Unification of theoretical approaches for epidemic spreading on complex networks

Markovian Approach for Exploring Competitive Diseases with Heterogeneity-Evidence from COVID-19 and Influenza in China

Due to the complex interactions between multiple infectious diseases, the spreading of diseases in human bodies can vary when people are exposed to multiple sources of infection at the same time. Typically, there is heterogeneity in individuals' responses to diseases, and the transmission routes of different diseases also vary. Therefore, this paper proposes an SIS disease spreading model with individual heterogeneity and transmission route heterogeneity under the simultaneous action of two competitive infectious diseases. We derive the theoretical epidemic spreading threshold using quenched mean-field theory and perform numerical analysis under the Markovian method. Numerical results confirm the reliability of the theoretical threshold and show the inhibitory effect of the proportion of fully competitive individuals on epidemic spreading. The results also show that the diversity of disease transmission routes promotes disease spreading, and this effect gradually weakens when the epidemic spreading rate is high enough. Finally, we find a negative correlation between the theoretical spreading threshold and the average degree of the network. We demonstrate the practical application of the model by comparing simulation outputs to temporal trends of two competitive infectious diseases, COVID-19 and seasonal influenza in China.

News Waves: Hard News, Soft News, Fake News, Rumors, News Wavetrains

We discuss the spread of a piece of news in a population. This is modeled by SIR model of epidemic spread. The model can be reduced to a nonlinear differential equation for the number of people affected by the news of interest. The differential equation has an exponential nonlinearity and it can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. Exact solutions to these equations can be obtained by the Simple Equations Method (SEsM). Some of these exact solutions can be used to model a class of waves associated with the spread of the news in a population. The presence of exact solutions allow to study in detail the dependence of the amplitude and the time horizon of the news waves on the wave parameters, such as the size of the population, initial number of spreaders of the piece of the news, transmission rate, and recovery rate. This allows for recommendations about the change of wave parameters in order to achieve a large amplitude or appropriate time horizon of the news wave. We discuss five types of news waves on the basis of the values of the transmission rate and recovery rate-types A, B, C, D, and E of news waves. In addition, we discuss the possibility of building wavetrains by news waves. There are three possible kinds of wavetrains with respect of the amplitude of the wave: increasing wavetrain, decreasing wavetrain, and mixed wavetrain. The increasing wavetrain is especially interesting, as it is connected to an increasing amplitude of the news wave with respect to the amplitude of the previous wave of the wavetrain. It can find applications in advertising, propaganda, etc.

Visibility graphs of critical and off-critical time series for absorbing state phase transitions

It is possible to investigate emergence in many real systems using time-ordered data. However, classical time series analysis is usually conditioned by data accuracy and quantity. A modern method is to map time series onto graphs and study these structures using the toolbox available in complex network analysis. An important practical problem to investigate the criticality in experimental systems is to determine whether an observed time series is associated with a critical regime or not. We contribute to this problem by investigating the mapping called visibility graph (VG) of a time series generated in dynamical processes with absorbing-state phase transitions. Analyzing degree correlation patterns of the VGs, we are able to distinguish between critical and off-critical regimes. One central hallmark is an asymptotic disassortative correlation on the degree for series near the critical regime in contrast with a pure assortative correlation observed for noncritical dynamics. We are also able to distinguish between continuous (critical) and discontinuous (noncritical) absorbing state phase transitions, the second of which is commonly involved in catastrophic phenomena. The determination of critical behavior converges very quickly in higher dimensions, where many complex system dynamics are relevant.

Scalable parallel and distributed simulation of an epidemic on a graph

We propose an algorithm to simulate Markovian SIS epidemics with homogeneous rates and pairwise interactions on a fixed undirected graph, assuming a distributed memory model of parallel programming and limited bandwidth. This setup can represent a broad class of simulation tasks with compartmental models. Existing solutions for such tasks are sequential by nature. We provide an innovative solution that makes trade-offs between statistical faithfulness and parallelism possible. We offer an implementation of the algorithm in the form of pseudocode in the Appendix. Also, we analyze its algorithmic complexity and its induced dynamical system. Finally, we design experiments to show its scalability and faithfulness. In our experiments, we discover that graph structures that admit good partitioning schemes, such as the ones with clear community structures, together with the correct application of a graph partitioning method, can lead to better scalability and faithfulness. We believe this algorithm offers a way of scaling out, allowing researchers to run simulation tasks at a scale that was not accessible before. Furthermore, we believe this algorithm lays a solid foundation for extensions to more advanced epidemic simulations and graph dynamics in other fields.

On epidemic spreading in metapopulation networks with time-varying contact patterns

Considering that people may change their face-to-face communication patterns with others depending on the season, we propose an epidemic model that incorporates a time-varying contact rate on a metapopulation network and its second-neighbor network. To describe the time-varying contact mode, we utilize a switched system and define two forms of the basic reproduction number corresponding to two different restrictions. We provide the theoretical proof for the stability of the disease-free equilibrium and confirm periodic stability conditions using simulations. The simulation results reveal that as the period of the switched system lengthens, the amplitude of the final infected density increases; however, the peak infected density within a specific period remains relatively unchanged. Interestingly, as the basic reproduction number grows, the amplitude of the final infected density within a period gradually rises to its maximum and then declines. Moreover, the contact rate that occupies a longer duration within a single period has a more significant influence on epidemic spreading. As the values of different contact rates progressively increase, the recovery rate, natural birth rate, and natural death rate all decrease, leading to a larger final infection density.

