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”.     

Socioeconomic Patterns of Twitter User Activity

Stratifying behaviors based on demographics and socioeconomic status is crucial for political and economic planning. Traditional methods to gather income and demographic information, like national censuses, require costly large-scale surveys both in terms of the financial and the organizational resources needed for their successful collection. In this study, we use data from social media to expose how behavioral patterns in different socioeconomic groups can be used to infer an individual’s income. In particular, we look at the way people explore cities and use topics of conversation online as a means of inferring individual socioeconomic status. Privacy is preserved by using anonymized data, and abstracting human mobility and online conversation topics as aggregated high-dimensional vectors. We show that mobility and hashtag activity are good predictors of income and that the highest and lowest socioeconomic quantiles have the most differentiated behavior across groups.     

基于手机大数据的中国人口迁徙模式及疫情影响研究

新型冠状病毒感染的肺炎(COVID-19)可通过人员接触与流动迅速传播,因此研究人类迁徙和出行模式的变化对疫情防控至关重要.本文基于手机运营商2020年春运及疫情暴发前后连续两个月的全国地级市之间的人口流动数据,运用时序网络分析方法构建人口流动网络拓扑结构指标,并通过引入地理衰减因子提出Spatial-Louvain社团检测算法,研究平时、春运、疫情防控隔离和生产复工四阶段的人口迁徙模式的时空演化规律.研究发现:受各地疫情防控措施影响,武汉封城后全国城市间人口流量急剧下降,并持续至2月中旬.疫情期间人口流动网络结构呈现四阶段的时空演化模式;本文提出的空间网络社团检测算法比传统Louvain算法平均模块度值提高了14%;中国城市分布以经济交互和地理位置为基础,形成了以核心城市为中心,向周边辐射的城市群格局;疫情因素仅能在短暂时间内改变部分城市的城市群归属,当该因素消失或减弱后,城市群能迅速恢复原有格局.     

Epidemic spreading in scale-free networks including the effect of individual vigilance

In this paper, we study the epidemic spreading in scale-free networks and propose a new susceptible-infected- recovered (SIR) model that includes the effect of individual vigilance. In our model, the effective spreading rate is dynamically adjusted with the time evolution at the vigilance period. Using the mean-field theory, an analytical result is derived. It shows that individual vigilance has no effect on the epidemic threshold. The numerical simulations agree well with the analytical result. Furthermore, we investigate the effect of individual vigilance on the epidemic spreading speed. It is shown that individual vigilance can slow the epidemic spreading speed effectively and delay the arrival of peak epidemic infection.     

Dynamical response of multi-patch, flux-based models to the input of infected people: Epidemic response to initiated events

The time course of an epidemic can be modeled using the differential equations that describe the spread of disease and by dividing people into “patches” of different sizes with the migration of people between these patches. We used these multi-patch, flux-based models to determine how the time course of infected and susceptible populations depends on the disease parameters, the geometry of the migrations between the patches, and the addition of infected people into a patch. We found that there are significantly longer lived transients and additional “ancillary” epidemics when the reproductive rate R is closer to 1, as would be typical of SARS (Severe Acute Respiratory Syndrome) and bird flu, than when R is closer to 10, as would be typical of measles. In addition we show, both analytical and numerical, how the time delay between the injection of infected people into a patch and the corresponding initial epidemic that it produces depends on R.     

Epidemic dynamics behavior in some bus transport networks

We abstract bus transport networks (BTNs) to complex networks using the Space P approach. First, we select three actual BTNs in three major cities in China, namely, Beijing, Shanghai and Hangzhou. Using the SIS model, we simulate and study the epidemic spreading in the three BTNs. We obtain the density of infected vertices varying with time and the stationary density of infected vertices varying with infection rate. Second, we simulate and study the epidemic spreading in a recently introduced BTN evolution model, the network properties of which correspond well with those of actual BTNs. Third, we use mean-field theory to analyze the epidemic dynamics behavior of the BTN evolution model and obtain the theoretical epidemic threshold of this model. The theoretical value agrees well with the simulation results. Based on the work in this paper, we provide the following possible forecasts for epidemic dynamics in actual BTNs. An actual BTN should have a finite positive epidemic threshold. If the effective infection rate is above this threshold, the epidemic spread in the network and the density of infected vertices finally stabilizes in a balanced state. Below this threshold, the number of infected vertices decays exponentially fast and the epidemic cannot spread on a large scale.     

