We consider the problem of identifying the source of an epidemic, spreading through a network, from a complete observation of the infected nodes in a snapshot of the network. Previous work on the problem has often employed geometric, spectral or heuristic approaches to identify the source, with the trees being the most studied network topology. We take a fully statistical approach and derive novel recursions to compute the Bayes optimal solution, under a susceptible-infected (SI) epidemic model. Our analysis is time and rate independent, and holds for general network topologies. We then provide two tractable algorithms for solving these recursions, a mean-field approximation and a greedy approach, and evaluate their performance on real and synthetic networks. Real networks are far from tree-like and an emphasis will be given to networks with high transitivity, such as social networks and those with communities. We show that on such networks, our approaches significantly outperform geometric and spectral centrality measures, most of which perform no better than random guessing. Both the greedy and mean-field approximation are scalable to large sparse networks.
The customary perspective to reason about epidemic mitigation in temporal networks hinges on the identification of nodes with specific features or network roles. The ensuing individual-based control strategies, however, are difficult to carry out in practice and ignore important correlations between topological and temporal patterns. Here we adopt a mesoscopic perspective and present a principled framework to identify collective features at multiple scales and rank their importance for epidemic spread. We use tensor decomposition techniques to build an additive representation of a temporal net…
We study the SIRS (Susceptible-Infected-Recovered-Susceptible) spreading processes over complex networks, by considering its exact $3^n$-state Markov chain model. The Markov chain model exhibits an interesting connection with its $2n$-state nonlinear "mean-field" approximation and the latter's corresponding linear approximation. We show that under the specific threshold where the disease-free state is a globally stable fixed point of both the linear and nonlinear models, the exact underlying Markov chain has an $O(\log n)$ mixing time, which means the epidemic dies out quickly. In fact, the ep…
We study the spread of discrete-time epidemics over arbitrary networks for well-known propagation models, namely SIS (susceptible-infected-susceptible), SIR (susceptible-infected-recovered), SIRS (susceptible-infected-recovered-susceptible) and SIV (susceptible-infected-vaccinated). Such epidemics are described by $2^n$- or $3^n$-state Markov chains. Ostensibly, because analyzing such Markov chains is too complicated, their $O(n)$-dimensional nonlinear "mean-field" approximation, and its linearization, are often studied instead. We provide a complete global analysis of the epidemic dynamics of…
An efficient strategy for the identification of influential spreaders that could be used to control epidemics within populations would be of considerable importance. Generally, populations are characterized by its community structures and by the heterogeneous distributions of weak ties among nodes bridging over communities. A strategy for community networks capable of identifying influential spreaders that accelerate the spread of disease is here proposed. In this strategy, influential spreaders serve as target nodes. This is based on the idea that, in k-shell decomposition, weak ties and stro…
In the context of Epidemic Intelligence, many Event-Based Surveillance (EBS) systems have been proposed in the literature to promote the early identification and characterization of potential health threats from online sources of any nature. Each EBS system has its own surveillance definitions and priorities, therefore this makes the task of selecting the most appropriate EBS system for a given situation a challenge for end-users. In this work, we propose a new evaluation framework to address this issue. It first transforms the raw input epidemiological event data into a set of normalized even…
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…
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…
The identification of which nodes are optimal seeds for spreading processes on a network is a non-trivial problem that has attracted much interest recently. While activity has mostly focused on non-recurrent type of dynamics, here we consider the problem for the Susceptible-Infected-Susceptible (SIS) spreading model, where an outbreak seeded in one node can originate an infinite activity avalanche. We apply the theoretical framework for avalanches on networks proposed by Larremore et al. [Phys. Rev. E 85, 066131 (2012)], to obtain detailed quantitative predictions for the spreading influence o…
The reconstruction of missing information in epidemic spreading on contact networks can be essential in the prevention and containment strategies. The identification and warning of infectious but asymptomatic individuals (i.e., contact tracing), the well-known patient-zero problem, or the inference of the infectivity values in structured populations are examples of significant epidemic inference problems. As the number of possible epidemic cascades grows exponentially with the number of individuals involved and only an almost negligible subset of them is compatible with the observations (e.g.,…
In this paper, we aim to understand the transient dynamics of a susceptible-infected (SI) epidemic spreading process on a large network. The SI model has been largely overlooked in the literature, while it is naturally a better fit for modeling the malware propagation in early times when patches/vaccines are not available, or over a wider range of timescales when massive patching is practically infeasible. Nonetheless, its analysis is simply non-trivial, as its important dynamics are all transient and the usual stability/steady-state analysis no longer applies. To this end, we develop a theore…
Epidemic processes are common out-of-equilibrium phenomena of broad interdisciplinary interest. Recently, dynamic message-passing (DMP) has been proposed as an efficient algorithm for simulating epidemic models on networks, and in particular for estimating the probability that a given node will become infectious at a particular time. To date, DMP has been applied exclusively to models with one-way state changes, as opposed to models like SIS (susceptible-infectious-susceptible) and SIRS (susceptible-infectious-recovered-susceptible) where nodes can return to previously inhabited states. Becaus…
