spreading over multi-level networks. By modeling spreading networks as interconnected subsystems and enforcing $X$-dissipativity through convex LMIs, it enables non-iterative, scalable design of inter-group connections that guarantee stability and disturbance robustness, including scalable mesh stability (SMS). The approach yields a sequence of LMIs at node and group levels, supporting necessary feasibility conditions and co-design of topology and control. Simulation results on a four-group network demonstrate superior infection containment and disturbance robustness compared with threshold-pruning and degree-based methods, while incurring moderate interconnection reductions. The framework offers a principled tool for epidemic control in complex, hierarchical networks and can be extended to time-varying settings and other spreading processes.

Abstract

Analyzing and controlling spreading processes are challenging problems due to the involved non-linear node (subsystem) dynamics, unknown disturbances, complex interconnections, and the large-scale and multi-level nature of the problems. The dissipativity concept provides a practical framework for addressing such concerns, thanks to the energy-based representation it offers for subsystems and the compositional properties it provides for the analysis and control of interconnected (networked) systems comprised of such subsystems. Therefore, in this paper, we utilize the dissipativity concept to analyze and control a spreading process that occurs over a hierarchy of nodes, groups, and a network (i.e., a spreading network). We start by generalizing several existing results on dissipativity-based topology design for networked systems. Next, we model the considered spreading network as a networked system and establish the dissipativity properties of its nodes. The generalized topology design method is then applied at multiple levels of the considered spreading network to formulate its analysis and control problems as Linear Matrix Inequality (LMI) problems. We identify and enforce localized necessary conditions to support the feasibility of the LMI problem solved at each subsequent hierarchical level of the spreading network. Consequently, the proposed method does not involve iterative multi-level optimization stages that are computationally inefficient. The proposed control solution ensures that the spreading network is not only stable but also dissipative and mesh-stable. Compared to conventional methods, such as threshold pruning and high-degree edge removal, our approach offers superior performance in terms of infection containment, control efficiency, and disturbance robustness. Extensive numerical results demonstrate the effectiveness of the proposed technique.

Hierarchical Analysis and Control of Epidemic Spreading over Networks using Dissipativity and Mesh Stability

TL;DR

The paper develops a hierarchical, dissipativity-based framework for analyzing and controlling epidemic spreading over multi-level networks. By modeling spreading networks as interconnected subsystems and enforcing -dissipativity through convex LMIs, it enables non-iterative, scalable design of inter-group connections that guarantee stability and disturbance robustness, including scalable mesh stability (SMS). The approach yields a sequence of LMIs at node and group levels, supporting necessary feasibility conditions and co-design of topology and control. Simulation results on a four-group network demonstrate superior infection containment and disturbance robustness compared with threshold-pruning and degree-based methods, while incurring moderate interconnection reductions. The framework offers a principled tool for epidemic control in complex, hierarchical networks and can be extended to time-varying settings and other spreading processes.

Abstract

Analyzing and controlling spreading processes are challenging problems due to the involved non-linear node (subsystem) dynamics, unknown disturbances, complex interconnections, and the large-scale and multi-level nature of the problems. The dissipativity concept provides a practical framework for addressing such concerns, thanks to the energy-based representation it offers for subsystems and the compositional properties it provides for the analysis and control of interconnected (networked) systems comprised of such subsystems. Therefore, in this paper, we utilize the dissipativity concept to analyze and control a spreading process that occurs over a hierarchy of nodes, groups, and a network (i.e., a spreading network). We start by generalizing several existing results on dissipativity-based topology design for networked systems. Next, we model the considered spreading network as a networked system and establish the dissipativity properties of its nodes. The generalized topology design method is then applied at multiple levels of the considered spreading network to formulate its analysis and control problems as Linear Matrix Inequality (LMI) problems. We identify and enforce localized necessary conditions to support the feasibility of the LMI problem solved at each subsequent hierarchical level of the spreading network. Consequently, the proposed method does not involve iterative multi-level optimization stages that are computationally inefficient. The proposed control solution ensures that the spreading network is not only stable but also dissipative and mesh-stable. Compared to conventional methods, such as threshold pruning and high-degree edge removal, our approach offers superior performance in terms of infection containment, control efficiency, and disturbance robustness. Extensive numerical results demonstrate the effectiveness of the proposed technique.

Paper Structure

This paper contains 24 sections, 15 theorems, 82 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Dissipativity
  5. Dissipativity Based Networked System Design
  6. Networked System Design Considerations
  7. Necessary Conditions to be Enforced at the Subsystems
  8. Enforcing Constraints on the Interconnection Matrix $M$
  9. Scalable Mesh Stability
  10. Dissipativity of a Generic Node
  11. Problem Formulation
  12. Spreading Network Model
  13. Hierarchical Networked System Representation
  14. Spreading Network Control Problem
  15. Dissipativity-Based Solution
  16. ...and 9 more sections

Key Result

Proposition 1

WelikalaP52022 Under As. As:SubsystemDissipativity, the networked system $\Sigma$ shown in Fig. Fig:NetworkedSystem can be made $\mathcal{X}$-dissipative (from input $w(t)$ to output $z(t)$, where $\mathcal{X} = \mathcal{X}^\top \triangleq [\mathcal{X}^{kl}]_{k,l\in\mathbb{N}_2}$ with $\mathcal{X}^{ where $\Psi$ is as defined in Eq:Prop:NetworkedSystem2, $\textbf{X}_p^{kl} \triangleq diag([p_iX_i^

Figures (7)

  • Figure 1: Networked System
  • Figure 2: Spreading network representation as a hierarchical networked system.
  • Figure 3: Considered spreading network: (a) without and (b) with inter-group interconnections. Proposed dissipativity-based design DissBC($1,1$): (c) interconnection topology and (d) histogram of percentage interconnection reductions.
  • Figure 4: Time evolution of average infection levels over the spreading network under different inter-group interconnection configurations: (1) No interconnections (blue), (2) Original uncontrolled network (orange), (3) Dissipativity-based design (magenta), (4) Threshold-based design (yellow), and (5) Degree-based design (green).
  • Figure 5: Spreading network designs (top row) and their percentage interconnection reductions histograms (bottom row) observed under the proposed dissipativity-based design approach with parameters $c_M = 10^{-9}$ (left column) and $c_M = 10^{9}$ (right column) with $\delta_M = 1$.
  • ...and 2 more figures

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • Corollary 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 2
  • proof
  • ...and 33 more