Collapse transition in epidemic spreading subject to detection with limited resources

Compartmental models are the most widely used framework for modeling infectious diseases. These models have been continuously refined to incorporate all the realistic mechanisms that can shape the course of an epidemic outbreak. Building on a compartmental model that accounts for early detection and isolation of infectious individuals through testing, in this article we focus on the viability of detection processes under limited availability of testing resources, and we study how the latter impacts on the detection rate. Our results show that, in addition to the well-known epidemic transition at R_{0}=1, a second transition occurs at R_{0}^{★}>1 pinpointing the collapse of the detection system and, as a consequence, the switch from a regime of mitigation to a regime in which the pathogen spreads freely. We characterize the epidemic phase diagram of the model as a function of the relevant control parameters: the basic reproduction number, the maximum detection capacity of the system, and the fraction of individuals in shelter. Our analysis thus provides a valuable tool for estimating the detection resources and the level of confinement needed to face epidemic outbreaks.

Large-deviations of disease spreading dynamics with vaccination

We numerically simulated the spread of disease for a Susceptible-Infected-Recovered (SIR) model on contact networks drawn from a small-world ensemble. We investigated the impact of two types of vaccination strategies, namely random vaccination and high-degree heuristics, on the probability density function (pdf) of the cumulative number C of infected people over a large range of its support. To obtain the pdf even in the range of probabilities as small as 10-80, we applied a large-deviation approach, in particular the 1/t Wang-Landau algorithm. To study the size-dependence of the pdfs within the framework of large-deviation theory, we analyzed the empirical rate function. To find out how typical as well as extreme mild or extreme severe infection courses arise, we investigated the structures of the time series conditioned to the observed values of C.

The Structure Entropy-Based Node Importance Ranking Method for Graph Data

Due to its wide application across many disciplines, how to make an efficient ranking for nodes in graph data has become an urgent topic. It is well-known that most classical methods only consider the local structure information of nodes, but ignore the global structure information of graph data. In order to further explore the influence of structure information on node importance, this paper designs a structure entropy-based node importance ranking method. Firstly, the target node and its associated edges are removed from the initial graph data. Next, the structure entropy of graph data can be constructed by considering the local and global structure information at the same time, in which case all nodes can be ranked. The effectiveness of the proposed method was tested by comparing it with five benchmark methods. The experimental results show that the structure entropy-based node importance ranking method performs well on eight real-world datasets.

Dimension reduction in higher-order contagious phenomena

We investigate epidemic spreading in a deterministic susceptible-infected-susceptible model on uncorrelated heterogeneous networks with higher-order interactions. We provide a recipe for the construction of one-dimensional reduced model (resilience function) of the N-dimensional susceptible-infected-susceptible dynamics in the presence of higher-order interactions. Utilizing this reduction process, we are able to capture the microscopic and macroscopic behavior of infectious networks. We find that the microscopic state of nodes (fraction of stable healthy individual of each node) inversely scales with their degree, and it becomes diminished due to the presence of higher-order interactions. In this case, we analytically obtain that the macroscopic state of the system (fraction of infectious or healthy population) undergoes abrupt transition. Additionally, we quantify the network's resilience, i.e., how the topological changes affect the stable infected population. Finally, we provide an alternative framework of dimension reduction based on the spectral analysis of the network, which can identify the critical onset of the disease in the presence or absence of higher-order interactions. Both reduction methods can be extended for a large class of dynamical models.

Epidemic Waves and Exact Solutions of a Sequence of Nonlinear Differential Equations Connected to the SIR Model of Epidemics

The SIR model of epidemic spreading can be reduced to a nonlinear differential equation with an exponential nonlinearity. This differential equation can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. The equations from the obtained sequence are treated by the Simple Equations Method (SEsM). This allows us to obtain exact solutions to some of these equations. We discuss several of these solutions. Some (but not all) of the obtained exact solutions can be used for the description of the evolution of epidemic waves. We discuss this connection. In addition, we use two of the obtained solutions to study the evolution of two of the COVID-19 epidemic waves in Bulgaria by a comparison of the solutions with the available data for the infected individuals.

Heterogeneous pair-approximation analysis for susceptible-infectious-susceptible epidemics on networks

The pair heterogeneous mean-field (PHMF) model has been used extensively in previous studies to investigate the dynamics of susceptible-infectious-susceptible epidemics on complex networks. However, the approximate treatment of the classical or reduced PHMF models lacks a rigorous theoretical analysis. By means of the standard and full PHMF models, we first derived the equivalent conditions for the approximate model treatment. Furthermore, we analytically derived a novel epidemic threshold for the PHMF model, and we demonstrated via numerical simulations that this threshold condition differs from all those reported in earlier studies. Our findings indicate that both the reduced and full PHMF models agree well with continuous-time stochastic simulations, especially when infection is spreading at considerably higher rates.

The Self-Information Weighting-Based Node Importance Ranking Method for Graph Data

Due to their wide application in many disciplines, how to make an efficient ranking for nodes, especially for nodes in graph data, has aroused lots of attention. To overcome the shortcoming that most traditional ranking methods only consider the mutual influence between nodes but ignore the influence of edges, this paper proposes a self-information weighting-based method to rank all nodes in graph data. In the first place, the graph data are weighted by regarding the self-information of edges in terms of node degree. On this base, the information entropy of nodes is constructed to measure the importance of each node and in which case all nodes can be ranked. To verify the effectiveness of this proposed ranking method, we compare it with six existing methods on nine real-world datasets. The experimental results show that our method performs well on all of these nine datasets, especially for datasets with more nodes.

