Individual dynamics and local heterogeneity provide a microscopic view of the epidemic spreading

The COVID-19 pandemic has caused severe global disasters,highlighting the importance of understanding the details and trends of epidemic transmission in order to introduce efficient intervention measures.While the widely used deterministic compartmental models have qualitatively presented continuous “analytical” insight and captured some transmission features,their treatment usually lacks spatiotemporal variation.Here,we propose a stochastic individual dynamical(SID)model to mimic the random and heterogeneous nature of epidemic propagation.The SID model provides a unifying framework for representing the spatiotemporal variations of epidemic development by tracking the movements of each individual.Using this model,we reproduce the infection curves for COVID-19 cases in different areas globally and find the local dynamics and heterogeneity at the individual level that affect the disease outbreak.The macroscopic trend of virus spreading is clearly illustrated from the microscopic perspective,enabling a quantitative assessment of different interventions.Seemingly,this model is also applicable to studying stochastic processes at the “meter scale”,e.g.,human society’s collective dynamics.

Dynamic properties of epidemic spreading on finite size complex networks

The Internet presents a complex topological structure, on which computer viruses can easily spread. By using theoretical analysis and computer simulation methods, the dynamic process of disease spreading on finite size networks with complex topological structure is investigated. On the finite size networks, the spreading process of SIS （susceptibleinfected-susceptible） model is a finite Markov chain with an absorbing state. Two parameters, the survival probability and the conditional infecting probability, are introduced to describe the dynamic properties of disease spreading on finite size networks. Our results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks. Also, knowledge about the dynamic character of virus spreading is helpful for adopting immunity policy.

Effect of incubation period on epidemic spreading in complex networks

We study the effect of incubation period on epidemic spreading in the Barabasi-Albert scale-free network and the Watts-Strogatz small world network by using a Suspectable-Incubated-Infected-Suspectable model. Our analytical investigations show that the epidemic threshold is independent of incubation period in both networks, which is verified by our large-scale simulation results. We also investigate the effect of incubation period on the epidemic dynamics in a supercritical regime. It is found that with the increase of incubation period Ω, a damped oscillation evolution of ρT （the ratio of persons in incubated state） appears and the time needed to reach a saturation value increases. Moreover, the steady value of ρT increases and approaches to an asymptotic constant with the value of Ω increasing. As a result, the infected ratio ρI decreases with the increase of Ω according to a power law.

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.

Epidemic Spreading Model Based on Social Active Degree in Social Networks

In this paper,an improved Susceptible-Infected-Susceptible(SIS) epidemic spreading model is proposed in order to provide a theoretical method to analyze and predict the spreading of diseases.This model is based on the following ideas:in social networks,the contact probability between nodes is decided by their social distances and their active degrees.The contact probability of two indirectly connected nodes is decided by the shortest path between them.Theoretical analysis and simulation experiment were conducted to evaluate the performance of this improved model.Because the proposed model is independent of the network structure,simulation experiments were done in several kinds of networks,namely the ER network,the random regular network,the WS small world network,and the BA scale-free network,in order to study the influences of certain factors have on the epidemic spreading,such as the social contact active degree,the network structure,the average degree,etc.This improved model provides an idea for studying the spreading rule of computer virus,attitudes,fashion styles and public opinions in social networks.

Comparative Effects of Avoidance and Immunization on Epidemic Spreading in a Dynamic Small-World Network with Community Structure

Considering the actual behavior of people’s short-term travel,we propose a dynamic small-world community network model with tunable community strength which has constant local links and time varying long-range jumps.Then an epidemic model of susceptible-infected-recovered is established based on the mean-field method to evaluate the inhibitory effects of avoidance and immunization on epidemic spreading.And an approximate formula for the epidemic threshold is obtained by mathematical analysis.The simulation results show that the epidemic threshold decreases with the increase of inner-community motivation rate and inter-community long-range motivation rate,while it increases with the increase of immunization rate or avoidance rate.It indicates that the inhibitory effect on epidemic spreading of immunization works better than that of avoidance.

