In most models of the spread of disease over contact networks it is assumed that the probabilities per unit time of disease transmission and recovery from disease are constant, implying exponential distributions of the time intervals for transmission and recovery. Time intervals for real diseases, however, have distributions that in most cases are far from exponential, which leads to disagreements, both qualitative and quantitative, with the models. In this paper, we study a generalized version of the SIR (susceptible-infected-recovered) model of epidemic disease that allows for arbitrary distributions of transmission and recovery times. Standard differential equation approaches cannot be used for this generalized model, but we show that the problem can be reformulated as a time-dependent message passing calculation on the appropriate contact network. The calculation is exact on trees (i.e., loopless networks) or locally tree-like networks (such as random graphs) in the large system size limit. On non-tree-like networks we show that the calculation gives a rigorous bound on the size of disease outbreaks. We demonstrate the method with applications to two specific models and the results compare favorably with numerical simulations.
We study the problem of estimating the origin of an epidemic outbreak -- given a contact network and a snapshot of epidemic spread at a certain time, determine the infection source. Finding the source is important in different contexts of computer or social networks. We assume that the epidemic spread follows the most commonly used susceptible-infected-recovered model. We introduce an inference algorithm based on dynamic message-passing equations, and we show that it leads to significant improvement of performance compared to existing approaches. Importantly, this algorithm remains efficient i…
We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time $t>0$. We specialise these results to a homogene…
We study the SIR ("susceptible, infected, removed/recovered") model on directed graphs with heterogeneous transmission probabilities within the message-passing approximation. We characterize the percolation transition, predict cluster size distributions and suggest vaccination strategies. All predictions are compared to numerical simulations on real networks. The percolation threshold which we predict is a rigorous lower bound to the threshold on real networks. For large, locally tree-like networks, our predictions agree very well with the numerical data.
The interaction among spreading processes on a complex network is a nontrivial phenomenon of great importance. It has recently been realized that cooperative effects among infective diseases can give rise to qualitative changes in the phenomenology of epidemic spreading, leading for instance to abrupt transitions and hysteresis. Here we consider a simple model for two interacting pathogens on a network and we study it by using the message-passing approach. In this way we are able to provide detailed predictions for the behavior of the model in the whole phase-diagram for any given network stru…
With the hit of new pandemic threats, scientific frameworks are needed to understand the unfolding of the epidemic. The use of mobile apps that are able to trace contacts is of utmost importance in order to control new infected cases and contain further propagation. Here we present a theoretical approach using both percolation and message--passing techniques, to the role of contact tracing, in mitigating an epidemic wave. We show how the increase of the app adoption level raises the value of the epidemic threshold, which is eventually maximized when high-degree nodes are preferentially targete…
The message passing approach of Karrer and Newman [Phys. Rev. E 82, 016101 (2010)] is an exact and practicable representation of susceptible-infected-recovered dynamics on finite trees. Here we show that, assuming Poisson contact processes, a pair-based moment closure representation [Sharkey, J. Math. Biol. 57, 311 (2008)] can be derived from their equations. We extend the applicability of both representations and discuss their relative merits. On arbitrary time-independent networks, as was shown for the message passing formalism, the pair-based moment closure equations also provide a rigorous…
Tensor network dynamical message passing for epidemic models
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院，UCAS）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）
Infectious diseases that incorporate pre-symptomatic transmission are challenging to monitor, model, predict and contain. We address this scenario by studying a variant of a stochastic susceptible-exposed-infected-recovered model on arbitrary network instances using an analytical framework based on the method of dynamic message-passing. This framework provides a good estimate of the probabilistic evolution of the spread on both static and contact networks, offering a significantly improved accuracy with respect to individual-based mean-field approaches while requiring a much lower computationa…
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 averag…
Identifying the most influential spreaders is one of outstanding problems in physics of complex systems. So far, many approaches have attempted to rank the influence of nodes but there is still the lack of accuracy to single out influential spreaders. Here, we directly tackle the problem of finding important spreaders by solving analytically the expected size of epidemic outbreaks when spreading originates from a single seed. We derive and validate a theory for calculating the size of epidemic outbreaks with a single seed using a message-passing approach. In addition, we find that the probabil…
Identifying influential subpopulations in metapopulation epidemic models has far-reaching potential implications for surveillance and intervention policies of a global pandemic. However, there is a lack of methods to determine influential nodes in metapopulation models based on a rigorous mathematical background. In this study, we derive the message-passing theory for metapopulation modeling and propose a method to determine influential spreaders. Based on our analysis, we identify the most dangerous city as a potential seed of a pandemic when applied to real-world data. Moreover, we particula…
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…
Hypergraph networks are closer to real life because they can reflect higher-order interactions, so researchers have begun using them to build models for real-world networks. The mean-field approach is the current tool for studying the percolation problem on hypergraph networks. However, we found that when there is a loop in the hypergraph network, the calculated results using this approach deviate from the real results. Therefore, in this paper, we rephrase the percolation on the hypergraph network as a message passing process, thus obtaining a message passing approach. Our proposed approach h…
Optimizing embedded systems, where the optimization of one depends on the state of another, is a formidable computational and algorithmic challenge, that is ubiquitous in real world systems. We study flow networks, where bilevel optimization is relevant to traffic planning, network control and design, and where flows are governed by an optimization requirement subject to the network parameters. We employ message-passing algorithms in flow networks with sparsely coupled structures to adapt network parameters that govern the network flows, in order to optimize a global objective. We demonstrate …