Distinct spreading patterns induced by coexisting channels in information spreading dynamics

In modern society, new communication channels and social platforms remarkably change the way of people receiving and sharing information, but the influences of these channels on information spreading dynamics have not been fully explored, especially in the aspects of outbreak patterns. To this end, based on a susceptible-accepted-recovered model, we examined the outbreak patterns of information spreading in a two-layered network with two coexisting channels: the intra-links within a layer and the inter-links across layers. Depending on the inter-layer coupling strength, i.e., average node degree and transmission probability between the two layers, we observed three different spreading patterns: (i) a localized outbreak with weak inter-layer coupling, (ii) two peaks with a time-delay outbreak appear for an intermediate coupling, and (iii) a synchronized outbreak for a strong coupling. Moreover, we showed that even though the average degree between the two layers is small, a large transmission probability still can compensate and promote the information spread from one layer to another, indicating by that the critical average degree decreases as a power law with transmission probability between the two layers. Additionally, we found that a large gap closed to the critical inter-layer average degree appears in the phase space of theoretical analysis, which indicates the emergence of a global large-scope outbreak. Our findings may, therefore, be of significance for understanding the outbreak behaviors of information spreading in real world.

Emotional Contagion in Physical-Cyber Integrated Networks: The Phase Transition Perspective

Understanding the emotional contagion process in the crowd will help to take measures in advance to avoid the large-scale spread of negative emotions in emergencies and reduce the loss of lives and properties. Studying the phase transition phenomenon is fundamental to analyzing and evaluating the crowd emotional contagion. However, it is a challenging issue since most people participate in both the physical and cyber networks at the same time. In this article, we focus on the emotional contagion in physical-cyber integrated networks from the phase transition perspective. To achieve this, we first construct a physical-cyber integrated network model to describe the interactions between physical and cyber networks. Second, we build an emotional contagion model to capture the characteristics of emotional contagion in the physical and cyber integrated networks accurately. Finally, we study the phase transition phenomenon of emotional contagion and identify the critical threshold by mapping the emotional contagion to the joint site/bond percolation model. Numerical simulations and experiments further support and enrich our conclusions. The proposed method is expected to provide guidance for controlling emotional contagion in emergencies.

Optimizing protection resource allocation for traffic-driven epidemic spreading

Optimizing the allocation of protection resources to control the spreading process in networks is a central problem in public health and network security. In this paper, we propose a comprehensive adjustable resource allocation mechanism in which the over allocation of resources can be also numerically reflected and study the effects of this mechanism on traffic-driven epidemic spreading. We observe that an inappropriate resource allocation scheme can induce epidemic spreading, while an optimized heterogeneous resource allocation scheme can significantly suppress the outbreak of the epidemic. The phenomenon can be explained by the role of nodes induced by the heterogeneous network structure and traffic flow distribution. Theoretical analysis also gives an exact solution to the epidemic threshold and reveals the optimal allocation scheme. Compared to the uniform allocation scheme, the increase in traffic flow will aggravate the decline of the epidemic threshold for the heterogeneous resource allocation scheme. This indicates that the uneven resource allocation makes the network performance of suppressing epidemic degrade with the traffic load level. Finally, it is demonstrated that real-world network topology also confirms the results.

Influence of heterogeneity of infection thresholds on epidemic spreading with neighbor resource supporting

The spread of disease on complex networks has attracted wide attention in physics, mathematics, and epidemiology. Recent works have demonstrated that individuals always exhibit different criteria for disease infection in a network that significantly influences the epidemic dynamics. In this paper, considering the heterogeneity of node susceptibility, we proposed an infection threshold model with neighbor resource support. The infection threshold of an individual is associated with the degree, and a parameter follows the normal distribution. Based on improved heterogeneous mean-field theory and extensive numerical simulations, we find that the mean and standard deviation of the infection threshold model can affect the phase transition and epidemic outbreak size. As the mean of the normal distribution parameter increases from a small value to a large value, the system shows a change from a continuous phase transition to a discontinuous phase transition, and the disease even stops spreading. The disease spreads from a discontinuous phase transition to continuous for the sizeable mean value as the standard deviation increases. Furthermore, the standard deviation also varies in the outbreak size.

J-SPACE: a Julia package for the simulation of spatial models of cancer evolution and of sequencing experiments

Background

The combined effects of biological variability and measurement-related errors on cancer sequencing data remain largely unexplored. However, the spatio-temporal simulation of multi-cellular systems provides a powerful instrument to address this issue. In particular, efficient algorithmic frameworks are needed to overcome the harsh trade-off between scalability and expressivity, so to allow one to simulate both realistic cancer evolution scenarios and the related sequencing experiments, which can then be used to benchmark downstream bioinformatics methods.

Result

We introduce a Julia package for SPAtial Cancer Evolution (J-SPACE), which allows one to model and simulate a broad set of experimental scenarios, phenomenological rules and sequencing settings.Specifically, J-SPACE simulates the spatial dynamics of cells as a continuous-time multi-type birth-death stochastic process on a arbitrary graph, employing different rules of interaction and an optimised Gillespie algorithm. The evolutionary dynamics of genomic alterations (single-nucleotide variants and indels) is simulated either under the Infinite Sites Assumption or several different substitution models, including one based on mutational signatures. After mimicking the spatial sampling of tumour cells, J-SPACE returns the related phylogenetic model, and allows one to generate synthetic reads from several Next-Generation Sequencing (NGS) platforms, via the ART read simulator. The results are finally returned in standard FASTA, FASTQ, SAM, ALN and Newick file formats.

Conclusion

J-SPACE is designed to efficiently simulate the heterogeneous behaviour of a large number of cancer cells and produces a rich set of outputs. Our framework is useful to investigate the emergent spatial dynamics of cancer subpopulations, as well as to assess the impact of incomplete sampling and of experiment-specific errors. Importantly, the output of J-SPACE is designed to allow the performance assessment of downstream bioinformatics pipelines processing NGS data. J-SPACE is freely available at: https://github.com/BIMIB-DISCo/J-Space.jl.