Spatial-temporal characteristics of epidemic spread in-out flow——Using SARS epidemic in Beijing as a case study

For better detecting the spatial-temporal change mode of individual susceptible-infected-symptomatic-treated-recovered epidemic progress and the characteristics of information/material flow in the epidemic spread network between regions,the epidemic spread mechanism of virus input and output was explored based on individuals and spatial regions.Three typical spatial information parameters including working unit/address,onset location and reporting unit were selected and SARS epidemic spread in-out flow in Beijing was defined based on the SARS epidemiological investigation data in China from 2002 to 2003 while its epidemiological characteristics were discussed.Furthermore,by the methods of spatial-temporal statistical analysis and network characteristic analysis,spatial-temporal high-risk hotspots and network structure characteristics of Beijing outer in-out flow were explored,and spatial autocorrelation/heterogeneity,spatial-temporal evolutive rules and structure characteristics of the spread network of Beijing inner in-out flow were comprehensively analyzed.The results show that(1)The outer input flow of SARS epidemic in Beijing concentrated on Shanxi and Guangdong provinces,but the outer output flow was disperse and mainly includes several north provinces such as Guangdong and Shandong.And the control measurement should focus on the early and interim progress of SARS breakout.(2)The inner output cases had significant positive autocorrelative characteristics in the whole studied region,and the high-risk population was young and middle-aged people with ages from 20 to 60 and occupations of medicine and civilian labourer.(3)The downtown districts were main high-risk hotspots of SARS epidemic in Beijing,the northwest suburban districts/counties were secondary high-risk hotspots,and northeast suburban areas were relatively safe.(4)The district/county nodes in inner spread network showed small-world characteristics and information/material flow had notable heterogeneity.The suburban Tongzhou and Changping districts were the underlying high-risk regions,and several suburban districts such as Shunyi and Huairou were the relatively low-risk safe regions as they carried out minority information/material flow.The exploration and analysis based on epidemic spread in-out flow help better detect and discover the potential spatial-temporal evolutive rules and characteristics of SARS epidemic,and provide a more effective theoretical basis for emergency/control measurements and decision-making.

Epidemic spreading on scale-free networks with diversity of node anti-attack abilities

In this article, a modified susceptible-infected-removed （SIR） model is proposed to study the influence of diversity of node anti-attack abilities on the threshold of propagation in scale-free networks. In particular, a vulnerability function related to node degree is introduced into the model to describe the diversity of a node anti-attack ability. Analytical results are derived using the mean-field theory and it is observed that the diversity of anti-attack of nodes in scale-free networks can increase effectively the threshold of epidemic propagation. The simulation results agree with the analytical results. The results show that the vulnerability functions can help adopt appropriate immunization strategies.

Modeling epidemic spread in transportation networks:A review

The emergence of novel infectious diseases has become a serious global problem.Convenient transportation networks lead to rapid mobilization in the context of globalization,which is an important factor underlying the rapid spread of infectious diseases.Transportation systems can cause the transmission of viruses during the epidemic period,but they also support the reopening of economies after the epidemic.Understanding the mechanism of the impact of mobility on the spread of infectious diseases is thus important,as is establishing the risk model of the spread of infectious diseases in transportation networks.In this study,the basic structure and application of various epidemic spread models are reviewed,including mathematical models,statistical models,network-based models,and simulation models.The advantages and limitations of model applications within transportation systems are analyzed,including dynamic characteristics of epidemic transmission and decision supports for management and control.Lastly,research trends and prospects are discussed.It is suggested that there is a need for more in-depth research to examine the mutual feedback mechanism of epidemics and individual behavior,as well as the proposal and evaluation of intervention measures.The findings in this study can help evaluate disease intervention strategies,provide decision supports for transport policy during the epidemic period,and ameliorate the deficiencies of the existing system.

