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

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 some 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 (i.e., it also minimizes the impact and propagation of disturbances affecting the spreading network). 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.

Index Terms—Dissipativity-Based Control, Spreading Processes, Epidemics Control, Spreading Networks Design.

I Introduction

Studying epidemic spreading processes over interconnected systems is crucial for multiple domains, including epidemiology, social services, and healthcare systems. In terms of underlying dynamics, the spreading processes of information and opinions that occur in computer networks and social systems also take a very similar form. Therefore, understanding how infections propagate across multiple groups in a network–due to nodes in those groups having some inter-group interconnections–is crucial for designing effective control mechanisms to prevent undesirable outbreaks by preventing such interactions. Typically, these systems are modeled as graphs where nodes represent entities (e.g., individuals, households, communities, etc.), and edges denote interactions through which the spreading process occurs. Recent research has highlighted the importance of modeling such processes in a hierarchical manner, particularly when dealing with large-scale multi-level networks such as local, regional, and global spread of epidemics [??]. However, designing control strategies for these networks is challenging due to the presence of nonlinear node dynamics, unknown disturbances, complex interconnections, and their inherently large-scale multi-level nature.

Traditional control techniques for spreading networks generally focus on modifying the network topology to mitigate infection levels or enhance robustness. Some work also investigate acceleration of local healing (recovery) rates. Two commonly employed methods are threshold pruning and high-degree edge removal. Threshold pruning aims to eliminate inter-group connections exceeding a specified threshold, thus removing the most impactful pathways for infection spread. High-degree edge removal, on the other hand, targets nodes with high inter-group connectivity and eliminates their outgoing connections, thereby limiting the influence of highly connected nodes [??]. Despite their practical appeal, these methods are typically heuristic and fail to adequately account for the complex dynamics and external disturbances present in real-world systems. Additionally, such approaches are often static and do not offer guarantees regarding the stability or robustness of the network under varying conditions.

Dissipativity theory provides a powerful alternative framework for analyzing and controlling networked systems. Unlike conventional methods that rely on heuristic pruning strategies, dissipativity offers a systematic approach to evaluating the energy transfer properties of systems, which can be directly related to stability and robustness criteria. By using an energy-based representation of subsystems, dissipativity theory facilitates a compositional approach to the analysis and design of large-scale systems. It has been successfully applied to various networked control problems, including consensus, synchronization, and disturbance rejection [??]. However, its application to epidemic spreading control remains largely unexplored. In this paper, we propose a dissipativity-based framework for the analysis and design of spreading networks. Our approach models the spreading network as a hierarchical system with disturbance inputs and performance outputs, enabling the use of robust networked systems techniques in a scalable manner. The proposed method provides stability and robustness guarantees through dissipativity conditions formulated as Linear Matrix Inequality (LMI) problems, allowing for efficient computation and practical implementation.

Our contributions can be summarized as follows.

1. We model epidemic spreading processes as hierarchical networked systems with disturbance inputs and performance outputs, enabling the application of robust networked systems techniques in a scalable manner.

2. We derive dissipativity properties of the nodes under disturbances in epidemic spreading processes, enabling the analysis of dissipativity of any group of such nodes and, subsequently, of any network of such groups and beyond.

3. Going beyond the analysis of spreading networks, we propose a systematic spreading network design technique (to ensure stability and robustness via dissipativity) with soft and hard constraints on deviations from a given nominal interconnection network.

4. The proposed approach does not involve solving iterative multi-level optimization problems as we include specifically designed necessary conditions to support the feasibility of the optimization problem solved in each subsequent layer. All optimization problems take the form of LMI problems, hence convex, and can be solved using existing software toolboxes.

5. While we propose a very generalized approach, we provide clear implementation details and extensive numerical results showcasing the benefits of the proposed method and comparing it with basic spreading network control methods.

This paper is structured as follows. Section II provides necessary preliminary results on dissipativity and networked systems. Section III presents the spreading network dynamics and formulates the analysis and control (design) problems. The proposed dissipativity-based analysis and control techniques are detailed in Section IV. Simulation results and conclusions are given in Sections V and VI, respectively.

II Preliminaries

In this section, we provide some preliminary results that will be utilized in the remainder of this paper.

II-A Notations

We use $\mathbb{R}$ and $\mathbb{N}$ to denote sets of real and natural numbers, respectively. For any $N\in\mathbb{N}$, we define $\mathbb{N}_{N}\triangleq\{1,2,..,N\}$. An $n\times m$ block matrix $A$ is denoted as $A=[A_{ij}]_{i\in\mathbb{N}_{n},j\in\mathbb{N}_{m}}$ or as $A=[A^{ij}]_{i\in\mathbb{N}_{n},j\in\mathbb{N}_{m}}$). We consider $[A_{ij}]_{j\in\mathbb{N}_{m}}$ and $\text{diag}([A_{ii}]_{i\in\mathbb{N}_{n}})$ to be a block row matrix and a block diagonal matrix, respectively. $\mathbf{0}$ and $\mathbf{I}$ respectively denote zero and identity matrices (their dimensions will be clear from the context). A symmetric positive definite (semi-definite) matrix $A\in\mathbb{R}^{n\times n}$ is denoted by $A>0\ (A\geq 0)$. We use the symbol $\star$ to represent conjugate matrices in symmetric matrices (their meaning will be clear from the context). We also define $\mathcal{H}(A)\triangleq A+A^{\top}$, $\mathbf{1}_{\{\cdot\}}$ as the indicator function and $\mathrm{e}_{ij}\triangleq\mathbf{1}_{\{i=j\}}$. The notations $\mathcal{K}_{\infty}$ and $\mathcal{KL}$ denote respective class-$\mathcal{K}$ functions [1]. We use $|\cdot|$ to denote the Euclidean (spectral) norm of a vector (matrix), and $\|\cdot\|$ to denote any standard norm (note $|\cdot|=\|\cdot\|_{2}$). For a time-varying vector (i.e., a signal) $x\triangleq\{x(t)\in\mathbb{R}^{n}\}_{t\geq 0}$, $\mathcal{L}_{2}$ and $\mathcal{L}_{\infty}$ norms are given by $\|x\|\triangleq\sqrt{\int_{0}^{\infty}|x(t)|^{2}dt}$ and $\|x\|_{\infty}\triangleq\sup_{t\geq 0}|x(t)|$, respectively. $\overline{\lambda}(\cdot)$, $\underline{\lambda}(\cdot)$ denotes the maximum and minimum eigenvalues of a matrix.

II-B Dissipativity

Consider the non-linear dynamical system described by

$$\dot{x}(t)=f(x(t),u(t)),$$ (1)

$$y(t)=h(x(t),u(t)),$$

where $x(t)\in\mathbb{R}^{n}$, $u(t)\in\mathbb{R}^{l}$ and $y(t)\in\mathbb{R}^{m}$ denote the state, input and output, respectively. Functions $f:\mathbb{R}^{n}\times\mathbb{R}^{l}\rightarrow\mathbb{R}^{n}$ and $h:\mathbb{R}^{n}\times\mathbb{R}^{q}\rightarrow\mathbb{R}^{m}$ are continuously differentiable, and satisfy $f(\mathbf{0},\mathbf{0})=\mathbf{0}$ and $h(\mathbf{0},\mathbf{0})=\mathbf{0}$.

Definition 1

The system (1) is called dissipative under supply rate $s:\mathbb{R}^{l}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$ if there is a continuously differentiable storage function $V:\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that $V(x)>0$ when $x\neq 0$, $V(0)=0$, and

$\dot{V}(x)=\nabla_{x}V(x)f(x,u)\leq s(u,y)$,

for all trajectories $(x,u)\in\mathbb{R}^{n}\times\mathbb{R}^{q}$ of the system.

Based on the used supply rate $s(.,.)$, the dissipativity property can be specialized. The $X$-dissipativity property (also known as $(Q,S,R)$-dissipativity in the literature), defined in the sequel, uses a quadratic supply rate determined by a coefficient matrix $X=X^{\top}\triangleq[X^{kl}]_{k,l\in\mathbb{N}_{2}}\in\mathbb{R}^{l+m}$ [2].

Definition 2

The system (1) is $X$-dissipative if it is dissipative under the quadratic supply rate function:

$$s(u,y)\triangleq\begin{bmatrix}u\\ y\end{bmatrix}^{\top}\begin{bmatrix}X^{11}&X^{12}\\ X^{21}&X^{22}\end{bmatrix}\begin{bmatrix}u\\ y\end{bmatrix}.$$

The notion of $X$-dissipativity captures several well-known properties of interest [3], as outlined in the following remark.

Remark 1

If the system (1) is $X$-dissipative with:

1. $X=\scriptsize\begin{bmatrix}\mathbf{0}&\frac{1}{2}\mathbf{I}\\ \frac{1}{2}\mathbf{I}&\mathbf{0}\end{bmatrix}\normalsize$, then it is passive;
   

2. $X=\scriptsize\begin{bmatrix}-\nu\mathbf{I}&\frac{1}{2}\mathbf{I}\\ \frac{1}{2}\mathbf{I}&-\rho\mathbf{I}\end{bmatrix}\normalsize$, then it is strictly passive ($\nu$ and $\rho$ are the input and output passivity indices, denoted as IF-OFP($\nu,\rho$));
   

3. $X=\scriptsize\begin{bmatrix}\gamma^{2}\mathbf{I}&\mathbf{0}\\ \mathbf{0}&-\mathbf{I}\end{bmatrix}\normalsize$, then it is $L_{2}$-stable ($\gamma$ is the $L_{2}$-gain, denoted as $L2G(\gamma)$).

II-C Node Dissipativity

Consider a scalar dynamic system $\Sigma_{s}:u(t)\rightarrow y(t)$ described by (for known and fixed $\bar{\gamma},\delta>0$):

$$\dot{x}(t)= \ -\gamma(t)x(t)+(1-x(t))u(t)$$ (2)

$$y(t)= \ x(t)$$

where $|\gamma(t)-\bar{\gamma}|\leq\delta$, $x(t)\in[0,1]$ and $u(t)\in\mathbb{R}_{\geq 0}$.