The interplay between disease spreading and awareness diffusion in multiplex networks with activity-driven structure

The interplay between disease and awareness has been extensively studied in static networks. However, most networks in reality will evolve over time. Based on this, we propose a novel epidemiological model in multiplex networks. In this model, the disease spreading layer is a time-varying network generated by the activity-driven model, while the awareness diffusion layer is a static network, and the heterogeneity of individual infection and recovery ability is considered. First, we extend the microscopic Markov chain approach to analytically obtain the epidemic threshold of the model. Then, we simulate the spread of disease and find that stronger heterogeneity in the individual activities of a physical layer can promote disease spreading, while stronger heterogeneity of the virtual layer network will hinder the spread of disease. Interestingly, we find that when the individual infection ability follows Gaussian distribution, the heterogeneity of infection ability has little effect on the spread of disease, but it will significantly affect the epidemic threshold when the individual infection ability follows power-law distribution. Finally, we find the emergence of a metacritical point where the diffusion of awareness is able to control the onset of the epidemics. Our research could cast some light on exploring the dynamics of epidemic spreading in time-varying multiplex networks.

Gravity-Law Based Critical Bots Identification in Large-Scale Heterogeneous Bot Infection Network

The explosive growth of botnets has posed an unprecedented potent threat to the internet. It calls for more efficient ways to screen influential bots, and thus precisely bring the whole botnet down beforehand. In this paper, we propose a gravity-based critical bots identification scheme to assess the influence of bots in a large-scale botnet infection. Specifically, we first model the propagation of the botnet as a Heterogeneous Bot Infection Network (HBIN). An improved SEIR model is embedded into HBIN to extract both heterogeneous spatial and temporal dependencies. Within built-up HBIN, we elaborate a gravity-based influential bots identification algorithm where intrinsic influence and infection diffusion influence are specifically designed to disclose significant bots traits. Experimental results based on large-scale sample collections from the implemented prototype system demonstrate the promising performance of our scheme, comparing it with other state-of-the-art baselines.

Hypergraph assortativity: A dynamical systems perspective

The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets.

Logistic growth on networks: Exact solutions for the susceptible-infected model

The susceptible-infected (SI) model is the most basic of all compartmental models used to describe the spreading of information through a population. Despite its apparent simplicity, the analytic solution of this model on networks is still lacking. We address this problem here using a novel formulation inspired by the mathematical treatment of many-body quantum systems. This allows us to organize the time-dependent expectation values for the state of individual nodes in terms of contributions from subgraphs of the network. We compute these contributions systematically and find a set of symmetry relations among subgraphs of differing topologies. We use our novel approach to compute the spreading of information on three different sample networks. The exact solution, which matches with Monte Carlo simulations, visibly departs from the mean-field results.

A general model of hierarchical fractal scale-free networks

We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs.

Characterization of Group Behavior of Corruption in Construction Projects Based on Contagion Mechanism

With the rapid development of construction projects, more and more engineering corruption problems have emerged. Therefore, this paper proposes a SEIR (susceptible-exposed-infected-recovered) based corruption model to better understand the propagation process of corruption cases in construction projects. In this model, the data samples are collected from the 2018 Engineering Corruption Case Judgment Document, the propagation parameters are obtained through actual case analysis with the help of complex networks, the change process and key influencing factors of actual nodes in engineering corruption cases are simulated by Python. The study results indicate that the personnel conforms to the “4–9 transmission law,” in which the early stage is a period of high incidence of corruption cases. The network of corruption cases is somewhat vulnerable, and its spread is about minus 8 times the change in crackdown rate and 10 times the change in infection rate. The variation range of the susceptible population S and the removed person R in the propagation simulation curve can predict the relationship between corruption infection rate and crackdown rate, which can provide theoretical guidance for preventing the occurrence of corruption.

Large deviations of a susceptible-infected-recovered model around the epidemic threshold

We numerically study the dynamics of the SIR disease model on small-world networks by using a large-deviation approach. This allows us to obtain the probability density function of the total fraction of infected nodes and of the maximum fraction of simultaneously infected nodes down to very small probability densities like 10^{-2500}. We analyze the structure of the disease dynamics and observed three regimes in all probability density functions, which correspond to quick mild, quick extremely severe, and sustained severe dynamical evolutions, respectively. Furthermore, the mathematical rate functions of the densities are investigated. The results indicate that the so-called large-deviation property holds for the SIR model. Finally, we measured correlations with other quantities like the duration of an outbreak or the peak position of the fraction of infections, also in the rare regions which are not accessible by standard simulation techniques.

Percolation of temporal hierarchical mobility networks during COVID-19

Percolation theory is essential for understanding disease transmission patterns on the temporal mobility networks. However, the traditional approach of the percolation process can be inefficient when analysing a large-scale, dynamic network for an extended period. Not only is it time-consuming but it is also hard to identify the connected components. Recent studies demonstrate that spatial containers restrict mobility behaviour, described by a hierarchical topology of mobility networks. Here, we leverage crowd-sourced, large-scale human mobility data to construct temporal hierarchical networks composed of over 175 000 block groups in the USA. Each daily network contains mobility between block groups within a Metropolitan Statistical Area (MSA), and long-distance travels across the MSAs. We examine percolation on both levels and demonstrate the changes of network metrics and the connected components under the influence of COVID-19. The research reveals the presence of functional subunits even with high thresholds of mobility. Finally, we locate a set of recurrent critical links that divide components resulting in the separation of core MSAs. Our findings provide novel insights into understanding the dynamical community structure of mobility networks during disruptions and could contribute to more effective infectious disease control at multiple scales. This article is part of the theme issue 'Data science approaches to infectious disease surveillance'.

