Containing epidemic outbreaks by message-passing techniques

The problem of targeted network immunization can be defined as the one of finding a subset of nodes in a network to immunize or vaccinate in order to minimize a tradeoff between the cost of vaccination and the final (stationary) expected infection under a given epidemic model. Although computing the expected infection is a hard computational problem, simple and efficient mean-field approximations have been put forward in the literature in recent years. The optimization problem can be recast into a constrained one in which the constraints enforce local mean-field equations describing the average stationary state of the epidemic process. For a wide class of epidemic models, including the susceptible-infected-removed and the susceptible-infected-susceptible models, we define a message-passing approach to network immunization that allows us to study the statistical properties of epidemic outbreaks in the presence of immunized nodes as well as to find (nearly) optimal immunization sets for a given choice of parameters and costs. The algorithm scales linearly with the size of the graph and it can be made efficient even on large networks. We compare its performance with topologically based heuristics, greedy methods, and simulated annealing

Recoverable prevalence in growing scale-free networks and the effective immunization

We study the persistent recoverable prevalence and the extinction of computer viruses via e-mails on a growing scale-free network with new users, which structure is estimated form real data. The typical phenomenon is simulated in a realistic model with the probabilistic execution and detection of viruses. Moreover, the conditions of extinction by random and targeted immunizations for hubs are derived through bifurcation analysis for simpler models by using a mean-field approximation without the connectivity correlations. We can qualitatively understand the mechanisms of the spread in linearly growing scale-free networks.Comment: 9 pages, 9 figures, 1 table. Update version after helpful referee comment

Centrality Measures for Networks with Community Structure

Understanding the network structure, and finding out the influential nodes is a challenging issue in the large networks. Identifying the most influential nodes in the network can be useful in many applications like immunization of nodes in case of epidemic spreading, during intentional attacks on complex networks. A lot of research is done to devise centrality measures which could efficiently identify the most influential nodes in the network. There are two major approaches to the problem: On one hand, deterministic strategies that exploit knowledge about the overall network topology in order to find the influential nodes, while on the other end, random strategies are completely agnostic about the network structure. Centrality measures that can deal with a limited knowledge of the network structure are required. Indeed, in practice, information about the global structure of the overall network is rarely available or hard to acquire. Even if available, the structure of the network might be too large that it is too much computationally expensive to calculate global centrality measures. To that end, a centrality measure is proposed that requires information only at the community level to identify the influential nodes in the network. Indeed, most of the real-world networks exhibit a community structure that can be exploited efficiently to discover the influential nodes. We performed a comparative evaluation of prominent global deterministic strategies together with stochastic strategies with an available and the proposed deterministic community-based strategy. Effectiveness of the proposed method is evaluated by performing experiments on synthetic and real-world networks with community structure in the case of immunization of nodes for epidemic control.Comment: 30 pages, 4 figures. Accepted for publication in Physica A. arXiv admin note: text overlap with arXiv:1411.627

Dynamical patterns of epidemic outbreaks in complex heterogeneous networks

We present a thorough inspection of the dynamical behavior of epidemic phenomena in populations with complex and heterogeneous connectivity patterns. We show that the growth of the epidemic prevalence is virtually instantaneous in all networks characterized by diverging degree fluctuations, independently of the structure of the connectivity correlation functions characterizing the population network. By means of analytical and numerical results, we show that the outbreak time evolution follows a precise hierarchical dynamics. Once reached the most highly connected hubs, the infection pervades the network in a progressive cascade across smaller degree classes. Finally, we show the influence of the initial conditions and the relevance of statistical results in single case studies concerning heterogeneous networks. The emerging theoretical framework appears of general interest in view of the recently observed abundance of natural networks with complex topological features and might provide useful insights for the development of adaptive strategies aimed at epidemic containment.Comment: 13 pages, 11 figure

Epidemic processes in complex networks

In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and sociotechnical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and nonequilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel analytical frameworks, and it has produced results of conceptual and practical relevance. A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, mathematicians, epidemiologists, computer, and social scientists share a common interest in studying epidemic spreading and rely on similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while focusing on the main results and the paradigmatic models in infectious disease modeling, the major results concerning generalized social contagion processes are also presented. Finally, the research activity at the forefront in the study of epidemic spreading in coevolving, coupled, and time-varying networks is reported.Comment: 62 pages, 15 figures, final versio

Invited review: Epidemics on social networks

Since its first formulations almost a century ago, mathematical models for disease spreading contributed to understand, evaluate and control the epidemic processes.They promoted a dramatic change in how epidemiologists thought of the propagation of infectious diseases.In the last decade, when the traditional epidemiological models seemed to be exhausted, new types of models were developed.These new models incorporated concepts from graph theory to describe and model the underlying social structure.Many of these works merely produced a more detailed extension of the previous results, but some others triggered a completely new paradigm in the mathematical study of epidemic processes. In this review, we will introduce the basic concepts of epidemiology, epidemic modeling and networks, to finally provide a brief description of the most relevant results in the field.Comment: 17 pages, 13 figure

Traffic Control for Network Protection Against Spreading Processes

Epidemic outbreaks in human populations are facilitated by the underlying transportation network. We consider strategies for containing a viral spreading process by optimally allocating a limited budget to three types of protection resources: (i) Traffic control resources, (ii), preventative resources and (iii) corrective resources. Traffic control resources are employed to impose restrictions on the traffic flowing across directed edges in the transportation network. Preventative resources are allocated to nodes to reduce the probability of infection at that node (e.g. vaccines), and corrective resources are allocated to nodes to increase the recovery rate at that node (e.g. antidotes). We assume these resources have monetary costs associated with them, from which we formalize an optimal budget allocation problem which maximizes containment of the infection. We present a polynomial time solution to the optimal budget allocation problem using Geometric Programming (GP) for an arbitrary weighted and directed contact network and a large class of resource cost functions. We illustrate our approach by designing optimal traffic control strategies to contain an epidemic outbreak that propagates through a real-world air transportation network.Comment: arXiv admin note: text overlap with arXiv:1309.627