A Dissipativity Approach to Analyzing Composite Spreading Networks

TL;DR

This paper addresses how disease spreading in multi-resolution networks can be understood when subnetworks are interconnected. It develops an input-output SIS model for each subnetwork, introduces a composition operation to assemble a composite network via a binary coupling matrix, and analyzes dissipativity using storage $V^k(x^k)$ and supply $S^k(u^k,y^k)$ functions. A key contribution is an LMI-based condition, $M^T\Psi M I < 0$, with weights $\alpha_k>0$, that ensures a unique disease-free equilibrium for the composite network, along with corollaries describing how scaling inter-network coupling can guarantee stability. The framework is illustrated with a primary-school influenza-style simulation showing that reducing average interaction time between classes to below 79% of the original level can prevent an outbreak, highlighting practical intervention policies. Overall, the work provides a bottom-up, dissipativity-based method to certify and guide control of large-scale spreading networks without requiring strong connectivity across all components.

Abstract

The study of spreading processes often analyzes networks at different resolutions, e.g., at the level of individuals or countries, but it is not always clear how properties at one resolution can carry over to another. Accordingly, in this work we use dissipativity theory from control system analysis to characterize composite spreading networks that are comprised by many interacting subnetworks. We first develop a method to represent spreading networks that have inputs and outputs. Then we define a composition operation for composing multiple spreading networks into a larger composite spreading network. Next, we develop storage and supply rate functions that can be used to demonstrate that spreading dynamics are dissipative. We then derive conditions under which a composite spreading network will converge to a disease-free equilibrium as long as its constituent spreading networks are dissipative with respect to those storage and supply rate functions. To illustrate these results, we use simulations of an influenza outbreak in a primary school, and we show that an outbreak can be prevented by decreasing the average interaction time between any pair of classes to less than 79% of the original interaction time.

Figures (8)

• Figure 1: A composite spreading network is comprised of four subnetworks. Nodes in red, yellow, blue, and green around the perimeter represent populations within different networks, and edges of the same color (drawn in the interior) indicate transmission interactions within each network. Black edges connecting nodes from different networks (i.e., nodes of different colors) illustrate how these subnetworks combine to form the composite spreading network.
• Figure 2: A plot of the sizes of the infected proportions for each network in Figure \ref{['fig:composite_network']}, each of which is modeled using an $SIS$ model. Before the $30^{th}$ time step, there are no transmissions between nodes of different colors, meaning these networks are isolated. However, at the $30^{th}$ time step (indicated by the dashed purple line), new transmission channels are introduced, leading to the presence of black edges in Figure \ref{['fig:composite_network']}. These new transmission channels result in a resurgence of the outbreak across the entire composite network, despite the fact that each individual network had only a negligible portion of its population infected.
• Figure 3: Composite Network of Three Subnetworks. Nodes with the same color are in the same subnetwork. We use orange edges to represent the transmissions from subnetworks 1 and 3 to subnetwork 2.
• Figure 4: A undirected contacting network in a French school involving children aged $6$-$12$stehle2011high. This network consists of $228$ nodes representing the students and $5,539$ edges representing their pairwise interactions over one day. Each student belongs to a specific class (indexed from $1$ to $10$) at the school, which is indicated by the color of the node. The edges between students represent the contact duration, with thicker edges indicating longer cumulative contact time throughout the day.
• Figure 5: $R_0$ of the ten classes. We present the basic reproduction numbers of the ten classes. The basic reproduction number of the $k^{th}$ spreading network is computed as $\rho((\Gamma^k)^{-1}B^k)$, $k\in\underline{10}$. We observe that the basic reproduction numbers for all classes are less than one, indicating that it is unnecessary to isolate any classes to avoid an outbreak.

Theorems & Definitions (25)

• Definition 1: Input Vector $u^{k}$
• Example 1
• Definition 2: Input Transmission Matrix
• Remark 1
• Example 2
• Definition 3: Dissipative Network $SIS$ Spreading Dynamics
• Remark 2
• Definition 4: Supply Rate Function for Network $SIS$ Dynamics
• Definition 5: Storage Function for Network $SIS$ Dynamics
• Remark 3