Proposition 1

The system $\Sigma_{s}$ (2) is $X$-dissipative for any $X=X^{\top}\triangleq\scriptsize\begin{bmatrix}a&b\\ b&c\end{bmatrix}\in\mathbb{R}^{2\times 2}$ that satisfies the LMI problem:

$$\mbox{Find: }\ p,\ a,\ b,\ c,\$$ (3)

$$\mbox{Sub. to: }\ p>0,$$

$\displaystyle$

Proof:

Note that, in this proof, we omit denoting the dependence on time $t$ of variables $x,u,\gamma$ and $y$ for notational convenience. As $x=0$ is an equilibrium point of $\Sigma_{s}$ (with $u=0$ and $y=0$), we consider a quadratic storage function $V(x)$ and a quadratic supply rate function $s(u,y)$ defined as V(x) = 12 p x^2   and  s(u,y) = [u y]^⊤[a bb c] [u y], respectively, where $p\in\mathbb{R}_{>0}$ and $a,b,c\in\mathbb{R}$ are design variables. For $X$-dissipativity of the system $\Sigma_{s}$, we require ˙V(x) ≤s(u,y) ⇔0 ≤Φ(u,x) ≜s(u,x) - ˙V(x) to hold for all possible trajectories of the system. Using $\dot{V}(x)=px\dot{x}$ with (2), $\Phi(u,x)$ can be expressed as

$$\Phi(u,x)= \ au^{2}+((2b-p)+px)xu+(c+p\gamma)x^{2}.$$

Since $\gamma\geq\bar{\gamma}-\delta$, we get $\Phi(u,x)\geq\bar{\Phi}(u,x)$ where ¯Φ(u,x) ≜au^2 + ((2b-p) + px)xu + (c + p(¯γ-δ))x^2. Therefore, we need to identify the conditions to ensure $\bar{\Phi}(u,x)\geq 0,\forall x\in[0,1],u\geq 0$.

To this end, note that $\bar{\Phi}(0,0)=0$, $\bar{\Phi}(0,x)=(c+p(\bar{\gamma}-\delta))x^{2}$ and $\bar{\Phi}(u,0)=au^{2}$. Therefore, it is clear that we require $a\geq 0$ and $(c+p(\bar{\gamma}-\delta))\geq 0$. Now, let us define Ψ_x(u) ≜¯Φ(u,x) ≡au^2 + ¯bu + ¯c, where $a\geq 0$, $\bar{b}\triangleq((2b-p)+px)x$, and $\bar{c}\triangleq(c+p(\bar{\gamma}-\delta))x^{2}\geq 0$.

First, let us consider the case $a>0$, for which $\Psi_{x}(u)$ is a convex quadratic function in $u$ for a given $x\in[0,1]$. Let $u^{*}$ be the unconstrained minimizer of $\Psi_{x}(u)$ for a given $x\in[0,1]$. Using the convexity and $\frac{\partial\Psi_{x}(u)}{\partial u}=0$, we get u^* = -¯b2a  and  Ψ_x(u^*) = -¯b24a + ¯c. Note that, if $u^{*}\leq 0$, the convexity of $\Psi_{x}(u)$ and $\Psi_{x}(0)\geq 0$ automatically implies that $\Psi_{x}(u)\geq 0,\forall u\geq 0$. On the other hand, if $u^{*}\geq 0$, we then require enforcing $\Psi_{x}(u^{*})\geq 0$ to guarantee $\Psi_{x}(u)\geq 0,\forall u\geq 0$. Using the $u^{*}$ and $\Psi_{x}(u^{*})$ expressions, these requirements translate to: If $a>0$, then $\bar{b}\geq 0$, or $\bar{b}\leq 0$ and $4a\bar{c}-\bar{b}^{2}\geq 0$. Finally, for the case $a=0$, $\Psi_{x}(u)=\bar{b}u+\bar{c}$ with $\bar{c}\geq 0$. Therefore, if $a=0$, we require $\bar{b}\geq 0$ to ensure $\Psi_{x}(u)\geq 0,\forall u\geq 0$. In all, we require $\bar{c}\geq 0$ and {a≥0, ¯b≥0,   or a¿0,¯b≤0,4a¯c-¯b2≥0.

Recall that $\bar{b}=((2b-p)+px)x=px^{2}+(2b-p)x$, $p>0$ and $x\in[0,1]$. Therefore, $\bar{b}\geq 0\iff(2b-p)\geq 0$, and $\bar{b}\leq 0\iff b\leq 0$. Further,

$$4a\bar{c}-\bar{b}^{2}= \ 4a(c+p(\bar{\gamma}-\delta))x^{2}-((2b-p)+px)^{2}x^{2}\geq 0$$

$$\iff 4a(c+p(\bar{\gamma}-\delta))-((2b-p)+px)^{2}\geq 0$$

$$\iff a(c+p(\bar{\gamma}-\delta))-b^{2}\geq 0.$$

Furthermore,

$$\bar{c}=(c+p(\bar{\gamma}-\delta))x^{2}\geq 0\iff(c+p(\bar{\gamma}-\delta))\geq 0.$$

Using these equivalences, we can arrive at the requirements as $(c+p(\bar{\gamma}-\delta))\geq 0$ and

$$\begin{cases}a\geq 0,(2b-p)\geq 0,\quad\mbox{or}\\ a>0,b\leq 0,a(c+p(\bar{\gamma}-\delta))-b^{2}\geq 0,\end{cases}$$

which leads to (3) using the Schur complement. ∎

Based on Prop. 1 and Rm. 1, the following corollary can be established regarding the passivity of the system $\Sigma_{s}$ (2).

Corollary 1

The system $\Sigma_{s}$ (2) with $\bar{\gamma}-\delta\geq 0$ is passive, and also IF-OFP($\nu,\rho$) with $\nu\leq 0,\rho\leq p(\bar{\gamma}-\delta)$ and $p\leq 1$.

Proof:

The proof follows directly from applying Prop. 1 for the first two $X$-dissipativity cases given in Rm. 1. ∎

II-D Networked System Dissipativity

Consider a networked system $\Sigma:w\rightarrow z$ as shown in Fig. 1 comprised of $N$ subsystems $\{\Sigma_{i}:i\in\mathbb{N}_{N}\}$ with an input $w\in\mathbb{R}^{q}$ (disturbance) and an output $z\in\mathbb{R}^{r}$ (performance). Each subsystem $\Sigma_{i}:u_{i}\rightarrow y_{i},i\in\mathbb{N}_{N}$ follows the dynamics

$$\dot{x}_{i}(t)= \ f_{i}(x_{i}(t),u_{i}(t)),$$ (4)

$$y_{i}(t)= \ g_{i}(x_{i}(t),u_{i}(t)),$$

where $x_{i}(t)\in\mathbb{R}^{n_{i}}$, $u_{i}(t)\in\mathbb{R}^{l_{i}}$ and $y_{i}(t)\in\mathbb{R}^{m_{i}}$ denotes the state, input and output, respectively, and $f_{i}(\mathbf{0},\mathbf{0})=\mathbf{0}$ and $h_{i}(\mathbf{0},\mathbf{0})=\mathbf{0}$. The subsystems are interconnected according to the relationship

$$\begin{bmatrix}u(t)\\ z(t)\end{bmatrix}=M\begin{bmatrix}y(t)\\ w(t)\end{bmatrix}=\begin{bmatrix}M_{uy}&M_{uw}\\ M_{zy}&M_{zw}\end{bmatrix}\begin{bmatrix}y(t)\\ w(t)\end{bmatrix},$$ (5)

where $u(t)=[u_{i}^{\top}(t)]_{i\in\mathbb{N}_{N}}^{\top}\in\mathbb{R}^{l}$ and $y(t)=[y_{i}^{\top}(t)]_{i\in\mathbb{N}_{N}}^{\top}\in\mathbb{R}^{m}$ are vectorized subsystem inputs and outputs, respectively, and $M\in\mathbb{R}^{(l+r)\times(m+q)}$ the interconnection matrix.

Figure 1: Networked System

In the context of this paper, we require the following simplifying assumption regarding the dissipativity properties of the subsystems.

Assumption 1

Each subsystem $\Sigma_{i},\ i\in\mathbb{N}_{N}$ (4) is $X_{i}$-dissipative (from input $u_{i}(t)$ to output $y_{i}(t)$) where $X_{i}=X_{i}^{\top}=[X_{i}^{kl}]_{k,l\in\mathbb{N}_{2}}$ is such that $X_{i}^{11}>0$.

In the following proposition, subsystem dissipativity properties $\{X_{i}:i\in\mathbb{N}_{N}\}$ are used to design the interconnection matrix $M$ so that the networked system is dissipative.

Proposition 2

Under As. 1, the networked system $\Sigma$ shown in Fig. 1 can be made $\mathcal{X}$-dissipative (from input $w(t)$ to output $z(t)$, where $\mathcal{X}=\mathcal{X}^{\top}=[\mathcal{X}^{kl}]_{k,l\in\mathbb{N}_{2}}$ with $\mathcal{X}^{22}<0$), by designing the interconnection matrix $M$ via the LMI problem

Find: $$L_{uy},L_{uw},M_{zy},M_{zw},\{p_{i}:i\in\mathbb{N}_{N}\}$$ (6)

Sub. to: $$p_{i}>0,\ \forall i\in\mathbb{N}_{N},\ \Psi>0$$

where $\Psi$ is as defined in (7), $\textbf{X}_{p}^{kl}\triangleq diag([p_{i}X_{i}^{kl}]_{i\in\mathbb{N}_{N}})$ for any $k,l\in\mathbb{N}_{2}$, $\textbf{X}^{12}\triangleq diag([(X_{i}^{11})^{-1}X_{i}^{12}]_{i\in\mathbb{N}_{N}})$, $\textbf{X}^{21}\triangleq(\textbf{X}^{12})^{\top}$ with $M_{uy}\triangleq(\textbf{X}_{p}^{11})^{-1}L_{uy}$ and $M_{uw}\triangleq(\textbf{X}_{p}^{11})^{-1}L_{uw}$.

 

$$\Psi\triangleq\begin{bmatrix}\textbf{X}_{p}^{11}&\textbf{0}&L_{uy}&L_{uw}\\ \textbf{0}&-\mathcal{X}^{22}&-\mathcal{X}^{22}M_{zy}&-\mathcal{X}^{22}M_{zw}\\ L_{uy}^{\top}&-M_{zy}^{\top}\mathcal{X}^{22}&-L_{uy}^{\top}\textbf{X}^{12}-\textbf{X}^{21}L_{uy}-\textbf{X}_{p}^{22}&-\textbf{X}^{21}L_{uw}+M_{zy}^{\top}\mathcal{X}^{21}\\ L_{uw}^{\top}&-M_{zw}^{\top}\mathcal{X}^{22}&-L_{uw}^{\top}\textbf{X}^{12}+\mathcal{X}^{12}M_{zy}&M_{zw}^{\top}\mathcal{X}^{21}+\mathcal{X}^{12}M_{zw}+\mathcal{X}^{11}\end{bmatrix}$$ (7)

$$\Phi\triangleq\begin{bmatrix}\textbf{X}_{p}^{11}&\textbf{0}&L_{uy}&\textbf{X}_{p}^{11}\\ \textbf{0}&-\mathcal{X}^{22}&-\mathcal{X}^{22}&\mathbf{0}\\ L_{uy}^{\top}&-\mathcal{X}^{22}&-L_{uy}^{\top}\textbf{X}^{12}-\textbf{X}^{21}L_{uy}-\textbf{X}_{p}^{22}&-\textbf{X}_{p}^{21}+\mathcal{X}^{21}\\ \textbf{X}_{p}^{11}&\mathbf{0}&-\textbf{X}_{p}^{12}+\mathcal{X}^{12}&\mathcal{X}^{11}\end{bmatrix}$$ (8)

If the interconnection matrix $M$ is constrained so that

$$M_{uw}\triangleq\mathbf{I},\quad M_{zy}\triangleq\mathbf{I},\quad M_{zw}\triangleq\mathbf{0},$$ (9)

the following corollary can be used to design the remaining interconnection matrix block $M_{uy}$ so that the networked system is dissipative.