The impact of information dissemination on vaccination in multiplex networks

The impact of information dissemination on epidemic control is essentially subject to individual behaviors. Vaccination is one of the most effective strategies against the epidemic spread, whose correlation with the information dissemination should be better understood. To this end, we propose an evolutionary vaccination game model in multiplex networks by integrating an information-epidemic spreading process into the vaccination dynamics, and explore how information dissemination influences vaccination. The spreading process is described by a two-layer coupled susceptible-alert-infected-susceptible (SAIS) model, where the strength coefficient between two layers characterizes the tendency and intensity of information dissemination. We find that the impact of information dissemination on vaccination decision-making depends on not only the vaccination cost and network topology, but also the stage of the system evolution. For instance, in a two-layer BA scale-free network, information dissemination helps to improve vaccination density only at the early stage of the system evolution, as well as when the vaccination cost is smaller. A counter-intuitive conclusion that more information transmission cannot promote vaccination is obtained when the vaccination cost is larger. Moreover, we study the impact of the strength coefficient and individual sensitivity on the fraction of infected individuals and social cost, and unveil the role of information dissemination in controlling the epidemic.

A new model to identify node importance in complex networks based on DEMATEL method

It is still a hot research topic to identify node importance in complex networks. Recently many methods have been proposed to deal with this problem. However, most of the methods only focus on local or path information, they do not combine local and global information well. In this paper, a new model to identify node importance based on Decision-making Trial and Evaluation Laboratory (DEMATEL) is presented. DEMATEL method is based on graph theory which takes the global information into full consideration so that it can effectively identify the importance of one element in the whole complex system. Some experiments based on susceptible-infected (SI) model are used to compare the new model with other methods. The applications in three different networks illustrate the effectiveness of the new model.

Network percolation reveals adaptive bridges of the mobility network response to COVID-19

Human mobility is crucial to understand the transmission pattern of COVID-19 on spatially embedded geographic networks. This pattern seems unpredictable, and the propagation appears unstoppable, resulting in over 350,000 death tolls in the U.S. by the end of 2020. Here, we create the spatiotemporal inter-county mobility network using 10 TB (Terabytes) trajectory data of 30 million smart devices in the U.S. in the first six months of 2020. We investigate the bond percolation process by removing the weakly connected edges. As we increase the threshold, the mobility network nodes become less interconnected and thus experience surprisingly abrupt phase transitions. Despite the complex behaviors of the mobility network, we devised a novel approach to identify a small, manageable set of recurrent critical bridges, connecting the giant component and the second-largest component. These adaptive links, located across the United States, played a key role as valves connecting components in divisions and regions during the pandemic. Beyond, our numerical results unveil that network characteristics determine the critical thresholds and the bridge locations. The findings provide new insights into managing and controlling the connectivity of mobility networks during unprecedented disruptions. The work can also potentially offer practical future infectious diseases both globally and locally.

Role of lurkers in threshold-driven information spreading dynamics

The threshold model as a classical paradigm for studying information spreading processes has been well studied. The main focuses are on how the underlying social network structure or the size of initial seeds can affect the cascading dynamics. However, the influence of node characteristics has been largely ignored. Here, inspired by empirical observations, we extend the threshold model by taking into account lurking nodes, who rarely interact with their neighbors. In particular, we consider two different scenarios: (i) Lurkers are absolutely silent and never interact with others and (ii) lurkers intermittently interact with their neighborhood with an activity rate p. In the first case, we demonstrate that lurkers may reduce the effective average degree of the underlying network, playing a dual role in spreading dynamics. In the latter case, we find that the stochastic dynamic behavior of lurkers could significantly promote the spread of information. Concretely, slightly raising the activity rate p of lurkers may result in a remarkable increase in the final cascade size. Further increasing p could make nodes become more stable on average, while it is still easy to observe global cascades due to the fluctuations of the effective degree of nodes.

Effects of confinement and vaccination on an epidemic outburst: A statistical mechanics approach

This work describes a simple agent model for the spread of an epidemic outburst, with special emphasis on mobility and geographical considerations, which we characterize via statistical mechanics and numerical simulations. As the mobility is decreased, a percolation phase transition is found separating a free-propagation phase in which the outburst spreads without finding spatial barriers and a localized phase in which the outburst dies off. Interestingly, the number of infected agents is subject to maximal fluctuations at the transition point, building upon the unpredictability of the evolution of an epidemic outburst. Our model also lends itself to testing vaccination schedules. Indeed, it has been suggested that if a vaccine is available but scarce it is convenient to carefully select the vaccination program to maximize the chances of halting the outburst. We discuss and evaluate several schemes, with special interest on how the percolation transition point can be shifted, allowing for higher mobility without epidemiological impact.

Antivax movement and epidemic spreading in the era of social networks: Nonmonotonic effects, bistability, and network segregation

In this work, we address a multicoupled dynamics on complex networks with tunable structural segregation. Specifically, we work on a networked epidemic spreading under a vaccination campaign with agents in favor and against the vaccine. Our results show that such coupled dynamics exhibits a myriad of phenomena such as nonequilibrium transitions accompanied by bistability. Besides we observe the emergence of an intermediate optimal segregation level where the community structure enhances negative opinions over vaccination but counterintuitively hinders-rather than favoring-the global disease spreading. Thus our results hint vaccination campaigns should avoid policies that end up segregating excessively antivaccine groups so that they effectively work as echo chambers in which individuals look to confirmation without jeopardizing the safety of the whole population.

