Contagion dynamics on adaptive multiplex networks with awareness-dependent rewiring

Over the last few years, the interplay between contagion dynamics of social influences (e.g., human awareness, risk perception, and information dissemination) and biological infections has been extensively investigated within the framework of multiplex networks. The vast majority of existing multiplex network spreading models typically resort to heterogeneous mean-field approximation and microscopic Markov chain approaches. Such approaches usually manifest richer dynamical properties on multiplex networks than those on simplex networks; however, they fall short of a subtle analysis of the variations in connections between nodes of the network and fail to account for the adaptive behavioral changes among individuals in response to epidemic outbreaks. To transcend these limitations, in this paper we develop a highly integrated effective degree approach to modeling epidemic and awareness spreading processes on multiplex networks coupled with awareness-dependent adaptive rewiring. This approach keeps track of the number of nearest neighbors in each state of an individual; consequently, it allows for the integration of changes in local contacts into the multiplex network model. We derive a formula for the threshold condition of contagion outbreak. Also, we provide a lower bound for the threshold parameter to indicate the effect of adaptive rewiring. The threshold analysis is confirmed by extensive simulations. Our results show that awareness-dependent link rewiring plays an important role in enhancing the transmission threshold as well as lowering the epidemic prevalence. Moreover, it is revealed that intensified awareness diffusion in conjunction with enhanced link rewiring makes a greater contribution to disease prevention and control. In addition, the critical phenomenon is observed in the dependence of the epidemic threshold on the awareness diffusion rate, supporting the metacritical point previously reported in literature. This work may shed light on understanding of the interplay between epidemic dynamics and social contagion on adaptive networks.     

Reverse-feeding effect of epidemic by propagators in two-layered networks

Epidemic spreading has been studied for a long time and is currently focused on the spreading of multiple pathogens,especially in multiplex networks. However, little attention has been paid to the case where the mutual influence between different pathogens comes from a fraction of epidemic propagators, such as bisexual people in two separated groups of heterosexual and homosexual people. We here study this topic by presenting a network model of two layers connected by impulsive links, in contrast to the persistent links in each layer. We let each layer have a distinct pathogen and their interactive infection is implemented by a fraction of propagators jumping between the corresponding pairs of nodes in the two layers. By this model we show that(i) the propagators take the key role to transmit pathogens from one layer to the other,which significantly influences the stabilized epidemics;(ii) the epidemic thresholds will be changed by the propagators;and(iii) a reverse-feeding effect can be expected when the infective rate is smaller than its threshold of isolated spreading.A theoretical analysis is presented to explain the numerical results.     

Imperfect targeted immunization in scale-free networks

Scale-free networks are prone to epidemic spreading. To provide cost-effective protection for such networks, targeted immunization was proposed to selectively immunize the hub nodes. In many real-life applications, however, the targeted immunization may not be perfect, either because some hub nodes are hidden and consequently not immunized, or because the vaccination simply cannot provide perfect protection. We investigate the effects of imperfect targeted immunization in scale-free networks. Analysis and simulation results show that there exists a linear relationship between the inverse of the epidemic threshold and the effectiveness of targeted immunization. Therefore, the probability of epidemic outbreak cannot be significantly lowered unless the protection is reasonably strong. On the other hand, even a relatively weak protection over the hub nodes significantly decreases the number of network nodes ever getting infected and therefore enhances network robustness against virus. We show that the above conclusions remain valid where there exists a negative correlation between nodal degree and infectiousness.     

多关系网络上的流行病传播动力学研究

多关系网络已经吸引了许多人的注意, 目前的研究主要涉及其拓扑结构及其演化的分析、 不同类型关系的挖掘、重叠社区的检测、级联失效动力学等. 然而,多关系网络上流行病传播的研究还相对较少. 由此提出一种双关系网络模型(工作-朋友关系网), 研究多关系对于流行病传播动力学行为的影响. 在全接触模式下, 多关系的存在会显著降低网络中的爆发阈值, 使得疾病更容易流行而难以控制. 对比ER (Erdös-Rènyi), WS (Watts-Strogatz), BA (Barabási-Albert)三种网络, 由于结构异质性的差异, WS网络受到的影响最大, ER网络次之, BA网络最小. 有趣的是, 其爆发阈值的相对变化大小与网络结构无关. 在单点接触模式下, 增加强关系的权重将显著提升爆发阈值, 降低感染密度; 随着强关系的比例变化将出现最优值现象: 极大的爆发阈值和极小的感染密度. 随着强关系的边权增加, 达到最优值的边比例将减少. 更为有趣的是, 三个网络中优值出现的位置几乎一致, 独立于网络结构. 这一研究不但有助于理解多关系网络上的病毒传播过程, 也为多关系网络研究提供了一个新的视角. 关键词： 多关系网络 流行病传播 接触模式 爆发阈值     