Corollary 2

Under As. 1 and constraints (9), the networked system $\Sigma$ shown in Fig. 1 can be made $\mathcal{X}$-dissipative (from input $w(t)$ to output $z(t)$, where $\mathcal{X}=\mathcal{X}^{\top}=[\mathcal{X}^{kl}]_{k,l\in\mathbb{N}_{2}}$ with $\mathcal{X}^{22}<0$), by designing the interconnection matrix $M$ via the LMI problem

Find: $$L_{uy},\{p_{i}:i\in\mathbb{N}_{N}\}$$ (10)

Sub. to: $$p_{i}>0,\ \forall i\in\mathbb{N}_{N},\ \Phi>0$$

where $\Phi$ is as defined in (8), and $M_{uy}\triangleq(\textbf{X}_{p}^{11})^{-1}L_{uy}$.

Proof:

As $M_{uw}\triangleq(\textbf{X}_{p}^{11})^{-1}L_{uw}$ and $M_{uw}\triangleq\mathbf{I}$, we get $L_{uw}=\textbf{X}_{p}^{11}$. Note also that $\textbf{X}^{12}\triangleq(\textbf{X}_{p}^{11})^{-1}\textbf{X}_{p}^{12}$, and thus, $\textbf{X}_{p}^{11}\textbf{X}^{12}=\textbf{X}_{p}^{12}$. Similarly, $\textbf{X}^{21}\textbf{X}_{p}^{11}=\textbf{X}_{p}^{21}$. Applying these results together with (9) in (6), we can obtain (10). ∎

II-E Subsystem Dissipativity

Note that the feasibility of the network-level LMI problem presented in Prop. 2 (or Co. 2) depends on the used subsystem dissipativity properties $\{X_{i}:i\in\mathbb{N}_{N}\}$. Often, such subsystem dissipativity properties result from subsystem-level LMI problems. Therefore, it is worth identifying any necessary conditions required of subsystem dissipativity properties to help the feasibility of the network-level LMI problem, as they can be enforced at such subsystem-level LMI problems. In the following proposition, we identify such necessary conditions to help the feasibility of the network-level LMI problem (10) given in Co. 2 for the possible inclusion in subsystem-level LMI problems.

Proposition 3

For the feasibility of the network-level LMI problem (10) given in Co. 2, it is necessary that each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ is $X_{i}$-dissipative such that $X_{i}$ satisfies the LMI problem:

Find: $$\ X_{i},\{\bar{\mathcal{X}}_{ii}^{kl}:k,l\in\mathbb{N}_{2}\},$$ (11)

Sub. to: $$\ \tilde{\Phi}_{ii}>0$$

where $\tilde{\Phi}_{ii}$ is as defined in (12), each $\bar{\mathcal{X}}_{ii}^{kl}$ is a free variable matrix with the format of the $i$^th diagonal block of $\mathcal{X}^{kl}$ in (6), and $M_{uy}^{ii}$ is the $i$-th diagonal block of $M_{uy}$ that describes the self-connection matrix of the subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$.

Proof:

For the block matrix $\Phi=[\Phi_{kl}]_{k,l\in\mathbb{N}_{4}}$ given in (8), the equivalence Φ¿ 0 ⇔¯Φ ≜[[Φ_kl^ij]_k,l∈N_4]_i,j∈N_N ¿ 0 holds, where $\bar{\Phi}$ is the “block element-wise” permutation [4] of $\Phi$. Using the diagonal blocks of $\bar{\Phi}$, we can identify a set of necessary conditions for $\Phi>0$ as Φ¿ 0 ⇔¯Φ ¿ 0 ⟹{ ¯Φ_ii ¿ 0, ∀i∈N_N}, where each $\bar{\Phi}_{ii}\triangleq[\Phi_{kl}^{ii}]_{k,l\in\mathbb{N}_{4}}$ takes the form in (13). Note that, the implication M_uy = (X_p^11)^-1 L_uy ⟹M_uy^ii = p_i^-1(X_i^11)^-1L_uy^ii, ∀i∈N_N, has also been used to replace $L_{uy}^{ii}$ terms in deriving (13). Finally, we use the equivalence ¯Φ_ii ¿ 0 ⇔~Φ_ii ≜1pi¯Φ_ii ¿ 0, and the notation $\bar{\mathcal{X}}^{kl}_{ii}\triangleq\frac{1}{p_{i}}\mathcal{X}^{kl}_{ii},\forall k,l\in\mathbb{N}_{2}$ divide the matrix $\bar{\Phi}_{ii}$ in (13) by $p_{i}$ to obtain (12). ∎

$$\tilde{\Phi}_{ii}\triangleq\begin{bmatrix}X_{i}^{11}&\textbf{0}&X_{i}^{11}M_{uy}^{ii}&X_{i}^{11}\\ \textbf{0}&-\bar{\mathcal{X}}^{22}_{ii}&-\bar{\mathcal{X}}^{22}_{ii}&\mathbf{0}\\ (M_{uy}^{ii})^{\top}X_{i}^{11}&-\bar{\mathcal{X}}^{22}_{ii}&-(M_{uy}^{ii})^{\top}X_{i}^{12}-X_{i}^{21}M_{uy}^{ii}-X_{i}^{22}&-X_{i}^{21}+\bar{\mathcal{X}}^{21}_{ii}\\ X_{i}^{11}&\mathbf{0}&-X_{i}^{12}+\bar{\mathcal{X}}^{12}_{ii}&\bar{\mathcal{X}}^{11}_{ii}\end{bmatrix}$$ (12)

$$\bar{\Phi}_{ii}\triangleq\begin{bmatrix}p_{i}X_{i}^{11}&\textbf{0}&p_{i}X_{i}^{11}M_{uy}^{ii}&p_{i}X_{i}^{11}\\ \textbf{0}&-\mathcal{X}^{22}_{ii}&-\mathcal{X}^{22}_{ii}&\mathbf{0}\\ p_{i}(M_{uy}^{ii})^{\top}X_{i}^{11}&-\mathcal{X}^{22}_{ii}&-p_{i}(M_{uy}^{ii})^{\top}X_{i}^{12}-p_{i}X_{i}^{21}M_{uy}^{ii}-p_{i}X_{i}^{22}&-p_{i}X_{i}^{21}+\mathcal{X}^{21}_{ii}\\ p_{i}X_{i}^{11}&\mathbf{0}&-p_{i}X_{i}^{12}+\mathcal{X}^{12}_{ii}&\mathcal{X}^{11}_{ii}\end{bmatrix}$$ (13)

Remark 2

To include the identified necessary condition (11) in the subsystem-level LMI problem solved at the subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ (to find its dissipativity $X_{i}$), it is required to know the self-connection matrix $M_{uy}^{ii}$. As we will see in the sequel, this information is known and fixed at the “subsystems” considered in the spreading networks application studied in this paper.

Remark 3

A further simplified necessary condition than (11) can be obtained if the matrices $X_{i}^{kl}$ and $\bar{\mathcal{X}}_{ii}^{kl}$ (i.e., subsystem and network level dissipativity measures) are limited to be scalar matrices, over all $k,l\in\mathbb{N}_{2}$ and $i\in\mathbb{N}_{N}$. The idea is to reapply the diagonal block extraction and “block element-wise” formulation technique used in Prop. 3, for (11). Note that this simplification will be further significant if $M_{uy}^{ii}$ is a zero matrix or has a zero in its diagonal.

II-F Constraints

When designing an interconnection matrix $M$ for a networked system, instead of designing it from scratch, it is often required to minimize (or constrain) deviations from an existing interconnection matrix $\bar{M}$. However, including such an objective (or constrain) in the design techniques presented in Prop. 2 and Co. 2 makes the problem bilinear due to the involved change of variables, typically of the form $L=XM\iff M=X^{-1}L$ where $X$ and $L$ are the actual design variables. Nevertheless, there are some approximate strategies that we can use.

Consider an objective function of the form

$$J\triangleq\|M-\bar{M}\|=\|X^{-1}L-X^{-1}X\bar{M}\|\|X\|\|X\|^{-1},$$

which leads to

$$\|X\|^{-1}\|L-X\bar{M}\|\leq J\leq\|X^{-1}\|\|L-X\bar{M}\|.$$

This motivates an objective function of the form:

$$\hat{J}=\|L-X\bar{M}\|+\lambda\|X\|$$

with $\lambda\geq 0$ for the design problems.

On the other hand, consider a scenario where we require the designed interconnection matrix $M$ to be constrained with respect to the nominal $\bar{M}$ so that

$$\frac{\|M-\bar{M}\|}{\|\bar{M}\|}\leq\delta\iff\|M-\bar{M}\|^{2}\leq\delta^{2}\|\bar{M}\|^{2}$$ (14)

holds where $\delta\in[0,1]$ represents the allowed deviation from the nominal values. If the involved matrix norm above is the 2-norm (i.e., the matrix spectral norm, a.k.a., the largest singular value), we can obtain an equivalent condition as

$$0\leq \ \delta^{2}\bar{M}^{\top}\bar{M}-(M-\bar{M})^{\top}(M-\bar{M})$$

$$\ \iff 0\leq\begin{bmatrix}\mathbf{I}&M-\bar{M}\\ M^{\top}-\bar{M}^{\top}&\delta^{2}\bar{M}^{\top}\bar{M}\end{bmatrix}$$

$$\ \iff 0\leq\begin{bmatrix}XX^{\top}&L-X\bar{M}\\ L^{\top}-\bar{M}^{\top}X^{\top}&\delta^{2}\bar{M}^{\top}\bar{M}\end{bmatrix}.$$

The last two steps above result from the Schur complement and congruence techniques. Note that the final result is a bilinear matrix inequality in design variables $X,L$ due to the involved $XX^{\top}$ term. However, a necessary condition for (14) can be obtained as

$$0\leq\begin{bmatrix}XY^{\top}+YX^{\top}-YY^{\top}&L-X\bar{M}\\ L^{\top}-\bar{M}^{\top}X^{\top}&\delta^{2}\bar{M}^{\top}\bar{M}\end{bmatrix},$$

using the fact that, for any matrix $Y$,

$$0\leq(X-Y)(X-Y)^{\top}\iff XY^{\top}+YX^{\top}-YY^{\top}\leq XX^{\top}.$$

$$\bar{\alpha}^{2}E^{\top}\mathcal{P}^{-1}E\succeq E+E^{\top}-\frac{1}{\bar{\alpha}^{2}}\mathcal{P}$$

Finally, reapplying the Schur complement technique, we get

$$0\leq\begin{bmatrix}\mathbf{I}&Y^{\top}&\mathbf{0}\\ Y&XY^{\top}+YX^{\top}&L-X\bar{M}\\ \mathbf{0}&L^{\top}-\bar{M}^{\top}X^{\top}&\delta^{2}\bar{M}^{\top}\bar{M}\end{bmatrix}.$$ (15)

Once a candidate solution for the design variables $X,L$ are known, as (15) is linear in $Y$, it can then be used to update the initial choice of $Y$. If such updates are executed iteratively, desired convergence condition is $Y=X$.