Comparative effects of avoidance and vaccination in disease spread on a dynamic small-world network

Dynamic small-world contact networks have fixed short range links and time-varying stochastic long range links. They are used to model mobile populations or as minimal models for traditional small-world networks. Here we study the relative effects of vaccinations and avoidance of infected individuals in a susceptible-infected-recovered (SIR) epidemic model on a dynamic small-world network. We derive the critical mobility required for an outbreak to occur as a function of the disease’s infectivity, recovery rate, avoidance rate, and vaccination rate. We also derive an expression that allows us to calculate the amount of vaccination and/or avoidance necessary to prevent an epidemic. Calculated quantities show excellent agreement with simulations.     

Epidemic spreading on random surfer networks with infected avoidance strategy

In this paper, we study epidemic spreading on random surfer networks with infected avoidance(IA) strategy. In particular, we consider that susceptible individuals' moving direction angles are affected by the current location information received from infected individuals through a directed information network. The model is mainly analyzed by discrete-time numerical simulations. The results indicate that the IA strategy can restrain epidemic spreading effectively. However,when long-distance jumps of individuals exist, the IA strategy's effectiveness on restraining epidemic spreading is heavily reduced. Finally, it is found that the influence of the noises from information transferring process on epidemic spreading is indistinctive.     

The impact of human activity patterns on asymptomatic infectious processes in complex networks

The study of the impact of human activity patterns on network dynamics has attracted a lot of attention in recent years. However, individuals’ knowledge of their own physical states has rarely been incorporated into modeling processes. In real life, for certain infectious processes, infected agents may not have any visible or physical signs and symptoms; therefore, they may believe that they are uninfected even when they have been infected asymptomatically. This infection awareness factor is covered neither in the classical epidemic models such as SIS nor in network propagation studies. In this article, we propose a novel infectious process model that differentiates between the infection awareness states and the physical states of individuals and extend the SIS model to deal with both asymptomatic infection characteristics and human activity patterns. With regards to the latter, we focus particularly on individuals’ testing action, which is to determine whether an individual is infected by an epidemic. The simulation results show that less effort is required in controlling the disease when the transmission probability is either very small or large enough and that Poisson activity patterns are more effective than heavy-tailed patterns in controlling and eliminating asymptomatic infectious diseases due to the long-tail characteristic.     

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.     

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.     

Disease spreading on fitness-rewired complex networks

As people travel, human contact networks may change topologically from time to time. In this paper, we study the problem of epidemic spreading on this kind of dynamic network, specifically the one in which the rewiring dynamics of edges are carried out to preserve the degree of each node (called fitness rewiring). We also consider the adaptive rewiring of edges, which encourages disconnections from and discourages connections to infected nodes and eventually leads to the isolation of the infected from the susceptible with only a small number of links between them. We find that while the threshold of epidemic spreading remains unchanged and prevalence increases in the fitness rewiring dynamics, meeting of the epidemic threshold is delayed and prevalence is reduced (if adaptive dynamics are included). To understand these different behaviors, we introduce a new measure called the “mean change of effective links” and find that creation and deletion of pathways for pathogen transmission are the dominant factors in fitness and adaptive rewiring dynamics, respectively.     

Epidemic prevalence on random mobile dynamical networks: individual heterogeneity and correlation

In this paper, we extend the susceptible-infected-susceptible (SIS) epidemiological model on a random dynamical network composed of mobile individuals, in which the infection is caused by the collisions between susceptible and infected individuals at the spreading rate proportional to their susceptibilities and infectivities. We analytically study the criticality of spreading dynamics under different distributions of individual susceptibility and infectivity, and numerically verify the cases of power-law and (or) Gaussian distributions. Our findings show that the heterogeneity of individual susceptibility and infectivity increases the epidemic threshold, and the positive correlation of individual susceptibility and infectivity avails to the epidemic prevalence.     

Epidemic spreading in complex networks

The study of epidemic spreading in complex networks is currently a hot topic and a large body of results have been achieved. In this paper, we briefly review our contributions to this field, which includes the underlying mechanism of rumor propagation, the epidemic spreading in community networks, the influence of varying topology, and the influence of mobility of agents. Also, some future directions are pointed out.      