This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate $ω$ (and reconnect to non-infectious individuals with probability $α$ or else simply drop the edge if $α=0$), so-called preventive rewiring. The models are denoted SIR-$ω$ and SEIR-$ω$, and we focus attention on the early stages of an outbreak, where we derive expression for the basic reproduction number $R_0$ and the expected degree of the infectious nodes $E(D_I)$ using two different approximation approaches. The first approach appr…
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 …
In this work, we introduce a novel mathematical network model for community level preventative health interventions. We develop algorithms to approximately solve this novel formulation at large scale and we rigorously explore their theoretical properties. We create a realistic simulation environment for interventions designed to curb the prevalence of obesity occurring in the region of Montreal, Canada, and use this environment to empirically evaluate the performance of the algorithms we develop. We find that our algorithms significantly outperform all baseline interventions. Moreover, for fix…
In social and biological systems, the structural heterogeneity of interaction networks gives rise to the emergence of a small set of influential nodes, or influencers, in a series of dynamical processes. Although much smaller than the entire network, these influencers were observed to be able to shape the collective dynamics of large populations in different contexts. As such, the successful identification of influencers should have profound implications in various real-world spreading dynamics such as viral marketing, epidemic outbreaks and cascading failure. In this chapter, we first summari…
Detection of patient-zero can give new insights to the epidemiologists about the nature of first transmissions into a population. In this paper, we study the statistical inference problem of detecting the source of epidemics from a snapshot of spreading on an arbitrary network structure. By using exact analytic calculations and Monte Carlo estimators, we demonstrate the detectability limits for the SIR model, which primarily depend on the spreading process characteristics. Finally, we demonstrate the applicability of the approach in a case of a simulated sexually transmitted infection spreadin…
Indian Statistical Institute（印度统计研究所）；IBM Research（IBM研究院）
Amidst an increasing number of infected cases during the Covid-19 pandemic, it is essential to trace, as early as possible, the susceptible people who might have been infected by the disease due to their close proximity with people who were tested positive for the virus. This early contact tracing is likely to limit the rate of spread of the infection within a locality. In this paper, we investigate how effectively and efficiently can such a list of susceptible people be found given a list of infected persons and their locations. To address this problem from an information retrieval (search) p…
Real networks exhibit heterogeneous nature with nodes playing far different roles in structure and function. To identify vital nodes is thus very significant, allowing us to control the outbreak of epidemics, to conduct advertisements for e-commercial products, to predict popular scientific publications, and so on. The vital nodes identification attracts increasing attentions from both computer science and physical societies, with algorithms ranging from simply counting the immediate neighbors to complicated machine learning and message passing approaches. In this review, we clarify the concep…
A pivotal idea in network science, marketing research and innovation diffusion theories is that a small group of nodes -- called influencers -- have the largest impact on social contagion and epidemic processes in networks. Despite the long-standing interest in the influencers identification problem in socio-economic and biological networks, there is not yet agreement on which is the best identification strategy. State-of-the-art strategies are typically based either on heuristic centrality metrics or on analytic arguments that only hold for specific network topologies or peculiar dynamical re…
Since the 21st century, the global outbreaks of infectious diseases such as SARS in 2003, H1N1 in 2009, and H7N9 in 2013, have become the critical threat to the public health and a hunting nightmare to the government. Understanding the propagation in large-scale metapopulations and predicting the future outbreaks thus become crucially important for epidemic control and prevention. In the literature, there have been a bulk of studies on modeling intra-city epidemic propagation but with the single population assumption (homogeneity). Some recent works on metapopulation propagation, however, focu…
The spread of an infection on a real-world social network is determined by the interplay of two processes: the dynamics of the network, whose structure changes over time according to the encounters between individuals, and the dynamics on the network, whose nodes can infect each other after an encounter. Physical encounter is the most common vehicle for the spread of infectious diseases, but detailed information about encounters is often unavailable because expensive, unpractical to collect or privacy sensitive. We asks whether the friendship ties between the individuals in a social network su…
We propose a mathematical framework, based on conic geometric programming, to control a susceptible-infected-susceptible viral spreading process taking place in a directed contact network with unknown contact rates. We assume that we have access to time series data describing the evolution of the spreading process observed by a collection of sensor nodes over a finite time interval. We propose a data-driven robust convex optimization framework to find the optimal allocation of protection resources (e.g., vaccines and/or antidotes) to eradicate the viral spread at the fastest possible rate. In …
Northeastern University（东北大学）；Northwestern University（西北大学）；Greenwich University（格林威治大学）；Harvard University（哈佛大学）；University of California San Diego（加利福尼亚大学圣地亚哥分校）
Locating the source of an epidemic, or patient zero (P0), can provide critical insights into the infection's transmission course and allow efficient resource allocation. Existing methods use graph-theoretic centrality measures and expensive message-passing algorithms, requiring knowledge of the underlying dynamics and its parameters. In this paper, we revisit this problem using graph neural networks (GNNs) to learn P0. We establish a theoretical limit for the identification of P0 in a class of epidemic models. We evaluate our method against different epidemic models on both synthetic and a rea…
Payame Noor University（帕亚梅努尔大学）；University of Virginia（弗吉尼亚大学）
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…