Now, if the constraint (14) is element-wise, i.e.,

$$-\delta\bar{M}\ll M-\bar{M}\ll\delta\bar{M},$$

where $\delta\in[0,1]$, we can obtain an equivalent condition as

$$-\delta X\bar{M}\ll L-X\bar{M}\ll\delta X\bar{M},$$

when $X>0$ and $X$ is diagonal.

II-G Scalable Mesh Stability

Enforcing dissipativity in a networked system ensures the dissipation of disturbances. In contrast, enforcing scalable mesh stability (SMS), as defined below, ensures the dispersion of the disturbances over the network.

For this discussion, let us consider the networked system $\Sigma$ shown in Fig. 1 under the constraints (9). In this setting, each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ (4) can be described by (with a slight abuse of notation)

$$\dot{x}_{i}(t)=f_{i}(x_{i}(t),u_{i}(t))=f_{i}(x_{i}(t),\{x_{j}(t):j\in\mathcal{N}_{i}\},w_{i}(t)),$$ (16)

were $u_{i}(t)=\sum_{j\in\mathbb{N}_{N}}M_{uy}^{ij}x_{j}(t)+w_{i}(t)$ and $\mathcal{N}_{i}\triangleq\{j\in\mathbb{N}_{N}:M_{uy}^{ij}\neq\mathbf{0},j\neq i\}$ denotes the set of neighbors of each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$.

Definition 3

[5] A subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ (16) is input-to-state stable (ISS) if there exists functions $\beta_{i}\in\mathcal{KL},\ \gamma_{i}\in\mathcal{K}_{\infty}$ and $\sigma_{i}\in\mathcal{K}_{\infty}$ such that

$$|x_{i}(t)|\leq\beta_{i}(|x_{i}(t_{0})|,t-t_{0})+\gamma_{i}(\max_{j\in\mathcal{N}_{i}}\|x_{i}\|_{\infty})+\sigma_{i}(\|w_{i}\|_{\infty}).$$

Definition 4

[5] The networked system comprised of the subsystems (16) is said to be scalable mesh stable (SMS) if the ISS property of each subsystem implies the existence of some functions $\beta\in\mathcal{KL}$ and $\sigma\in\mathcal{K}_{\infty}$ such that

$$\max_{i\in\mathbb{N}_{N}}|x_{i}(t)|\leq\beta(\max_{i\in\mathbb{N}_{N}}|x_{i}(t_{0})|,t-t_{0})+\sigma(\max_{i\in\mathbb{N}_{N}}\|w_{i}\|_{\infty}),$$

for any initial condition $x_{i}(t_{0})$ and any disturbance function $w_{i},i\in\mathbb{N}_{N}$.

Proposition 4

[5] The networked system comprised of the subsystems (16) is SMS (see Def. 4) if each subsystem is ISS such that there exists $\tilde{\gamma}\in(0,1)$ that satisfies

$$\gamma_{i}(s)\leq\tilde{\gamma}s,\quad\forall s\geq 0,i\in\mathbb{N}_{N}.$$

The following proposition provides the additional conditions that should be included in the networked system design problem so as to ensure the SMS property. However, we require the following slightly stronger assumption than As. 1 regarding the nature of the subsystem dissipativity.

Assumption 2

Each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ (16) is $X_{i}$-dissipative (from input $u_{i}(t)$ to output $x_{i}(t)$) where $X_{i}=X_{i}^{\top}=[X_{i}^{kl}]_{k,l\in\mathbb{N}_{2}}$ is such that $X_{i}^{11}>0$, and the corresponding storage function is $V_{i}(x_{i})\triangleq x_{i}^{\top}P_{i}x_{i}$ where $P_{i}>0$.

Proposition 5

Under As. 2 and constraints (9), the networked system comprised of the subsystems (16) can be made $\mathcal{X}$-dissipative (from input $w(t)$ to output $z(t)$, where $\mathcal{X}=\mathcal{X}^{\top}=[\mathcal{X}^{kl}]_{k,l\in\mathbb{N}_{2}}$ with $\mathcal{X}^{22}<0$) as well as SMS by designing the interconnection matrix $M$ via the LMI problem

Find: $$L_{uy},\{p_{i}:i\in\mathbb{N}_{N}\}$$ (17)

Sub. to: $$p_{i}>0,\ \forall i\in\mathbb{N}_{N},\ \Phi>0,$$

$$\tilde{\lambda}_{i1}\sum_{j\in\mathbb{N}_{N}}|(X_{i}^{11})^{-1}L_{uy}^{ij}|<p_{i},$$

where $\Phi$ is as defined in (8), and $M_{uy}\triangleq(\textbf{X}_{p}^{11})^{-1}L_{uy}$.

Proof:

Compared to Co. 2, here we only have to establish the SMS property for the networked system. Note that we will omit explicitly denoting the time dependence of variables for notational convenience.

According to As. 2, as each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ is $X_{i}$-dissipative, we have

$$\dot{V}_{i}(x_{i}) \leq x_{i}^{\top}X_{i}^{22}x_{i}+x_{i}^{\top}X_{i}^{21}u_{i}+x_{i}^{\top}X_{i}^{21}u_{i}+u_{i}^{\top}X_{i}^{11}u_{i}$$

$$\leq x_{i}(X_{i}^{22}+X_{i}^{21})x_{i}+u_{i}^{\top}(X_{i}^{11}+X_{i}^{12})u_{i}$$

$$\leq-x_{i}Q_{i}x_{i}+u_{i}^{\top}R_{i}u_{i},$$

$$\leq-\underline{\lambda}(Q_{i})|x_{i}|^{2}+\overline{\lambda}(R_{i})|u_{i}|^{2}$$ (18)

where we select $Q_{i}$ and $R_{i}$ so that

$$(X_{i}^{22}+X_{i}^{21})\leq-Q_{i}<0,\mbox{ and }(X_{i}^{11}+X_{i}^{12})\leq R_{i}.$$ (19)

Moreover, as each subsystem $\Sigma_{i},i\in\mathbb{N}_{N}$ has a quadratic storage function of the form $V_{i}(x_{i})=x_{i}^{\top}P_{i}x_{i}$, we have

$$\underline{\lambda}(P_{i})|x_{i}|^{2}\leq V_{i}(x_{i})\leq\overline{\lambda}(P_{i})|x_{i}|^{2}.$$ (20)

Applying (20) in (18) leads to

$$\dot{V}_{i}(x_{i}) \leq-\frac{\underline{\lambda}(Q_{i})}{\overline{\lambda}(P_{i})}V_{i}(x_{i})+\overline{\lambda}(R_{i})|u_{i}|^{2}$$

$$=-\mu_{i}V_{i}(x_{i})+\theta_{i}$$ (21)

where we define $\mu_{i}\triangleq\frac{\underline{\lambda}(Q_{i})}{\overline{\lambda}(P_{i})}$ and $\theta_{i}\triangleq\overline{\lambda}(R_{i})|u_{i}|^{2}$. Note that (21) implies that

$$V_{i}(x_{i}) \leq(1-e^{-\mu_{i}t})\frac{\theta_{i}}{\mu_{i}}+V_{i}(x_{i}(0))e^{-\mu_{i}t}$$

$$\leq\frac{\theta_{i}}{\mu_{i}}+\overline{\lambda}(P_{i})|x_{i}(0)|^{2}e^{-\mu_{i}t}.$$ (22)

where we have further used the fact $(1-e^{-\mu_{i}t})\leq 1$ and (20). Using (20) and the definitions of $\theta_{i}$ and $\mu_{i}$, (22) implies

$$|x_{i}(t)|^{2}\leq\frac{\overline{\lambda}(R_{i})\overline{\lambda}(P_{i})}{\underline{\lambda}(P_{i})\underline{\lambda}(Q_{i})}|u_{i}|^{2}+\frac{\overline{\lambda}(P_{i})}{\underline{\lambda}(P_{i})}|x_{i}(0)|^{2}e^{-\mu_{i}t},$$

which, using the fact that $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$, leads to

$$|x_{i}(t)|\leq\sqrt{\frac{\overline{\lambda}(R_{i})\overline{\lambda}(P_{i})}{\underline{\lambda}(P_{i})\underline{\lambda}(Q_{i})}}|u_{i}|+\sqrt{\frac{\overline{\lambda}(P_{i})}{\underline{\lambda}(P_{i})}}|x_{i}(0)|e^{-\frac{\mu_{i}}{2}t},$$ (23)

We will next bound the term $|u_{i}|$. Using the relationship $u_{i}=\sum_{j\in\mathbb{N}_{N}}M_{uy}^{ij}x_{j}+w_{i}$, we get

$$|u_{i}| \leq\sum_{j\in\mathbb{N}_{N}}|M_{uy}^{ij}x_{j}|+|w_{i}|,$$

$$\leq\sum_{j\in\mathbb{N}_{N}}|M_{uy}^{ij}||x_{j}|+|w_{i}|$$

$$\leq\sum_{j\in\mathbb{N}_{N}}|M_{uy}^{ij}|\max_{j\in\mathbb{N}_{N}}\|x_{j}\|_{\infty}+\|w_{i}\|_{\infty}.$$

Applying this result in (23) gives

$$|x_{i}(t)|\leq \tilde{\lambda}_{i1}\sum_{j\in\mathbb{N}_{N}}|M_{uy}^{ij}|\max_{j\in\mathbb{N}_{N}}\|x_{j}\|_{\infty}+\tilde{\lambda}_{i1}\|w_{i}\|_{\infty}$$ (24)

$$+\tilde{\lambda}_{i2}|x_{i}(0)|e^{-\frac{\mu_{i}}{2}t},$$

where ~λ_i1 ≜λ(Ri)λ(Pi)λ(Pi)λ(Qi) and ~λ_i2 ≜λ(Pi)λ(Pi).

