Networks are landmarks of many complex phenomena where interweaving interactions between different agents transform simple local rule-sets into nonlinear emergent behaviors. While some recent studies unveil associations between the network structure and the underlying dynamical process, identifying stochastic nonlinear dynamical processes continues to be an outstanding problem. Here we develop a simple data-driven framework based on operator-theoretic techniques to identify and control stochastic nonlinear dynamics taking place over large-scale networks. The proposed approach requires no prior knowledge of the network structure and identifies the underlying dynamics solely using a collection of two-step snapshots of the states. This data-driven system identification is achieved by using the Koopman operator to find a low dimensional representation of the dynamical patterns that evolve linearly. Further, we use the global linear Koopman model to solve critical control problems by applying to model predictive control (MPC)--typically, a challenging proposition when applied to large networks. We show that our proposed approach tackles this by converting the original nonlinear programming into a more tractable optimization problem that is both convex and with far fewer variables.
We study the class of SIS epidemics on temporal networks and propose a new activity-driven and adaptive epidemic model that captures the impact of asymptomatic and infectious individuals in the network. In the proposed model, referred to as the A-SIYS epidemic, each node can be in three possible states: susceptible, infected without symptoms or asymptomatic and infected with symptoms or symptomatic. Both asymptomatic and symptomatic individuals are infectious. We show that the proposed A-SIYS epidemic captures several well-established epidemic models as special cases and obtain sufficient cond…
In this work, we use the spectral properties of graphons to study stability and sensitivity to noise of deterministic SIS epidemics over large networks. We consider the presence of additive noise in a linearized SIS model and we derive a noise index to quantify the deviation from the disease-free state due to noise. For finite networks, we show that the index depends on the adjacency eigenvalues of its graph. We then assume that the graph is a random sample from a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of the associated graphon operator, we find an approximatio…
In this paper, we investigate the distributed link removal strategy for networked meta-population epidemics. In particular, a deterministic networked susceptible-infected-recovered (SIR) model is considered to describe the epidemic evolving process. In order to curb the spread of epidemics, we present the spectrum-based optimization problem involving the Perron-Frobenius eigenvalue of the matrix constructed by the network topology and transition rates. A modified distributed link removal strategy is developed such that it can be applied to the SIR model with heterogeneous transition rates on w…
We study the influence of heterogeneous mobility patterns in a population on the SIS epidemic model. In particular, we consider a patchy environment in which each patch comprises individuals belonging the different classes, e.g., individuals in different socio-economic strata. We model the mobility of individuals of each class across different patches through an associated Continuous Time Markov Chain (CTMC). The topology of these multiple CTMCs constitute the multi-layer network of mobility. At each time, individuals move in the multi-layer network of spatially-distributed patches according t…
In this work, we aim to understand the influence of the heterogeneity of infection rates on the Susceptible-Infected-Susceptible (SIS) epidemic spreading. Employing the classic SIS model as the benchmark, we study the influence of the independently identically distributed infection rates on the average fraction of infected nodes in the metastable state. The log-normal, gamma and a newly designed distributions are considered for infection rates. We find that, when the recovery rate is small, i.e.\ the epidemic spreads out in both homogeneous and heterogeneous cases: 1) the heterogeneity of infe…
In previous work, we developed the scaled SIS process, which models the dynamics of SIS epidemics over networks. With the scaled SIS process, we can consider networks that are finite-sized and of arbitrary topology (i.e., we are not restricted to specific classes of networks). We derived for the scaled SIS process a closed-form expression for the time-asymptotic probability distribution of the states of all the agents in the network. This closed-form solution of the equilibrium distribution explicitly exhibits the underlying network topology through its adjacency matrix. This paper determines …
We consider a system in which two viruses of the Susceptible-Infected-Susceptible (SIS) type compete over general, overlaid graphs. While such systems have been the focus of many recent works, they have mostly been studied in the sense of convergence analysis, with no existing results quantifying the non-trivial coexistence equilibria (CE) - that is, when both competing viruses maintain long term presence over the network. In this paper, we prove monotonicity of the CE with respect to effective infection rates of the two viruses, and provide the first quantitative analysis of such equilibria i…
In this paper, we analyze dynamic switching networks, wherein the networks switch arbitrarily among a set of topologies. For this class of dynamic networks, we derive an epidemic threshold, considering the SIS epidemic model. First, an epidemic probabilistic model is developed assuming independence between states of nodes. We identify the conditions under which the epidemic dies out by linearizing the underlying dynamical system and analyzing its asymptotic stability around the origin. The concept of joint spectral radius is then used to derive the epidemic threshold, which is later validated …
Minimizing cumulative infections in SIS epidemic models over networks via an edge deletion algorithm
Posts and Telecommunications Institute of Technology（邮电大学）；Faculty of Fundamental Sciences - Lab Blockchain, Posts and Telecommunications Institute of Technology（邮电大学基础科学与区块链实验室）；Faculty of Information Technology, Posts and Telecommunications Institute of Technology（邮电大学信息技术学院）
In this work we study a modified Susceptible-Infected-Susceptible (SIS) model in which the infection rate $λ$ decays exponentially with the number of reinfections $n$, saturating after $n=l$. We find a critical decaying rate $ε_{c}(l)$ above which a finite fraction of the population becomes permanently infected. From the mean-field solution and computer simulations on hypercubic lattices we find evidences that the upper critical dimension is 6 like in the SIR model, which can be mapped in ordinary percolation.