Modeling the COVID-19 epidemic and awareness diffusion on multiplex networks

The coronavirus disease 2019(COVID-19)has been widely spread around the world,and the control and behavior dynamics are still one of the important research directions in the world.Based on the characteristics of COVID-19’s spread,a coupled disease-awareness model on multiplex networks is proposed in this paper to study and simulate the interaction between the spreading behavior of COVID-19 and related information.In the layer of epidemic spreading,the nodes can be divided into five categories,where the topology of the network represents the physical contact relationship of the population.The topological structure of the upper network shows the information interaction among the nodes,which can be divided into aware and unaware states.Awareness will make people play a positive role in preventing the epidemic diffusion,influencing the spread of the disease.Based on the above model,we have established the state transition equation,through the microscopic Markov chain approach(MMCA),and proposed the propagation threshold calculation method under the epidemic model.Furthermore,MMCA iteration and the Monte Carlo method are simulated on the static network and dynamic network,respectively.The current results will be beneficial to the study of COVID-19,and propose a more rational and effective model for future research on epidemics.     

Stochastic epidemics and rumours on finite random networks

In this paper, we investigate the stochastic spread of epidemics and rumours on networks. We focus on the general stochastic (SIR) epidemic model and a recently proposed rumour model on networks in Nekovee et al. (2007) [3], and on networks with different random structures, taking into account the structure of the underlying network at the level of the degree–degree correlation function. Using embedded Markov chain techniques and ignoring density correlations between neighbouring nodes, we derive a set of equations for the final size of the epidemic/rumour on a homogeneous network that can be solved numerically, and compare the resulting distribution with the solution of the corresponding mean-field deterministic model. The final size distribution is found to switch from unimodal to bimodal form (indicating the possibility of substantial spread of the epidemic/rumour) at a threshold value that is higher than that for the deterministic model. However, the difference between the two thresholds decreases with the network size, n, following a n^−1/3 behaviour. We then compare results (obtained by Monte Carlo simulation) for the full stochastic model on a homogeneous network, including density correlations at neighbouring nodes, with those for the approximating stochastic model and show that the latter reproduces the exact simulation results with great accuracy. Finally, further Monte Carlo simulations of the full stochastic model are used to explore the effects on the final size distribution of network size and structure (using homogeneous networks, simple random graphs and the Barabasi–Albert scale-free networks).     

Statistical model for the effects of phase and momentum randomization on electron transport

In this paper, the study of epidemic spreading of mobile individuals on networks focuses on the system in which each node of the network may be occupied by either one individual or a void, and each individual could move to a neighbour void node. It is found that for the susceptible-infected-susceptible (SIS) model, the diffusion increases the epidemic threshold for arbitrary heterogeneous networks having the degree fluctuations, and the diffusion doesn??t affect the epidemic threshold for regular random networks. In the SI model, the diffusion suppresses the epidemic spread at the early outbreak stage, which indicates that the growth time scale of outbreaks is monotonically increasing with diffusion rate d. The heterogeneous mean-field analysis is in good agreement with the numerical simulations on annealed networks.     

异质自适应网络中的核心-边缘结构及其对疾病传播的抑制作用

节点属性异质自适应网络中疾病传播的研究表明节点属性异质性可以很大程度上增大传播阈值, 并且自组织形成一个更鲁棒的度异质网络结构. 本文从数值模拟方面研究鲁棒的度分布异质结构的自组织形成过程, 分析发现核心-边缘结构的形成才是导致传播阈值增大的根本原因. 鉴于此, 提出一种重连策略, 能够促进核心-边缘结构的形成, 从而达到增大传播阈值的目的. 这不仅有助于深入认识节点属性异质自适应网络中的流行病传播过程, 而且为疾病传播控制策略的提出提供了新思路.     