Based on Prop. 4 and Def. 3, the networked system to be SMS, we require the existence of functions $\beta_{i}\in\mathcal{KL}$ and $\sigma_{i}\in\mathcal{K}_{\infty}$ such that —x_i(t) —≤β_i(—x_i(t_0)—, t-t_0) + ~γ max_j∈N_N ∥x_i ∥_∞+ σ_i(∥w_i ∥_∞), for all $i\in\mathbb{N}_{N}$ where $\tilde{\gamma}$ is a scalar $\tilde{\gamma}\in(0,1)$. Comparing this with (24) we can obtain the condition for the SMS of the networked system as

$$\tilde{\lambda}_{i1}\sum_{j\in\mathbb{N}_{N}}|M_{uy}^{ij}|<1,\quad\forall i\in\mathbb{N}_{N}.$$

Finally, under the change of variables relationship, M_uy≜(X_p^11)^-1 L_uy ⇔M_uy^ij = (p_iX_i^11)^-1 L_uy^ij, ∀i∈N_N, the condition for the SMS of the networked system becomes

$$\tilde{\lambda}_{i1}\sum_{j\in\mathbb{N}_{N}}|(X_{i}^{11})^{-1}L_{uy}^{ij}|<p_{i},\quad\forall i\in\mathbb{N}_{N}.$$ (25)

∎

III Problem Formulation

III-A Spreading Network

Consider an epidemic spreading over a network $\Sigma$ consisting of $N\in\mathbb{N}$ groups denoted by $\{\Sigma_{i}:i\in\mathbb{N}_{N}\}$. Each group $\Sigma_{i},i\in\mathbb{N}_{N}$ can be represented by an open graph $\mathcal{G}_{i}\equiv(\mathcal{V}_{i},\mathcal{E}_{i})$, where vertices $\mathcal{V}_{i}\triangleq\{\Sigma_{i,k}:k\in\mathbb{N}_{N_{i}}\}$ represent the nodes, edges $\mathcal{E}_{i}\subseteq\mathcal{V}_{i}\times\mathcal{V}_{i}$ represent the epidemic spreading interactions between the nodes, and $N_{i}$ is the total number of nodes in the group $\mathcal{G}_{i}$.

For a group $\Sigma_{i},i\in\mathbb{N}_{N}$, an intra-group transmission matrix $M_{ii}\triangleq[M_{ii,kl}]_{k,l\in\mathbb{N}_{N_{i}}}\in\mathbb{R}^{N_{i}\times N_{i}}_{\geq 0}$ is used to contain all the transmission rate values corresponding to the node interactions within the group. In particular, having a positive transmission rate value $M_{ii,kl}>0$ implies the existence of a directed edge $(\Sigma_{i,l},\Sigma_{i,k})\in\mathcal{V}_{i}$, indicating that node $\Sigma_{i,l}$ can infect the node $\Sigma_{i,k}$, where $k,l\in\mathbb{N}_{N_{i}}$ (allowing self-infections, i.e., $k=l$) and $i\in\mathbb{N}_{N}$. Clearly, the intra-group transmission matrix $M_{ii}$ fully defines (implies) the epidemic spreading interactions $\mathcal{V}_{i}$ within the group $\Sigma_{i},i\in\mathbb{N}_{N}$.

Extending this notion of intra-group transmission matrices, we define inter-group transmission matrices to contain transmission rate values corresponding to node interactions across different groups. For example, the inter-group transmission matrix $M_{ij}\triangleq[M_{ij,kl}]_{k\in\mathbb{N}_{N_{i}},l\in\mathbb{N}_{N_{j}}}\in\mathbb{R}^{N_{i}\times N_{j}}_{\geq 0}$, represents the spreading interactions directed from the nodes $\{\Sigma_{j,l},l\in\mathbb{N}_{N_{j}}\}$ in the group $\Sigma_{j}$ towards the nodes $\{\Sigma_{i,k}:k\in\mathbb{N}_{N_{i}}\}$ in the group $\Sigma_{i}$ where $i,j\in\mathbb{N}_{N}$ and $i\neq j$. In particular, having a positive transmission rate value $M_{ij,kl}>0$ implies the existence of a global directed edge $(\Sigma_{j,l},\Sigma_{i,k})$, indicating that node $\Sigma_{j,l}$ can infect the node $\Sigma_{i,k}$, where $k\in\mathbb{N}_{N_{i}},l\in\mathbb{N}_{N_{j}}$ and $i,j\in\mathbb{N}_{N}$ with $i\neq j$.

The network Susceptible-Infected-Susceptible (SIS) model is used to model the dynamics of the infected proportion (also called the “state”) $x_{i,k}(t)\in[0,1]$ in each node $\Sigma_{i,k},k\in\mathbb{N}_{N_{i}},i\in\mathbb{N}_{N}$, as

$$\Sigma_{i,k}:\Big\{\dot{x}_{i,k}(t)=-\gamma_{i,k}(t)\,x_{i,k}(t)\,+\,(1-x_{i,k}(t))u_{i,k}(t),$$ (26)

where $\gamma_{i,k}(t)$ is the total recovery rate and $u_{i,k}(t)$ is the external infection effect (also called the “input”). In particular, $\gamma_{i,k}(t)\triangleq\bar{\gamma}_{i,k}+\tilde{\gamma}_{i,k}(t)$, where $\bar{\gamma}_{i,k}$ is the mean recovery rate and $\tilde{\gamma}_{i,k}(t)$ accounts for the uncertainty/modeling errors in the recovery rate so that $|\tilde{\gamma}_{i,k}(t)|\leq\delta_{i,k}$ (both $\bar{\gamma}_{i,k}$ and $\delta_{i,k}$ are assumed as known and fixed, and satisfies $\bar{\gamma}_{i,k}-\delta_{i,k}>0$). On the other hand, in (26), $u_{i,k}(t)$ is defined as

$$u_{i,k}(t)\triangleq\sum_{\begin{subarray}{c}j\in\mathbb{N}_{N}\end{subarray}}\sum_{l\in\mathbb{N}_{N_{j}}}M_{ij,kl}x_{j,l}(t)+w_{i,k}(t),$$ (27)

and $w_{i,k}(t)$ is a disturbance input that accounts for modeling errors related to transmission rates.

III-B Networked System Representation

The considered spreading network $\Sigma$ (26)-(27) can be viewed as a networked system, more precisely, as a hierarchy of networked systems, as shown in Fig. 2.

To see this, we first split the input $u_{i,k}(t),k\in\mathbb{N}_{N_{i}},i\in\mathbb{N}_{N}$ in (27) into local and global components, respectively denoted by $u_{i,k}^{L}(t)$ and $u_{i,k}^{G}(t)$, as

$$u_{i,k}(t)=u_{i,k}^{L}(t)+u_{i,k}^{G}(t),$$

where $u_{i,k}^{L}(t)$ is defined using the intra-group interactions as

$$u_{i,k}^{L}(t)\triangleq\sum_{l\in\mathbb{N}_{N_{i}}}M_{ii,kl}x_{i,l}(t),$$ (28)

and $u_{i,k}^{G}(t)$ is defined using the inter-group interactions as

$$u_{i,k}^{G}(t)\triangleq\sum_{\begin{subarray}{c}j\in\mathbb{N}_{N}\\ j\neq i\end{subarray}}\sum_{l\in\mathbb{N}_{N_{j}}}M_{ij,kl}x_{j,l}(t)+w_{i,k}(t).$$ (29)

We next vectorized (28) over $k,l\in\mathbb{N}_{N_{i}}$ to obtain

$$u_{i}^{L}(t)=M_{ii}x_{i}(t),$$ (30)

where $u_{i}^{L}(t)\triangleq[u_{i,k}^{L}(t)]_{k\in\mathbb{N}_{N_{i}}}^{\top}$ and $x_{i}(t)\triangleq[x_{i,k}]_{k\in\mathbb{N}_{N_{i}}}^{\top}$ are the vectorized local input and state of the group $\Sigma_{i},i\in\mathbb{N}_{N}$, respectively. Subsequently, the vectorized version of (29) over $k,l\in\mathbb{N}_{N_{i}}$ can be obtained as

$$u_{i}^{G}(t)=\sum_{\begin{subarray}{c}j\in\mathbb{N}_{N}\\ j\neq i\end{subarray}}M_{ij}x_{j}(t)+w_{i}(t),$$ (31)

where $u_{i}^{G}(t)\triangleq[u_{i,k}^{G}(t)]_{k\in\mathbb{N}_{N_{i}}}^{\top}$ and $w_{i}(t)\triangleq[w_{i,k}(t)]_{k\in\mathbb{N}_{N_{i}}}^{\top}$ are the vectorized global input and disturbance of the group $\Sigma_{i},i\in\mathbb{N}_{N}$, respectively. We also define

$$u_{i}(t)\triangleq u_{i}^{L}(t)+u_{i}^{G}(t)=M_{ii}x_{i}(t)+u_{i}^{G}(t)$$

as the vectorized total input of the group $\Sigma_{i},i\in\mathbb{N}_{N}$.

Each group $\Sigma_{i},i\in\mathbb{N}_{N}$ now can be seen as a networked system $\Sigma_{i}:u_{i}^{G}(t)\rightarrow x_{i}(t)$ (see Fig. 1), comprised of the nodes $\{\Sigma_{i,k}:k\in\mathbb{N}_{N_{i}}\}$ interconnected according to the interconnection matrix

$$\mathbf{M}_{i}=\begin{bmatrix}M_{ii}&\mathbf{I}\\ \mathbf{I}&\mathbf{0}\end{bmatrix},$$ (32)

where the external input is $u_{i}^{G}(t)$ and the output is $x_{i}(t)$.

Next, we continue vectorizing (31) over $i\in\mathbb{N}_{N}$, to obtain

$$u^{G}(t)=\bar{M}x(t)+w(t),$$

where $u^{G}(t)\triangleq[(u_{i}^{G}(t))^{\top}]_{i\in\mathbb{N}_{N_{i}}}^{\top}$, $x(t)\triangleq[x_{i}^{\top}(t)]_{k\in\mathbb{N}_{N_{i}}}^{\top}$ and $w(t)\triangleq[w_{i}^{\top}(t)]_{k\in\mathbb{N}_{N_{i}}}^{\top}$ are the vectorized global input, state and disturbance, respectively, and $\bar{M}\triangleq[\bar{M}_{ij}]_{i,j\in\mathbb{N}_{N}}$ with $\bar{M}_{ij}\triangleq M_{ij}\mathbf{1}_{\{i\neq j\}}$ is the transmission matrix representing all inter-group interactions over the spreading network.

Finally, we can view the spreading network as a networked system $\Sigma:w(t)\rightarrow x(t)$ (see Fig. 1), comprised of the groups $\{\Sigma_{i}:i\in\mathbb{N}_{N}\}$ interconnected according to the interconnection matrix

$$\textbf{M}=\begin{bmatrix}\bar{M}&\mathbf{I}\\ \mathbf{I}&\mathbf{0}\end{bmatrix},$$ (33)

where the external input is $w(t)$ that represents disturbances due to unmodeled dynamics, and the output is $x(t)$ that acts as a performance input for the spreading network.

Figure 2: Spreading Network as a Networked System

Problem

Our goal is to determine a set of optimal alterations for the inter-group interconnections (i.e., $\bar{M}$) to ensure the stability and robustness of the infections-free state (i.e., $x=\mathbf{0}$) for the spreading network. In this pursuit, we also provide recommendations for improving inter-group interactions (i.e., $M_{ii},\forall i\in\mathbb{N}_{N}$), if it is imperative to achieving stability and robustness of the infections-free state for the spreading network.

IV Dissipativity-Based Solution

IV-A Dissipativity Analysis and Design

We begin by establishing the dissipativity properties of the nodes.