It has been shown in the past that many real-world networks exhibit community structures and non-trivial clustering which comes with the occurrence of a notable number of triangular connections. Yet the influence of such connection patterns on the dynamics of disease transmission is not fully understood. In order to study their role in the context of Susceptible-Infected-Susceptible (SIS) epidemics we use the Triadic Random Graph (TRG) model to construct small networks (N=49) from distinct, closed, directed triadic subpatterns. We compare various global properties of TRGs and use the N-intertw…
An accurate approximate formula of the die-out probability in a SIS epidemic process on a network is proposed. The formula contains only three essential parameters: the largest eigenvalue of the adjacency matrix of the network, the effective infection rate of the virus, and the initial number of infected nodes in the network. The die-out probability formula is compared with the exact die-out probability in complete graphs, Erdős-Rényi graphs, and a power-law graph. Furthermore, as an example, the formula is applied to the $N$-Intertwined Mean-Field Approximation, to explicitly incorporate the …
The SIS (susceptible-infected-susceptible) epidemic model on an arbitrary network, without making approximations, is a $2^n$-state Markov chain with a unique absorbing state (the all-healthy state). This makes analysis of the SIS model and, in particular, determining the threshold of epidemic spread quite challenging. It has been shown that the exact marginal probabilities of infection can be upper bounded by an $n$-dimensional linear time-invariant system, a consequence of which is that the Markov chain is "fast-mixing" when the LTI system is stable, i.e. when $\fracβδ<\frac{1}{λ_{\max}(A)}$ …
The decay rate of SIS epidemics on the complete graph $K_{N}$ is computed analytically, based on a new, algebraic method to compute the second largest eigenvalue of a stochastic three-diagonal matrix up to arbitrary precision. The latter problem has been addressed around 1950, mainly via the theory of orthogonal polynomials and probability theory. The accurate determination of the second largest eigenvalue, also called the \emph{decay parameter}, has been an outstanding problem appearing in general birth-death processes and random walks. Application of our general framework to SIS epidemics sh…
We study a coupled epidemic-mobility model in which, at each time, individuals move in a network of spatially-distributed regions (sub-populations) according to a Continuous Time Markov Chain (CTMC) and subsequently interact with the local sub-population according to an SIS model. We derive a deterministic continuum limit model describing these interactions. We prove the existence of a disease-free equilibrium and an endemic equilibrium under different parameter regimes and establish their (almost) global asymptotic stability using Lyapunov techniques. For the stability of disease-free equilib…
In this brief, we study epidemic spreading dynamics taking place in complex networks. We specifically investigate the effect of synergy, where multiple interactions between nodes result in a combined effect larger than the simple sum of their separate effects. Although synergistic effects play key roles in various biological and social phenomena, their analyses have been often performed by means of approximation techniques and for limited types of networks. In order to address this limitation, this paper proposes a rigorous approach to quantitatively understand the effect of synergy in the Sus…
The nodes in communication networks are possibly and most likely equipped with different recovery resources, which allow them to recover from a virus with different rates. In this paper, we aim to understand know how to allocate the limited recovery resources to efficiently prevent the spreading of epidemics. We study the susceptible-infected-susceptible (SIS) epidemic model on directed scale-free networks. In the classic SIS model, a susceptible node can be infected by an infected neighbor with the infection rate $β$ and an infected node can be recovered to be susceptible again with the recov…
Even though transitivity is a central structural feature of social networks, its influence on epidemic spread on coevolving networks has remained relatively unexplored. Here we introduce and study an adaptive SIS epidemic model wherein the infection and network coevolve with non-trivial probability to close triangles during edge rewiring, leading to substantial reinforcement of network transitivity. This new model provides a unique opportunity to study the role of transitivity in altering the SIS dynamics on a coevolving network. Using numerical simulations and Approximate Master Equations (AM…