疾病传播与级联失效相互作用的研究:度不相关网络中疾病扩散条件的分析

在网络科学中,对疾病传播和级联失效的研究分属两个独立的领域,但在实际中存在许多两个过程相互耦合的情况. 比如在通信网络中,病毒传播会对数据传输造成影响,导致网络中负载变化,进而可能引发级联失效. 这个现象已被观察到. 通过建立两个动态过程相互作用的模型及针对该模型的分析,本文给出了计入节点的负载和容量时疾病爆发的条件. 这一条件是由描述疾病传播速率的传播概率与描述节点容量大小的冗余系数共同决定的. 进一步探讨表明,当疾病传播速率一定而冗余系数变化时,疾病恰好开始传播的临界点附近未感染且未失效的节点的数量是最大的,即在此点上网络处于最佳工作状态. 因此给出疾病爆发的临界条件具有重要意义. 关键词： 复杂网络 疾病传播 级联失效     

具有局部结构的增长无标度网络中传染病传播机制研究

为了研究人群中的一些基本的社会关系结构,如家庭、室友、同事等,对传染病传播过程的影响机制,本文建立了一个具有局部结构的增长无标度网络模型.研究表明,局部结构的引入使得该网络模型能够同时再现社会网络的两个重要特征：节点度分布的不均匀性以及节点度之间的相关性.首先,该网络的节点度和局部结构度均服从幂律分布,且度分布指数依赖于局部结构的大小.此外,局部结构的存在还导致网络节点度之间具有正相关特性,而这种正相关正是社会网络所特有的一个重要特性.接着,通过理论分析和数值模拟,我们进一步研究了该网络结构对易感者-感染 关键词： 复杂网络 无标度网络 局部结构 传染病建模     

Thresholds for epidemic spreading in networks

We study the threshold of epidemic models in quenched networks with degree distribution given by a power-law. For the susceptible-infected-susceptible model the activity threshold λ(c) vanishes in the large size limit on any network whose maximum degree k(max) diverges with the system size, at odds with heterogeneous mean-field (HMF) theory. The vanishing of the threshold has nothing to do with the scale-free nature of the network but stems instead from the largest hub in the system being active for any spreading rate λ>1/√k(max) and playing the role of a self-sustained source that spreads the infection to the rest of the system. The susceptible-infected-removed model displays instead agreement with HMF theory and a finite threshold for scale-rich networks. We conjecture that on quenched scale-rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state.     

Finite size effects in epidemic spreading: the problem of overpopulated systems

In this paper we analyze the impact of network size on the dynamics of epidemic spreading. In particular, we investigate the pace of infection in overpopulated systems. In order to do that, we design a model for epidemic spreading on a finite complex network with a restriction to at most one contamination per time step, which can serve as a model for sexually transmitted diseases spreading in some student communes. Because of the highly discrete character of the process, the analysis cannot use the continuous approximation widely exploited for most models. Using a discrete approach, we investigate the epidemic threshold and the quasi-stationary distribution. The main results are two theorems about the mixing time for the process: it scales like the logarithm of the network size and it is proportional to the inverse of the distance from the epidemic threshold.     

Epidemic spreading on weighted complex networks

Nowadays, the emergence of online services provides various multi-relation information to support the comprehensive understanding of the epidemic spreading process. In this Letter, we consider the edge weights to represent such multi-role relations. In addition, we perform detailed analysis of two representative metrics, outbreak threshold and epidemic prevalence, on SIS and SIR models. Both theoretical and simulation results find good agreements with each other. Furthermore, experiments show that, on fully mixed networks, the weight distribution on edges would not affect the epidemic results once the average weight of whole network is fixed. This work may shed some light on the in-depth understanding of epidemic spreading on multi-relation and weighted networks.     

Effects of City-Size Heterogeneity on Epidemic Spreading in a Metapopulation: A Reaction-Diffusion Approach

We review and introduce a generalized reaction-diffusion approach to epidemic spreading in a metapopulation modeled as a complex network. The metapopulation consists of susceptible and infected individuals that are grouped in subpopulations symbolizing cities and villages that are coupled by human travel in a transportation network. By analytic methods and numerical simulations we calculate the fraction of infected people in the metapopulation in the long time limit, as well as the relevant parameters characterizing the epidemic threshold that separates an epidemic from a non-epidemic phase. Within this model, we investigate the effect of a heterogeneous network topology and a heterogeneous subpopulation size distribution. Such a system is suited for epidemic modeling where small villages and big cities exist simultaneously in the metapopulation. We find that the heterogeneous conditions cause the epidemic threshold to be a non-trivial function of the reaction rates (local parameters), the network’s topology (global parameters) and the cross-over population size that separates “village dynamics” from “city dynamics”.     

Modelling the effect of heterogeneous vaccination on metapopulation epidemic dynamics

Vaccination as an epidemic control strategy has a significant effect on epidemic spreading. In this paper, we propose a novel epidemic spreading model on metapopulation networks to study the impact of heterogeneous vaccination on epidemic dynamics, where nodes represent geographical areas and links connecting nodes correspond to human mobility between areas. Using a mean-field approach, we derive the theoretical spreading threshold revealing a non-trivial dependence on the heterogeneity of vaccination. Extensive Monte Carlo simulations validate the theoretical threshold and also show the complex temporal epidemic behaviours above the threshold.     

Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks

The study compares the epidemic spread on static and dynamic small-world networks. They are constructed as a 2-dimensional Newman and Watts model (500 × 500 square lattice with additional shortcuts), where the dynamics involves rewiring shortcuts in every time step of the epidemic spread. We assume susceptible-infectious-removed (SIR) model of the disease. We study the behaviour of the epidemic over the range of shortcut probability per underlying bond ϕ = 0–0.5. We calculate percolation thresholds for the epidemic outbreak, for which numerical results are checked against an approximate analytical model. We find a significant lowering of percolation thresholds on the dynamic network in the parameter range given. The result shows the behaviour of the epidemic on dynamic network is that of a static small world with the number of shortcuts increased by 20.7±1.4 %, while the overall qualitative behaviour stays the same. We derive corrections to the analytical model which account for the effect. For both dynamic and static small worlds we observe suppression of the average epidemic size dependence on network size in comparison with the finite-size scaling known for regular lattice. We also study the effect of dynamics for several rewiring rates relative to infectious period of the disease.     

自适应网络中病毒传播的稳定性和分岔行为研究

自适应复杂网络是以节点状态与拓扑结构之间存在反馈回路为特征的网络. 针对自适应网络病毒传播模型, 利用非线性微分动力学系统研究病毒传播行为; 通过分析非线性系统对应雅可比矩阵的特征方程, 研究其平衡点的局部稳定性和分岔行为, 并推导出各种分岔点的计算公式. 研究表明, 当病毒传播阈值小于病毒存在阈值, 即R₀0^c时, 网络中病毒逐渐消除, 系统的无病毒平衡点是局部渐近稳定的; R₀^c0<1时, 网络出现滞后分岔, 产生双稳态现象, 系统存在稳定的无病毒平衡点、较大稳定的地方病平衡点和较小不稳定的地方病平衡点; R₀>1时, 网络中病毒持续存在, 系统唯一的地方病平衡点是局部渐近稳定的. 研究发现, 系统先后出现了鞍结分岔、跨临界分岔、霍普夫分岔等分岔行为. 最后通过数值仿真验证所得结论的正确性. 关键词： 自适应网络 稳定性 分岔 基本再生数     

具有非均匀传输和抗攻击差异的网络病毒传播模型

针对实际网络中节点存在抗攻击差异以及边的非均匀传输等情况,基于平均场理论,提出具有抗攻击差异和非均匀传输特性的网络病毒传播平均场SIR模型.该模型中,通过引入脆弱性函数和传输函数,分别描述节点的抗攻击差异以及边的非均匀传输能力.通过对所提模型的分析,得到传播阈值的理论结果.理论分析和仿真表明,节点的抗攻击差异以及边的非均匀传输,都可导致出现正的传播阈值,使得病毒传播风险有效降低.     

Epidemic centrality — is there an underestimated epidemic impact of network peripheral nodes?

In the study of disease spreading on empirical complex networks in SIR model, initially infected nodes can be ranked according to some measure of their epidemic impact. The highest ranked nodes, also referred to as “superspreaders”, are associated to dominant epidemic risks and therefore deserve special attention. In simulations on studied empirical complex networks, it is shown that the ranking depends on the dynamical regime of the disease spreading. A possible mechanism leading to this dependence is illustrated in an analytically tractable example. In systems where the allocation of resources to counter disease spreading to individual nodes is based on their ranking, the dynamical regime of disease spreading is frequently not known before the outbreak of the disease. Therefore, we introduce a quantity called epidemic centrality as an average over all relevant regimes of disease spreading as a basis of the ranking. A recently introduced concept of phase diagram of epidemic spreading is used as a framework in which several types of averaging are studied. The epidemic centrality is compared to structural properties of nodes such as node degree, k-cores and betweenness. There is a growing trend of epidemic centrality with degree and k-cores values, but the variation of epidemic centrality is much smaller than the variation of degree or k-cores value. It is found that the epidemic centrality of the structurally peripheral nodes is of the same order of magnitude as the epidemic centrality of the structurally central nodes. The implications of these findings for the distributions of resources to counter disease spreading are discussed.     

A Weighted Configuration Model and Inhomogeneous Epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight (reflecting how closely ‘connected’ the corresponding vertices are). It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that R ₀ can be substantially over- or underestimated if the correlation between degree and weight is not taken into account.