Lemma 1

Each node $\Sigma_{i,k},\,k\in\mathbb{N}_{N_{i}},\,i\in\mathbb{N}_{N}$ (26) in the spreading network is $X_{i,k}$-dissipative (from $u_{i,k}(t)$ to $x_{i,k}(t)$) where $X_{i,k}\triangleq\scriptsize\begin{bmatrix}a&b\\ b&c\end{bmatrix}$ is given by LMI problem:

$$\mbox{Find: }\ p,\ a,\ b,\ c,\$$ (34)

$$\mbox{Sub. to: }\ p>0,$$

$\displaystyle$

Proof:

The proof follows directly from applying Prop. 1 for the node dynamics (26) as it is identical to (2). ∎

Next, we use the established node dissipativity properties to identify the group dissipativity properties.

Lemma 2

Each group $\Sigma_{i},\,i\in\mathbb{N}_{N}$ (see Fig. 2) in the spreading network is $\mathcal{X}_{i}$-dissipative (from $u_{i}^{G}(t)$ to $x_{i}(t)$) where $\mathcal{X}_{i}\triangleq[\mathcal{X}_{i}^{lm}]_{l,m\in\mathbb{N}_{2}}$ is given by the LMI problem:

Find: $$\mathcal{X}_{i},\{p_{i,k}:k\in\mathbb{N}_{N_{i}}\}$$ (35)

Sub. to: $$p_{i,k}>0,\ \forall k\in\mathbb{N}_{N_{i}},\ \Phi_{i}>0,$$

where $\Phi_{i}$ is as defined in (42), and $\textbf{X}_{p_{i}}^{lm}$ for any $l,m\in\mathbb{N}_{2}$ is defined using the known node dissipativity properties $\{X_{i,k}^{lm}:k\in\mathbb{N}_{N_{i}}\}$ as $\textbf{X}_{p_{i}}^{lm}\triangleq diag([p_{i,k}X_{i,k}^{lm}]_{k\in\mathbb{N}_{N_{i}}})$.

Proof:

Using the networked system representation (32) of each group $\Sigma_{i},\,i\in\mathbb{N}_{N}$, for its $\mathcal{X}_{i}$-dissipativity analysis, we can apply Co. 2. To this end, we first verify that As. 1 holds for each group $\Sigma_{i},\,i\in\mathbb{N}_{N}$ as dissipativity properties $X_{i,k}$ of each node $\Sigma_{i,k},\,k\in\mathbb{N}_{N_{i}}$ identified in Lm. 1 satisfies $X_{i,k}^{11}>0$. Next, we introduce an extra constraint for the design variable $L_{uy}$ in Co. 2 as $L_{uy}=\textbf{X}_{p}^{11}M_{uy}$ as Co. 2 is originally intended for design problems (not for analysis problems). Consequently, we can obtain (35). ∎

Next, we use the established group dissipativity properties to optimize the inter-group interconnections to ensure (or optimize) the dissipativity of the spreading network.

Lemma 3

The spreading network $\Sigma$ shown in Fig. 2 can be made $\mathfrak{X}$-dissipative (from input $w(t)$ to output $x(t)$) where $\mathfrak{X}\triangleq[\mathfrak{X}^{lm}]_{l,m\in\mathbb{N}_{2}}$, by designing the interconnection matrix $\bar{M}$ via the LMI problem

Find: $$L,\{p_{i}:i\in\mathbb{N}_{N}\}$$ (36)

Sub. to: $$p_{i}>0,\ \forall i\in\mathbb{N}_{N},\ \Phi>0$$

where $\Phi$ is as defined in (43), $\mathbfcal{X}_{p}^{lm}$ for any $l,m\in\mathbb{N}_{2}$ is defined using the known group dissipativity properties $\{\mathcal{X}_{i}^{lm}:i\in\mathbb{N}_{N}\}$ as $\mathbfcal{X}_{p}^{lm}\triangleq diag([p_{i}\mathcal{X}_{i}^{lm}]_{i\in\mathbb{N}_{N_{i}}})$, $\mathbfcal{X}^{12}\triangleq diag([(\mathcal{X}_{i}^{11})^{-1}\mathcal{X}_{i}^{12}]_{i\in\mathbb{N}_{N}})$, $\mathbfcal{X}^{21}\triangleq(\mathbfcal{X}^{12})^{\top}$, $L$ shares the structure of $\bar{M}$, and $\bar{M}\triangleq(\mathbfcal{X}_{p}^{11})^{-1}L$.

Proof:

Based on the networked system representation (33) of the spreading network $\Sigma$, to synthesize its interconnection matrix $\bar{M}$ to ensure/optimize its $\mathfrak{X}$-dissipativity, we can apply Co. 2. In this pursuit, we first verify that As. 1 holds for $\Sigma$ as dissipativity properties $\mathcal{X}_{i}$ of each group $\Sigma_{i},i\in\mathbb{N}_{N}$ identified in Lm. 2 satisfies $\mathcal{X}_{i}^{11}>0$. Consequently, starting from (10), we can obtain (36). ∎

When designing the interconnection matrix $\bar{M}$, to optimize the deviations from a given nominal interconnection matrix $\bar{M}_{0}$ (i.e., to optimize $\|\bar{M}-\bar{M}_{0}\|$), in the LMI problem (36), one can use an objective function of the form

$$J=\|L-\mathbfcal{X}_{p}^{11}\bar{M}_{0}\|+\alpha\|\mathbfcal{X}_{p}^{11}\|+\beta\|\mathfrak{X}\|$$ (37)

with some scaling coefficients $\alpha,\beta\geq 0$. While the first two terms aim to minimize the deviations from the nominal interconnection matrix, the last term ensures the rate of the stored energy is minimized.

On the other hand, when the designed interconnection matrix $\bar{M}$ needs to be constrained with respect to a given nominal interconnection matrix $\bar{M}_{0}$ so that $\|\bar{M}-\bar{M}_{0}\|\leq\delta\|\bar{M}\|,\delta\in(0,1),$ one can include an additional LMI constraint in the LMI problem (36) as

$$0\leq\begin{bmatrix}\mathbf{I}&Y^{\top}&\mathbf{0}\\ Y&\mathbfcal{X}_{p}^{11}Y^{\top}+Y\mathbfcal{X}_{p}^{11}&L-\mathbfcal{X}_{p}^{11}\bar{M}_{0}\\ \mathbf{0}&L^{\top}-\bar{M}_{0}^{\top}\mathbfcal{X}_{p}^{11}&\delta^{2}\bar{M}_{0}^{\top}\bar{M}_{0}\end{bmatrix},$$ (38)

with some $Y>0$ (e.g., $Y=\mathbf{I}$, or ideally, $Y\simeq\mathbfcal{X}_{p}^{11}$). Note also that, in the above proposed LMI constraint, $\delta$ can be treated as a design variable to be minimized.

If the matrix $\mathbfcal{X}_{p}^{11}$ takes a diagonal form (i.e., if each $\mathcal{X}_{i}^{11}$ is diagonal), an element-wise version of the above constraint $\|\bar{M}-\bar{M}_{0}\|\leq\delta\|\bar{M}\|,\delta\in(0,1),$ can be enforced by including the LMI constraint

$$-\delta\mathbfcal{X}_{p}^{11}\bar{M}_{0}\ll L-\mathbfcal{X}_{p}^{11}\bar{M}_{0}\ll\delta\mathbfcal{X}_{p}^{11}\bar{M}_{0},$$ (39)

in the LMI problem (36).

IV-B Supporting Feasibility with Necessary Conditions

The following lemma presents a necessary condition to be included in each node dissipativity analysis to support the feasibility of the corresponding group dissipativity analysis.

Lemma 4

At each group $\Sigma_{i},i\in\mathbb{N}_{N}$, for the feasibility of its $\mathcal{X}_{i}$-dissipativity analysis problem (in Lm. 2), a necessary condition that can be enforced at each node $\Sigma_{i,k},k\in\mathbb{N}_{N_{i}}$ in its $X_{i,k}$-dissipativity analysis problem (in Lm. 1, $X_{i,k}\triangleq\scriptsize\begin{bmatrix}a&b\\ b&c\end{bmatrix}\normalsize$) is the feasibility of the LMI problem:

Find: $$\ a,\,b,\,c,\,\bar{a},\,\bar{b},\,\bar{c},$$ (40)

Sub. to: $$\ \tilde{\Phi}_{i,k}>0,$$

$$\tilde{\Phi}_{i,k}\triangleq\begin{bmatrix}a&0&a\,m&a\\ 0&-\bar{c}&-\bar{c}&0\\ a\,m&-\bar{c}&-2b\,m-c&-b+\bar{b}\\ a&0&-b+\bar{b}&\bar{a}\end{bmatrix},$$

where $\scriptsize\begin{bmatrix}\bar{a}&\bar{b}\\ \bar{b}&\bar{c}\end{bmatrix}$ shadows $[\mathcal{X}_{i,k}^{lm}]_{l,m\in\mathbb{N}_{2}}$ (recall that each $\mathcal{X}_{i,k}^{lm}$ is the $k$^th diagonal element of $\mathcal{X}^{lm}_{i}$ in (35)), and $m\triangleq M_{ii,kk}$ is the $k$-th diagonal element of $M_{ii}$.

Proof:

The proof follows directly from applying Prop. 3 for the networked system representation (32) of each group $\Sigma_{i},\,i\in\mathbb{N}_{N}$, for its $\mathcal{X}_{i}$-dissipativity analysis. ∎

The following lemma presents a necessary condition to be included in each group dissipativity analysis to support the feasibility of the spreading network design.

Lemma 5

For the feasibility of $\mathfrak{X}$-dissipative spreading network design problem (in Lm. 3), a necessary condition that can be enforced at each group $\Sigma_{i},i\in\mathbb{N}_{N}$ in its $\mathcal{X}_{i}$-dissipativity analysis problem (in Lm. 2) is the feasibility of the LMI problem:

Find: $$\ \mathcal{X}_{i},\bar{\mathfrak{X}}_{i}$$ (41)

Sub. to: $$\ \tilde{\Phi}_{i}>0,$$

$$\tilde{\Phi}_{i}\triangleq\begin{bmatrix}\mathcal{X}_{i}^{11}&\mathbf{0}&\mathbf{0}&\mathcal{X}_{i}^{11}\\ \mathbf{0}&-\bar{\mathfrak{X}}^{22}_{i}&-\bar{\mathfrak{X}}^{22}_{i}&\mathbf{0}\\ \mathbf{0}&-\bar{\mathfrak{X}}^{22}_{i}&-\mathcal{X}_{i}^{22}&-\mathcal{X}_{i}^{21}+\bar{\mathfrak{X}}^{21}_{i}\\ \mathcal{X}_{i}^{11}&\mathbf{0}&-\mathcal{X}_{i}^{12}+\bar{\mathfrak{X}}^{12}_{i}&\bar{\mathfrak{X}}^{11}_{i}\end{bmatrix},$$

where $\bar{\mathfrak{X}}_{i}\triangleq[\bar{\mathfrak{X}}_{i}^{lm}]_{l,m\in\mathbb{N}_{2}}$ shadows $[\mathfrak{X}_{i}^{lm}]_{l,m\in\mathbb{N}_{2}}$ (recall that each $\mathfrak{X}_{i}^{lm}$ is the $i$^th diagonal block of $\mathfrak{X}^{lm}$ in (35)).