Microscopic edge-based compartmental modeling method for analyzing the susceptible-infected-recovered epidemic spreading on networks

The edge-based compartmental modeling (EBCM) approach has been used widely to characterize the nonrecurrent epidemic spreading dynamics (e.g., the susceptible-infected-recovered model) in complex networks. By using the probability theory, we derived an individual-based formulation for this approach, which we herein refer to as the microscopic EBCM method. We found that both for small and large initial infection numbers, the epidemic evolution agreed well with the ensemble averages of our stochastic simulations on different complex networks. Moreover, we showed that the dynamical message passing model, the standard EBCM system, and the pair quenched mean-field equations can be deduced by our microscopic EBCM method. In addition, the microscopic EBCM method was used to analyze the effect of epidemic awareness on networks. Importantly, the simple EBCM model for exponential awareness was developed. Our method provides a way for handling nontrivial disease transmission processes with irreversible dynamics.

Scalable Estimation of Epidemic Thresholds via Node Sampling

Infectious or contagious diseases can be transmitted from one person to another through social contact networks. In today's interconnected global society, such contagion processes can cause global public health hazards, as exemplified by the ongoing Covid-19 pandemic. It is therefore of great practical relevance to investigate the network transmission of contagious diseases from the perspective of statistical inference. An important and widely studied boundary condition for contagion processes over networks is the so-called epidemic threshold. The epidemic threshold plays a key role in determining whether a pathogen introduced into a social contact network will cause an epidemic or die out. In this paper, we investigate epidemic thresholds from the perspective of statistical network inference. We identify two major challenges that are caused by high computational and sampling complexity of the epidemic threshold. We develop two statistically accurate and computationally efficient approximation techniques to address these issues under the Chung-Lu modeling framework. The second approximation, which is based on random walk sampling, further enjoys the advantage of requiring data on a vanishingly small fraction of nodes. We establish theoretical guarantees for both methods and demonstrate their empirical superiority.

A new SEIAR model on small-world networks to assess the intervention measures in the COVID-19 pandemics

A new susceptible-exposed-infected-asymptomatically infected-removed (SEIAR) model is developed to depict the COVID-19 transmission process, considering the latent period and asymptomatically infected. We verify the suppression effect of typical measures, cultivating human awareness, and reducing social contacts. As for cutting off social connections, the feasible measures encompass social distancing policy, isolating infected communities, and isolating hub nodes. Furthermore, it is found that implementing corresponding anti-epidemic measures at different pandemic stages can achieve significant results at a low cost. In the beginning, global lockdown policy is necessary, but isolating infected wards and hub nodes could be more beneficial as the situation eases. The proposed SEIAR model emphasizes the latent period and asymptomatically infected, thus providing theoretical support for subsequent research.

Impact of presymptomatic transmission on epidemic spreading in contact networks: A dynamic message-passing analysis

Infectious diseases that incorporate presymptomatic 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 computational cost compared to numerical simulations. It facilitates the derivation of epidemic thresholds, which are phase boundaries separating parameter regimes where infections can be effectively contained from those where they cannot. These have clear implications on different containment strategies through topological (reducing contacts) and infection parameter changes (e.g., social distancing and wearing face masks), with relevance to the recent COVID-19 pandemic.

Ring vaccination strategy in networks: A mixed percolation approach

Ring vaccination is a mitigation strategy that consists in seeking and vaccinating the contacts of a sick patient, in order to provide immunization and halt the spread of disease. We study an extension of the susceptible-infected-recovered (SIR) epidemic model with ring vaccination in complex and spatial networks. Previously, a correspondence between this model and a link percolation process has been established, however, this is only valid in complex networks. Here, we propose that the SIR model with ring vaccination is equivalent to a mixed percolation process of links and nodes, which offers a more complete description of the process. We verify that this approach is valid in both complex and spatial networks, the latter being built according to the Waxman model. This model establishes a distance-dependent cost of connection between individuals arranged in a square lattice. We determine the epidemic-free regions in a phase diagram based on the wiring cost and the parameters of the epidemic model (vaccination and infection probabilities and recovery time). In addition, we find that for long recovery times this model maps into a pure node percolation process, in contrast to the SIR model without ring vaccination, which maps into link percolation.

Employing Fuzzy Logic to Analyze the Structure of Complex Biological and Epidemic Spreading Models

Complex networks constitute a new field of scientific research that is derived from the observation and analysis of real-world networks, for example, biological, computer and social ones. An important subset of complex networks is the biological, which deals with the numerical examination of connections/associations among different nodes, namely interfaces. These interfaces are evolutionary and physiological, where network epidemic models or even neural networks can be considered as representative examples. The investigation of the corresponding biological networks along with the study of human diseases has resulted in an examination of networks regarding medical supplies. This examination aims at a more profound understanding of concrete networks. Fuzzy logic is considered one of the most powerful mathematical tools for dealing with imprecision, uncertainties and partial truth. It was developed to consider partial truth values, between completely true and completely false, and aims to provide robust and low-cost solutions to real-world problems. In this manuscript, we introduce a fuzzy implementation of epidemic models regarding the Human Immunodeficiency Virus (HIV) spreading in a sample of needle drug individuals. Various fuzzy scenarios for a different number of users and different number of HIV test samples per year are analyzed in order for the samples used in the experiments to vary from case to case. To the best of our knowledge, analyzing HIV spreading with fuzzy-based simulation scenarios is a research topic that has not been particularly investigated in the literature. The simulation results of the considered scenarios demonstrate that the existence of fuzziness plays an important role in the model setup process as well as in analyzing the effects of the disease spread.

Quantum Contagion: A Quantum-Like Approach for the Analysis of Social Contagion Dynamics with Heterogeneous Adoption Thresholds

Modeling the information of social contagion processes has recently attracted a substantial amount of interest from researchers due to its wide applicability in network science, multi-agent-systems, information science, and marketing. Unlike in biological spreading, the existence of a reinforcement effect in social contagion necessitates considering the complexity of individuals in the systems. Although many studies acknowledged the heterogeneity of the individuals in their adoption of information, there are no studies that take into account the individuals’ uncertainty during their adoption decision-making. This resulted in less than optimal modeling of social contagion dynamics in the existence of phase transition in the final adoption size versus transmission probability. We employed the Inverse Born Problem (IBP) to represent probabilistic entities as complex probability amplitudes in edge-based compartmental theory, and demonstrated that our novel approach performs better in the prediction of social contagion dynamics through extensive simulations on random regular networks.