We investigate the sensitivity of epidemic behavior to a bounded susceptibility constraint -- susceptible nodes are infected by their neighbors via the regular SI/SIS dynamics, but subject to a cap on the infection rate. Such a constraint is motivated by modern social networks, wherein messages are broadcast to all neighbors, but attention spans are limited. Bounded susceptibility also arises in distributed computing applications with download bandwidth constraints, and in human epidemics under quarantine policies. Network epidemics have been extensively studied in literature; prior work chara…
Recent work has shown that different theoretical approaches to the dynamics of the Susceptible-Infected-Susceptible (SIS) model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks. Here we present large-scale numerical simulations of the SIS dynamics on various types of networks, allowing the precise determination of the effective threshold for systems of finite size N. We compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean-field theory and of the quenched mean-field theory. We show th…
Most real networks are characterized by connectivity patterns that evolve in time following complex, non-Markovian, dynamics. Here we investigate the impact of this ubiquitous feature by studying the Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS) epidemic models on activity driven networks with and without memory (i.e., Markovian and non-Markovian). We show that while memory inhibits the spreading process in SIR models, where the epidemic threshold is moved to larger values, it plays the opposite effect in the case of the SIS, where the threshold is lowered. Th…
Recent violent political extremist events in Brazil and epidemic modeling: The role of a SIS-like model on the understanding of spreading and control of radicalism
We study the effect of external infection sources on phase transitions in epidemic processes. In particular, we consider an epidemic spreading on a network via the SIS/SIR dynamics, which in addition is aided by external agents - sources unconstrained by the graph, but possessing a limited infection rate or virulence. Such a model captures many existing models of externally aided epidemics, and finds use in many settings - epidemiology, marketing and advertising, network robustness, etc. We provide a detailed characterization of the impact of external agents on epidemic thresholds. In particul…
The susceptible--infected--susceptible (SIS) epidemic process on complex networks can show metastability, resembling an endemic equilibrium. In a general setting, the metastable state may involve a large portion of the network, or it can be localized on small subgraphs of the contact network. Localized infections are not interesting because a true outbreak concerns network--wide invasion of the contact graph rather than localized infection of certain sites within the contact network. Existing approaches to localization phenomenon suffer from a major drawback: they fully rely on the steady--sta…
Time-varying network topologies can deeply influence dynamical processes mediated by them. Memory effects in the pattern of interactions among individuals are also known to affect how diffusive and spreading phenomena take place. In this paper we analyze the combined effect of these two ingredients on epidemic dynamics on networks. We study the susceptible-infected-susceptible (SIS) and the susceptible-infected-removed (SIR) models on the recently introduced activity-driven networks with memory. By means of an activity-based mean-field approach we derive, in the long time limit, analytical pre…
The spread of a disease, a computer virus or information is discussed in a directed complex network. We are concerned with a steady state of the spread for the SIR and SIS dynamic models. In a scale-free directed network it is shown that the threshold of its outbreak in both models approaches zero under a high correlation between nodal indegrees and outdegrees.
In the study of infectious diseases on networks, researchers calculate epidemic thresholds to help forecast whether a disease will eventually infect a large fraction of a population. Because network structure typically changes in time, which fundamentally influences the dynamics of spreading processes on them and in turn affects epidemic thresholds for disease propagation, it is important to examine epidemic thresholds in temporal networks. Most existing studies of epidemic thresholds in temporal networks have focused on models in discrete time, but most real-world networked systems evolve con…