Proof:

The proof follows directly from applying Prop. 3 for the networked system representation (32) of the spreading network, for its $\mathfrak{X}$-dissipative spreading network design problem. In this pursuit, the fact that $\bar{M}_{ii}=\mathbf{0},i\in\mathbb{N}_{N}$ has been used to simplify the derived necessary condition. ∎

$$\Phi_{i}\triangleq\begin{bmatrix}\textbf{X}_{p_{i}}^{11}&\textbf{0}&\textbf{X}_{p_{i}}^{11}M_{ii}&\textbf{X}_{p_{i}}^{11}\\ \textbf{0}&-\mathcal{X}^{22}_{i}&-\mathcal{X}^{22}_{i}&\mathbf{0}\\ M_{ii}^{\top}\textbf{X}_{p_{i}}^{11}&-\mathcal{X}^{22}_{i}&-M_{ii}^{\top}\textbf{X}_{p_{i}}^{12}-\textbf{X}_{p_{i}}^{21}M_{ii}-\textbf{X}_{p_{i}}^{22}&-\textbf{X}_{p_{i}}^{21}+\mathcal{X}^{21}_{i}\\ \textbf{X}_{p_{i}}^{11}&\mathbf{0}&-\textbf{X}_{p_{i}}^{12}+\mathcal{X}^{12}_{i}&\mathcal{X}^{11}_{i}\end{bmatrix}$$ (42)

$$\Phi\triangleq\begin{bmatrix}\mathbfcal{X}_{p}^{11}&\textbf{0}&L&\mathbfcal{X}_{p}^{11}\\ \textbf{0}&-\mathfrak{X}^{22}&-\mathfrak{X}^{22}&\mathbf{0}\\ L^{\top}&-\mathfrak{X}^{22}&-L^{\top}\mathbfcal{X}^{12}-\mathbfcal{X}^{21}L-\mathbfcal{X}_{p}^{22}&-\mathbfcal{X}_{p}^{21}+\mathfrak{X}^{21}\\ \mathbfcal{X}_{p}^{11}&\mathbf{0}&-\mathbfcal{X}_{p}^{12}+\mathfrak{X}^{12}&\mathfrak{X}^{11}\end{bmatrix}$$ (43)

IV-C Overall Process

Combining the above results, we provide the following unified theorem to outline the overall task of designing a dissipative spreading network.

Theorem 1

Given the node recovery rate information, intra-group interconnections and nominal inter-group interconnections, we can enforce the spreading network $\Sigma$ to be be $\mathfrak{X}$-dissipative by designing the interconnection matrix $bar{M}$ following the process outlined in Alg. 1.

Proof:

The proof is complete by noticing that each problem $\mathbb{P}_{i,k}$ is a joint version of LMI problems (34) and (40), each problem $\mathbb{P}_{i}$ is a joint version of LMI problems (35) and (41), and the problem $\mathbb{P}$ is a combination pf the LMI problem (36) with consitraints (38), (39) and the objective (37). ∎

Algorithm 1 Spreading Network Design Process

1:Input: $\{(\bar{\gamma}_{i,k},\delta_{i,k}):k\in\mathbb{N}_{N_{i}},i\in\mathbb{N}_{N}\}$, $\{M_{ii}:i\in\mathbb{N}_{N}\}$, $\bar{M}_{0}$

2:for $i=1,2,\ldots,N$ do $\triangleright$ Analyzing Groups

3:  for $k=1,2,\ldots,N_{i}$ do $\triangleright$ Analyzing Nodes

4:   Find $X_{i,k}$ (and $\bar{\mathcal{X}}_{i,k}$) via solving $\mathbb{P}_{i,k}$ (44).

5:  end for

6:  Find $\mathcal{X}_{i}$ (and $\bar{\mathfrak{X}}_{i,k}$) via solving the problem $\mathbb{P}_{i}$ (45).

7:end for

8:Find $\bar{M}$, $\mathfrak{X}$ via solving the problem $\mathbb{P}$ (46).

Problem $\mathbb{P}_{i,k}$

Find: $$\ p,\ a,\,b,\,c,\,\bar{a},\,\bar{b},\,\bar{c},$$ (44)

Sub. to: $$\ p>0,\,\tilde{\Phi}_{i,k}>0,$$

$$\begin{cases}a\geq 0,\ (2b-p)\geq 0,\ (c+p(\bar{\gamma}_{i,k}-\delta_{i,k}))\geq 0,\ \mbox{or}\\ a>0,\ b\leq 0,\ \begin{bmatrix}a&b\\ b&c+p(\bar{\gamma}_{i,k}-\delta_{i,k})\end{bmatrix}\geq 0.\end{cases}$$

$$\tilde{\Phi}_{i,k}\triangleq\begin{bmatrix}a&0&a\,m&a\\ 0&-\bar{c}&-\bar{c}&0\\ a\,m&-\bar{c}&-2b\,m-c&-b+\bar{b}\\ a&0&-b+\bar{b}&\bar{a}\end{bmatrix},$$

Problem $\mathbb{P}_{i}$

Find: $$\ \mathcal{X}_{i},\ \bar{\mathfrak{X}}_{i},\ \{p_{i,k}:k\in\mathbb{N}_{N_{i}}\}$$ (45)

Sub. to: $$\ p_{i,k}>0,\ \forall k\in\mathbb{N}_{N_{i}},\ \Phi_{i}>0,\ \tilde{\Phi}_{i}>0,$$

$$\tilde{\Phi}_{i}\triangleq\begin{bmatrix}\mathcal{X}_{i}^{11}&\mathbf{0}&\mathbf{0}&\mathcal{X}_{i}^{11}\\ \mathbf{0}&-\bar{\mathfrak{X}}^{22}_{i}&-\bar{\mathfrak{X}}^{22}_{i}&\mathbf{0}\\ \mathbf{0}&-\bar{\mathfrak{X}}^{22}_{i}&-\mathcal{X}_{i}^{22}&-\mathcal{X}_{i}^{21}+\bar{\mathfrak{X}}^{21}_{i}\\ \mathcal{X}_{i}^{11}&\mathbf{0}&-\mathcal{X}_{i}^{12}+\bar{\mathfrak{X}}^{12}_{i}&\bar{\mathfrak{X}}^{11}_{i}\end{bmatrix},$$

where $\Phi_{i}$ is as defined in (42).

Problem $\mathbb{P}$

$$\min_{\begin{subarray}{c}L,\mathfrak{X},\\ \{p_{i}:i\in\mathbb{N}_{N}\}\end{subarray}} \ J=\|L-\mathbfcal{X}_{p}^{11}\bar{M}_{0}\|+\alpha\|\mathbfcal{X}_{p}^{11}\|+\beta\|\mathfrak{X}\|,$$ (46)

Sub. to: $$p_{i}>0,\ \forall i\in\mathbb{N}_{N},\ \Phi>0$$

$$\ \begin{bmatrix}\mathbf{I}&Y^{\top}&\mathbf{0}\\ Y&\mathbfcal{X}_{p}^{11}Y^{\top}+Y\mathbfcal{X}_{p}^{11}&L-\mathbfcal{X}_{p}^{11}\bar{M}_{0}\\ \mathbf{0}&L^{\top}-\bar{M}_{0}^{\top}\mathbfcal{X}_{p}^{11}&\delta^{2}\bar{M}_{0}^{\top}\bar{M}_{0}\end{bmatrix}\geq 0,$$

$$\ -\delta\mathbfcal{X}_{p}^{11}\bar{M}_{0}\ll L-\mathbfcal{X}_{p}^{11}\bar{M}_{0}\ll\delta\mathbfcal{X}_{p}^{11}\bar{M}_{0},$$

where $\Phi$ is as defined in (43).

Remark 4

While we have only provided the theoretical result for the design of a dissipative spreading network, the same result can be used for dissipativity analysis of the spreading network given the inter-group interconnections. The approach is analogous to how we analyzed dissipativity of individual groups using the dissipativity-based topology design result in Pr. 2 at the group level given the intra-group interconnections.

Remark 5

The proposed framework’s ability to conduct both dissipativity analysis and dissipative design is beneficial as we can use it to reconfigure intra-group interconnections if a group is not sufficiently dissipative compromising the performance of the spreading network. Moreover, the same capability will play a major role when there are higher layers beyond the network level (e.g., a network of networks layer) in the considered spreading network.

V Simulation Results

To provide operational details and showcase the benefits of the proposed dissipativity-based spreading network design technique, we implement it along with several of its variants and two standard techniques for a randomly generated spreading network. In particular, we report the variation of the average infection level observed under different design methods over an extended period where different disturbances were injected into the spreading network. We also report the percentage reductions in the inter-group interconnections exercised by different design methods to measure the exerted spreading network design effort.

V-A Considered Spreading Network

As shown in Fig. 3(a)-3(b), the considered spreading network consists of four groups $G_{1},G_{2},G_{3},G_{4}$ with $5,6,7,4$ nodes, respectively. The mean recovery rates (i.e., $\bar{\gamma}_{i,k}$) for the nodes were selected from the uniform random distribution in the interval $[0.4,0.9]$. The recovery rate deviation bound (i.e., $\delta_{i,k}$) was set as $5\%$ of the selected recovery rate at each node. The initial infection level for each node was selected from the uniform random distribution in the interval $[0,1]$.

The intra-group interconnections were created randomly with a probability of $0.3$ for any node pair in a group, and the interconnection levels were selected from the uniform random distribution in the interval $[0.1,0.4]$. While we made no assumption regarding the group connectivity (like strong connectivity), we scaled down the intra-group interconnections to ensure group stability of the infections-free state when isolated from other groups. This scaling was done only to create an ideal reference behavior to compare with other spreading network design methods. On the other hand, the inter-group interconnections were created randomly with a probability of $0.2$ for any node pair in the spreading network, and the interconnection levels were selected from the uniform random distribution in the internal $[0.1,0.4]$ without any scaling.

The disturbance input (i.e., $w_{i,k}(t)$) affecting each node was chosen to include four distinct components, each active during specific time windows and triggered asynchronously at different nodes: a brief, low-amplitude sinusoidal fluctuation during $t\in[40,45]$, a step offset during $t\in[80,85]$, a toggling component during $t\in[120,130]$, and mild random noise added during $t\in[0,160]$. These short-lived perturbations combine to create a rich disturbance profile to evaluate the different design approaches.

We simulated this spreading network under different inter-group interconnection configurations given by different spreading network design methods, including the two cases where we used none of and all of the original inter-group interconnections. The observed variations of the average infection level (fraction) of the spreading network over a period $t\in[0,200]$ are shown in Fig. 4. Notice that when no inter-group interconnections are used, by design, the spreading network stabilizes to the infections-free state very rapidly overcoming all disturbance conditions. However, when there are inter-group interconnections, the spreading network does not stabilize to the infections-free state. Hence these two average infections profiles provide the best and worst case scenarios for the spreading process to compare other spreading network design methods.