Scaling behavior of threshold epidemics

We study the classic Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease. In this stochastic process, there are two competing mechanism: infection and recovery. Susceptible individuals may contract the disease from infected individuals, while infected ones recover from the disease at a constant rate and are never infected again. Our focus is the behavior at the epidemic threshold where the rates of the infection and recovery processes balance. In the infinite population limit, we establish analytically scaling rules for the time-dependent distribution functions that characterize the sizes of the infected and the recovered sub-populations. Using heuristic arguments, we also obtain scaling laws for the size and duration of the epidemic outbreaks as a function of the total population. We perform numerical simulations to verify the scaling predictions and discuss the consequences of these scaling laws for near-threshold epidemic outbreaks.     

Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks

Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks.     

Variations in epidemic distribution with some characteristic parameters

Considering the spread of an epidemic among a population of mobile agents that can get infected and maintain the infection for a period, we investigate the variation in the homogeneity of the distribution of the epidemic with the remaining time of infection τ, the velocity modulus of the agent v, and the infection rate α. We find that the distribution of the infected cluster size is always exponential. By analyzing the variation of the characteristic infected cluster size coefficient, we show that the inhomogeneity of epidemic distribution increases with an increase in τ for very low v, while it decreases with an increase in τ for moderate v. The epidemic distribution also tends to a homogeneous state as both v and α increase.     

Modelling of discrete-time SIS models with awareness interactions on degree-uncorrelated networks

Many epidemic models ignored the impact of awareness on epidemics in a population, though it is not the case from the real viewpoints. In this paper, a discrete-time SIS model with awareness interactions on degree-uncorrelated networks is considered. We study three kinds of awareness, including local awareness and global awareness which are originated from the epidemic-dependent information, and individual awareness which is epidemic-independent and determined by the individual information. We demonstrate analytically that awareness of the epidemic-dependent information cannot change the epidemic threshold regardless of the global or local spreading information. In contrast, epidemic-independent awareness to individual information increases the epidemic threshold in finite scale-free networks, but cannot halt the absence of epidemic threshold in an infinite scale-free network. By numerical simulations, we find that local awareness has a stronger impact on epidemic prevalence than global awareness. Our findings explore the effects of various types of awareness on epidemic spreading and address their roles in the epidemic control.     

Prediction of epidemics dynamics on networks with partial differential equations: A case study for COVID-19 in China

Since December 2019, the COVID-19 epidemic has repeatedly hit countries around the world due to various factors such as trade, national policies and the natural environment. To closely monitor the emergence of new COVID-19 clusters and ensure high prediction accuracy, we develop a new prediction framework for studying the spread of epidemic on networks based on partial differential equations (PDEs), which captures epidemic diffusion along the edges of a network driven by population flow data. In this paper, we focus on the effect of the population movement on the spread of COVID-19 in several cities from different geographic regions in China for describing the transmission characteristics of COVID-19. Experiment results show that the PDE model obtains relatively good prediction results compared with several typical mathematical models. Furthermore, we study the effectiveness of intervention measures, such as traffic lockdowns and social distancing, which provides a new approach for quantifying the effectiveness of the government policies toward controlling COVID-19 via the adaptive parameters of the model. To our knowledge, this work is the first attempt to apply the PDE model on networks with Baidu Migration Data for COVID-19 prediction.     

SIR model with general distribution function in the infectious period

The Susceptible-Infected-Removed or SIR model, as it was formulated by Kermack and McKendrick, is the key model for epidemic dynamics. Most applications of such a basic scheme use a constant rate for the removal term. However, that assumption corresponds to the rather unrealistic exponential distribution of infectious times. On the other hand, recent approaches, like numerical simulations, frequently assume a fixed and uniform duration for the infectious state—which is unrealistic too. The extreme assumptions in those different schemes are a hurdle that can frustrate any intention of drawing comparison between results from them. In the present contribution we study the delay equations for the SIR model, comparing the solutions for many typical cases with the simulation counterpart and with the standard SIR model. Using delay equations, where each infected individual is removed at a specific time after being infected, the dynamics for the infected and susceptible agree almost exactly with the numerical implementation. Even in the general case of distributed infective periods, the agreement is excellent.