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

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.