V-B Dissipativity-Based Design

We implemented the proposed dissipativity-based spreading network design method as follows. First, each node $\Sigma_{i,k},k\in\mathbb{N}_{N_{i}},i\in\mathbb{N}_{N}$ was considered to be IF-OFP($\nu,\rho$) (see Rm. 1), and hence, the problem $\mathbb{P}_{i,k}$ was solved by setting $b=0.5$ and optimizing for decision variables $a$ and $c$, representative of the passivity indices $-\nu$ and $-\rho$, respectively. As we plan to establish similar IF-OFP properties at the group level, in problem $\mathbb{P}_{i,k}$, we also set $\bar{b}=0.5$ and included $\bar{a}$ and $\bar{c}$ as decision variables. Overall, when solving the problem $\mathbb{P}_{i,k}$, we optimize the objective function $J_{i,k}\triangleq a+c+\bar{a}+\bar{c}$ to identify the maximum feasible node passivity indices while helping to establish similarly high passivity indices at the group level.

Next, each group $\Sigma_{i},i\in\mathbb{N}_{N}$ was considered to have a vectored IF-OFP property, and hence the problem $\mathbb{P}_{i}$ was solved by setting $\mathcal{X}_{i}^{12}=0.5\mathbf{I}$ and optimizing for diagonal matrices (decision variables) $\mathcal{X}_{i}^{11}$ and $\mathcal{X}_{i}^{22}$. As we plan to establish the network L2G($\gamma$) (see Rm. 1), in problem $\mathbb{P}_{i}$, we also set $\bar{\mathfrak{X}}_{i}^{12}=\mathbf{0}$, $\bar{\mathfrak{X}}_{i}^{22}=-\mathbf{I}$ and $\bar{\mathfrak{X}}_{i}^{11}=\bar{\gamma}_{i}\mathbf{I}$ and include $\bar{\gamma}_{i}$ as a decision variable. Overall, when solving the problem $\mathbb{P}_{i}$, we optimize the objective function $J_{i}\triangleq\mbox{trace}(\mathcal{X}_{i}^{11})+\mbox{trace}(\mathcal{X}_{i}^{22})+\bar{\gamma}_{i}$ to identify maximum feasible group passivity indices while helping to achieve lowest possible $L_{2}$ gain values for the spreading network.

Finally, we set out to design the inter-group interconnections $\bar{M}$ in the spreading network $\Sigma$ to make it L2G($\gamma$), i.e., finite-gain $L_{2}$ stable with a gain $\gamma$. For this, the problem $\mathbb{P}$ was solved by setting $\mathfrak{X}^{12}=\mathbf{0}$, $\mathfrak{X}^{22}=-\mathbf{I}$ and $\mathfrak{X}^{11}=\bar{\gamma}\mathbf{I}$, and optimizing for $\bar{M}$ and $\bar{\gamma}$ as decision variables. In particular, when solving problem $\mathbb{P}$, we optimized the objective function:

$$J=c_{M}\|L-\mathbfcal{X}_{p}^{11}\bar{M}_{0}\|_{1}+\bar{\gamma}$$

subject to a limit $\delta_{M}$ on the fractional deviation allowed for the magnitude of any inter-group interconnection level, i.e.,

$$(1-\delta_{M})\bar{M}_{0}<<\bar{M}\iff(1-\delta_{M})\mathbfcal{X}_{p}^{11}\bar{M}_{0}<<L.$$

In the baseline dissipativity-based design approach, we used the problem parameters $c_{M}=1$ and $\delta_{M}=1$. For comparison purposes, we also evaluate the dissipativity-based design for $c_{M}\in\{10^{-9},10^{9}\}$ (effectively, $c_{M}\in\{0,\infty\}$) and $\delta_{M}\in\{0.9,0.95\}$. Henceforth, these dissipativity-based methods are denoted as DissBC($c_{M},\delta_{M}$), e.g., the baseline dissipativity-based method is denoted as DisBC($1,1$).

V-C Threshold and Degree Based Methods

To compare the performance of the dissipativity-based spreading network designs, we also examine two basic, practical spreading network design methods (see also [6]). The first approach prunes inter-group interconnections with weights beyond a given threshold $t_{M}$. By removing the strongest inter-group links, this method reduces the most impactful pathways for disease propagation. Henceforth, this threshold-based approach is denoted as TBC($t_{M}$). The second approach identifies the top $d_{M}$ fraction of the nodes in terms of their inter-group out-degree and removes all their outgoing connections to nodes in other groups. By isolating these highly connected nodes, this method limits the spread of epidemics through critical inter-group links. Henceforth, this degree-based approach is denoted as DegBC($t_{M}$).

To make the comparison of different spreading network design methods fair, we first identified a “design effort” metric:

$$J_{M}\triangleq\sum_{\begin{subarray}{c}\forall i,j\in\mathbb{N}_{N},i\neq j,\\ \forall k\in\mathbb{N}_{N_{i}},l\in\mathbb{N}_{N_{j}}\end{subarray}}\frac{\bar{M}_{0}^{ij,kl}-\bar{M}^{ij,kl}}{\bar{M}_{0}^{ij,kl}}$$

representing the average fractional cutdown of the interconnections, observed under each method. The baseline DissBC($1,1$) method observed an average fractional cutdown of $J_{c}=87.1\%$, and hence the parameters $t_{M}$ and $d_{M}$ respectively of TBC($t_{M}$) and DegBC($d_{M}$) methods were tuned to achieve the similar $J_{M}$ level. The resulting parameters were $t_{M}=0.18$ and $d_{M}=0.72$.

V-D Results

Based on the reported average infection levels in Fig. 4, the baseline DissBC($1,1$) method provided the closest behavior to the best case (i.e., the case without interconnections). The TBC($0.18$) and DegBC($0.72$) methods while achieved stability, were considerably slow to respond to various disturbances. This effectiveness of the proposed DissBC($1,1$) method can be attributed to the systematic dynamics-aware LMI-based design process, which optimally tailors the inter-group interconnections while optimizing the a robust stability measure.

As stated in Rm. 4, the proposed dissipativity-based spreading networks design approach can also be used to analyze the dissipativity of a given spreading network. Consequently, it can be used to analyze the $L_{2}$ gain level $\gamma$ of any given spreading network. We used this $\gamma$ level as a performance metric besides the time-averaged infection level (fraction)

$$J_{x}\triangleq\frac{1}{T}\int_{0}^{T}x(t)dt,$$

and the average interconnection reductions fraction $J_{M}$ as measures of performance when comparing different spreading network design methods. The observed performance metrics are summarized in Tab. I.

Based on Tab. I, it is clear that the dissipativity-based spreading network design have superior (i.e., lower) $L_{2}$ gain $\gamma$ levels compared to other methods. In particular, DissBC($10^{-}9,1$) (i.e., when $c_{M}=10^{-9}$ and thus, when the design objective is effectively $\gamma$) has obtained the lowest $\gamma$ value, at $\gamma=24.10$. Interestingly, as can be seen in Figs. 5(a) and 5(c), the average reduction of interconnections in this DissBC($10^{-9},1$) case is also the lowest, at $J_{M}=0.8168$. This implies that it is not necessary to reduce interconnections significantly to obtain guaranteed theoretical robust stability measures.

Based on both Fig. 4 and Tab. I, dissipativity based spreading network design methods also have superior experimental average infections level performance (i.e., lower $J_{x}$ metrics). In particular, DissBC($10^{9},1$) (i.e., when $c_{M}=10^{9}$ and thus when the design objective is effectively the interconnections) has obtained the lowered $J_{x}$ value, at $J_{x}=0.0638$. This is because, due to its focus on optimizing only the interconnections, the solution has drastically cutdown interconnections. This behavior is evident from Figs. 5(b) and 5(d) and also from the fact that this method achieves the highest $J_{M}$ values, at $J_{M}=0.9575$.

As shown in Fig. 6, in dissipativity based designs, by reducing the $\delta_{M}$ parameter from $1$ to $0.95$ and $0.9$ (enforcing stricter constraints on interconnection deviations), DissBC($1,0.95$) and Diss($1,0.9$) methods have sacrificed both experimental performance $J_{x}$ and theoretical performance $\gamma$.

Considering these behaviors, it is clear that the selected baseline dissipativity-based design approach DissBC($1,1$) considers a moderate case where both the robust stability measure $\gamma$ and interconnection cost are treated equally, without forced constraints of interconnection deviations. As can be seen in Fig. 3(c) and 3(d), this leads to a spreading network with a close to the best: (i) robust stability guarantee with $\gamma=24.15$, (ii) experimental performance $J_{x}=0.0380$, and (iii) interconnection reductions $J_{M}=0.8710$ highlighting the benefit of co-designing networks while using a well-balanced multiple objective function.

Overall, the results indicate that the proposed dissipativity-based approach provides a structured and efficient means of spreading networks. Its robustness against asynchronous disturbances while using specifically designed minimal alterations to the nominal network further highlights its suitability for practical applications where system dynamics are complex and uncertain.

(a)

(b)

(c)

(d)

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).

(a)

(b)

(c)

(d)

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 $\delta_{M}=1$.

(a)

(b)

(c)

(d)

Figure 6: Spreading network designs (top row) and their percentage interconnection reductions histograms (bottom row) observed under the proposed dissipativity-based design approach with parameters $c_{M}=1$ with $\delta_{M}=0.95$ (left column) and $\delta_{M}=0.9$ (right column).

(a)

(b)

Figure 7: Spreading network designs observed under: (a) threshold-based approach with the threshold parameter $t_{M}=0.18$ and (b) degree-based approach with the degree parameter $d_{M}=0.72$

TABLE I: Observed performance metrics: $J_{x}$ (average infections), $J_{M}$ (average reductions), and $\gamma$ ($L_{2}$-gain) under different spreading network design techniques.

Design Method	$J_{x}$	$J_{M}$	$\gamma$
Without interconnections	0.0616	1.0000	-
With Interconnections	0.3603	0.0000	-
DissBC($1,1$)	0.0680	0.8710	24.15
DissBC($1,0.95$)	0.0645	0.9497	26.16
DissBC($1,0.9$)	0.0677	0.9000	35.08
DissBC($10^{-9},1$)	0.0709	0.8168	24.10
DissBC($10^{9},1$)	0.0638	0.9575	30.70
TBC($0.18$)	0.0700	0.8726	-
DegBC($0.72$)	0.0730	0.8854	249.6

VI Conclusion

In this paper, we presented a dissipativity-based framework for controlling epidemic spreading over networked systems. By formulating the control problem using dissipativity theory, we established a systematic method for designing inter-group interconnections that ensure stability and robustness against disturbances. Comparative analysis with traditional control techniques, including threshold pruning and high-degree edge removal, demonstrates the superior performance of the proposed approach in reducing infection levels with lower control effort. Future work includes extending the framework to account for adaptive or time-varying networks and exploring its application to other spreading processes beyond epidemic control.