Network dynamic model of epidemic transmission introducing a heterogeneous control factor

The COVID-19 epidemic is not only a medical issue but also a sophisticated social problem. We propose a network dynamics model of epidemic transmission introducing a heterogeneous control factor. The proposed model applied the classical susceptible- exposed-infectious-recovered model to the network based on effective distance and was modified by introducing a heterogeneous control factor with temporal and spatial characteristics. International aviation data were approximately used to estimate the flux fraction matrix, and the effective distance was calculated. Through parameter estimation and simulation, the theoretical values of the modified model fit well with practical values. By adjusting the parameters and observing the change of the results, we found that the modified model is more in line with the actual needs and has higher credibility in the comprehensive analysis. The assessment shows that the number of confirmed cases worldwide will reach about 20 million optimistically. In severe cases, the peak value will exceed 80 million, and the late stage of the epidemic shows a long tail shape, lasting more than one and a half years. The effective way to control the global epidemic is to strengthen international cooperation and to impose international travel restrictions and other measures.

Explicit and implicit network connectivity: Analytical formulation and application to transport processes

Connectivity is a fundamental structural feature of a network that determines the outcome of any dynamics that happens on top of it. However, an analytical approach to obtain connection probabilities between nodes associated with to paths of different lengths is still missing. Here, we derive exact expressions for random-walk connectivity probabilities across any range of numbers of steps in a generic temporal, directed, and weighted network. This allows characterizing explicit connectivity realized by causal paths as well as implicit connectivity related to motifs of three nodes and two links called here pitchforks. We directly link such probabilities to the processes of tagging and sampling any quantity exchanged across the network, hence providing a natural framework to assess transport dynamics. Finally, we apply our theoretical framework to study ocean transport features in the Mediterranean Sea. We find that relevant transport structures, such as fluid barriers and corridors, can generate contrasting and counterintuitive connectivity patterns bringing novel insights into how ocean currents drive seascape connectivity.

Exact Maximal Reduction Of Stochastic Reaction Networks By Species Lumping

MOTIVATION

Stochastic reaction networks are a widespread model to describe biological systems where the presence of noise is relevant, such as in cell regulatory processes. Unfortunately, in all but simplest models the resulting discrete state-space representation hinders analytical tractability and makes numerical simulations expensive. Reduction methods can lower complexity by computing model projections that preserve dynamics of interest to the user.

RESULTS

We present an exact lumping method for stochastic reaction networks with mass-action kinetics. It hinges on an equivalence relation between the species, resulting in a reduced network where the dynamics of each macro-species is stochastically equivalent to the sum of the original species in each equivalence class, for any choice of the initial state of the system. Furthermore, by an appropriate encoding of kinetic parameters as additional species, the method can establish equivalences that do not depend on specific values of the parameters. The method is supported by an efficient algorithm to compute the largest species equivalence, thus the maximal lumping. The effectiveness and scalability of our lumping technique, as well as the physical interpretability of resulting reductions, is demonstrated in several models of signaling pathways and epidemic processes on complex networks.

AVAILABILITY

The algorithms for species equivalence have been implemented in the software tool ERODE, freely available for download from https://www.erode.eu.

Dynamic measures for transportation networks

Most complex network analyses of transportation systems use simplified static representations obtained from existing connections in a time horizon. In static representations, travel times, waiting times and compatibility of schedules are neglected, thus losing relevant information. To obtain a more accurate description of transportation networks, we use a dynamic representation that considers synced paths and that includes waiting times to compute shortest paths. We use the shortest paths to define dynamic network, node and edge measures to analyse the topology of transportation networks, comparable with measures obtained from static representations. We illustrate the application of these measures with a toy model and a real transportation network built from schedules of a low-cost carrier. Results show remarkable differences between measures of static and dynamic representations, demonstrating the limitations of the static representation to obtain accurate information of transportation networks.

Critical properties of the SIS model on the clustered homophilic network

The spreading of epidemics in complex networks has been a subject of renewed interest of several scientific branches. In this regard, we have focused our attention on the study of the susceptible-infected-susceptible (SIS) model, within a Monte Carlo numerical simulation approach, representing the spreading of epidemics in a clustered homophilic network. The competition between infection and recovery that drives the system either to an absorbing or to an active phase is analyzed. We estimate the static critical exponents β ∕ ν , 1 ∕ ν and γ ∕ ν , through finite-size scaling (FSS) analysis of the order parameter ρ and its fluctuations, showing that they differ from those associated with the contact process on a scale-free network, as well as those predicted by the heterogeneous mean-field theory.

Effective control of SARS-CoV-2 transmission in Wanzhou, China

The effectiveness of control measures to contain coronavirus disease 2019 (COVID-19) in Wanzhou, China was assessed. Epidemiological data were analyzed for 183 confirmed COVID-19 cases and their close contacts from five generations of transmission of severe acute respiratory syndrome coronavirus 2 throughout the entire COVID-19 outbreak in Wanzhou. Approximately 67.2% and 32.8% of cases were symptomatic and asymptomatic, respectively. Asymptomatic and presymptomatic transmission accounted for 75.9% of the total recorded transmission. The reproductive number was 1.64 (95% confidence interval: 1.16-2.40) for G1-to-G2 transmission, decreasing to 0.31-0.39 in later generations, concomitant with implementation of rigorous control measures. Substantially higher infection risk was associated with contact within 5 d after the infectors had been infected, frequent contact and ≥8 h of contact duration. The spread of COVID-19 was effectively controlled in Wanzhou by breaking the transmission chain through social distancing, extensive contact tracing, mass testing and strict quarantine of close contacts.

Epidemic spreading and control strategies in spatial modular network

Epidemic spread on networks is one of the most studied dynamics in network science and has important implications in real epidemic scenarios. Nonetheless, the dynamics of real epidemics and how it is affected by the underline structure of the infection channels are still not fully understood. Here we apply the susceptible-infected-recovered model and study analytically and numerically the epidemic spread on a recently developed spatial modular model imitating the structure of cities in a country. The model assumes that inside a city the infection channels connect many different locations, while the infection channels between cities are less and usually directly connect only a few nearest neighbor cities in a two-dimensional plane. We find that the model experience two epidemic transitions. The first lower threshold represents a local epidemic spread within a city but not to the entire country and the second higher threshold represents a global epidemic in the entire country. Based on our analytical solution we proposed several control strategies and how to optimize them. We also show that while control strategies can successfully control the disease, early actions are essentials to prevent the disease global spread.

Effects of heterogeneous self-protection awareness on resource-epidemic coevolution dynamics

Recent studies have demonstrated that the allocation of individual resources has a significant influence on the dynamics of epidemic spreading. In the real scenario, individuals have a different level of awareness for self-protection when facing the outbreak of an epidemic. To investigate the effects of the heterogeneous self-awareness distribution on the epidemic dynamics, we propose a resource-epidemic coevolution model in this paper. We first study the effects of the heterogeneous distributions of node degree and self-awareness on the epidemic dynamics on artificial networks. Through extensive simulations, we find that the heterogeneity of self-awareness distribution suppresses the outbreak of an epidemic, and the heterogeneity of degree distribution enhances the epidemic spreading. Next, we study how the correlation between node degree and self-awareness affects the epidemic dynamics. The results reveal that when the correlation is positive, the heterogeneity of self-awareness restrains the epidemic spreading. While, when there is a significant negative correlation, strong heterogeneous or strong homogeneous distribution of the self-awareness is not conducive for disease suppression. We find an optimal heterogeneity of self-awareness, at which the disease can be suppressed to the most extent. Further research shows that the epidemic threshold increases monotonously when the correlation changes from most negative to most positive, and a critical value of the correlation coefficient is found. When the coefficient is below the critical value, an optimal heterogeneity of self-awareness exists; otherwise, the epidemic threshold decreases monotonously with the decline of the self-awareness heterogeneity. At last, we verify the results on four typical real-world networks and find that the results on the real-world networks are consistent with those on the artificial network.

Network temporality can promote and suppress information spreading

Temporality is an essential characteristic of many real-world networks and dramatically affects the spreading dynamics on networks. In this paper, we propose an information spreading model on temporal networks with heterogeneous populations. Individuals are divided into activists and bigots to describe the willingness to accept the information. Through a developed discrete Markov chain approach and extensive numerical simulations, we discuss the phase diagram of the model and the effects of network temporality. From the phase diagram, we find that the outbreak phase transition is continuous when bigots are relatively rare, and a hysteresis loop emerges when there are a sufficient number of bigots. The network temporality does not qualitatively alter the phase diagram. However, we find that the network temporality affects the spreading outbreak size by either promoting or suppressing, which relies on the heterogeneities of population and of degree distribution. Specifically, in networks with homogeneous and weak heterogeneous degree distribution, the network temporality suppresses (promotes) the information spreading for small (large) values of information transmission probability. In networks with strong heterogeneous degree distribution, the network temporality always promotes the information spreading when activists dominate the population, or there are relatively fewer activists. Finally, we also find the optimal network evolution scale, under which the network information spreading is maximized.

Spreading Control in Two-Layer Multiplex Networks

The problem of controlling a spreading process in a two-layer multiplex networks in such a way that the extinction state becomes a global attractor is addressed. The problem is formulated in terms of a Markov-chain based susceptible-infected-susceptible (SIS) dynamics in a complex multilayer network. The stabilization of the extinction state for the nonlinear discrete-time model by means of appropriate adaptation of system parameters like transition rates within layers and between layers is analyzed using a dominant linear dynamics yielding global stability results. An answer is provided for the central question about the essential changes in the step from a single to a multilayer network with respect to stability criteria and the number of nodes that need to be controlled. The results derived rigorously using mathematical analysis are verified using statical evaluations about the number of nodes to be controlled and by simulation studies that illustrate the stability property of the multilayer network induced by appropriate control action.

Disease spreading with social distancing: A prevention strategy in disordered multiplex networks

The frequent emergence of diseases with the potential to become threats at local and global scales, such as influenza A(H1N1), SARS, MERS, and recently COVID-19 disease, makes it crucial to keep designing models of disease propagation and strategies to prevent or mitigate their effects in populations. Since isolated systems are exceptionally rare to find in any context, especially in human contact networks, here we examine the susceptible-infected-recovered model of disease spreading in a multiplex network formed by two distinct networks or layers, interconnected through a fraction q of shared individuals (overlap). We model the interactions through weighted networks, because person-to-person interactions are diverse (or disordered); weights represent the contact times of the interactions. Using branching theory supported by simulations, we analyze a social distancing strategy that reduces the average contact time in both layers, where the intensity of the distancing is related to the topology of the layers. We find that the critical values of the distancing intensities, above which an epidemic can be prevented, increase with the overlap q. Also we study the effect of the social distancing on the mutual giant component of susceptible individuals, which is crucial to keep the functionality of the system. In addition, we find that for relatively small values of the overlap q, social distancing policies might not be needed at all to maintain the functionality of the system.