Spreading behavior of SIS epidemic model on networks with dynamical topology

Based on the two-dimensional regular lattice,a modified SIS(Susceptible-Infected-Susceptible)epidemic model with motion rules is presented to study the spreading behavior on networks with dynamical topology.The mean-field theory is utilized to analyze the critical threshold(λc)of epidemic spreading under the randomly mixing conditions.It is found that λc is only related with the population density within the lattice.Large-scale numerical simulations are carried out to verify the mean-field results,and it is observed that the long-range probability p largely affects the epidemic spreading behavior.In addition,the effect of the dual time scales on epidemic spreading is also investigated by the simulations,and it is shown that the dual time scales accelerate the dynamic spreading behavior.The results indicate that the model with motion can help us to further understand the real epidemics.

Epidemic spreading behavior with time delay on local-world evolving networks

An improved susceptible-infected-susceptible(SIS)model in the local-world evolving network model is presented to study the epidemic spreading behavior with time delay,which is added into the infected phase.The local-world evolving model displays a transition from the exponential network to the scale-free network with respect to the degree distribution.Two typical delay regimes,i.e.,uniform and degree-dependent delays are incorporated into the SIS epidemic model to investigate the epidemic infection processes in the local-world net-work model.The results indicate that the infection delay will promote the epidemic outbreaks,increase the prevalence and reduce the critical threshold of epidemic spreading.It is also found that local-world size M will considerably influence the epidemic spreading behavior with time delay in the local-world network through large-scale numerical simulations.Simulation results are also of relevance to fight epidemic outbreaks.

A cellular automata model of epidemics of a heterogeneous susceptibility

In this paper we present a model with spatial heterogeneity based on cellular automata （CA）. In the model we consider the relevant heterogeneity of host （susceptible） mixing and the natural birth rate. We divide the susceptible population into three groups according to the immunity of each individual based on the classical susceptible-infectedremoved （SIR） epidemic models, and consider the spread of an infectious disease transmitted by direct contact among humans and vectors that have not an incubation period to become infectious. We test the local stability and instability of the disease-free equilibrium by the spectrum radii of Jacobian. The simulation shows that the structure of the nearest neighbour size of the cell （or the degree of the scale-free networks） plays a very important role in the spread properties of infectious disease. The positive equilibrium of the infections versus the neighbour size follows the third power law if an endemic equilibrium point exists. Finally, we analyse the feature of the infection waves for the homogeneity and heterogeneous cases respectively.

Global stability of a susceptible-infected-susceptible epidemic model on networks with individual awareness

Recent research results indicate that individual awareness can play an important influence on epidemic spreading in networks. By local stability analysis, a significant conclusion is that the embedded awareness in an epidemic network can increase its epidemic threshold. In this paper, by using limit theory and dynamical system theory, we further give global stability analysis of a susceptible-infected-susceptible （SIS） epidemic model on networks with awareness. Results show that the obtained epidemic threshold is also a global stability condition for its endemic equilibrium, which implies the embedded awareness can enhance the epidemic threshold globally. Some numerical examples are presented to verify the theoretical results.

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.

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 % the velocity modulus of the agent v, and the infection rate a. 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 a increase.

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.

Random walk immunization strategy on scale-free networks

A novel immunization strategy called the random walk immunization strategy on scale-free networks is proposed. Different from other known immunization strategies, this strategy works as follows： a node is randomly chosen from the network. Starting from this node, randomly walk to one of its neighbor node; if the present node is not immunized, then immunize it and continue the random walk; otherwise go back to the previous node and randomly walk again. This process is repeated until a certain fraction of nodes is immunized. By theoretical analysis and numerical simulations, we found that this strategy is very effective in comparison with the other known immunization strategies.

A two-phase fluid model for epidemic flow

We propose a new mathematical and computational modeling framework that in-corporates fluid dynamics to study the spatial spread of infectious diseases.We model the susceptible and infected populations as two inviscid fluids which interact with each other.Their motion at the macroscopic level characterizes the progression and spread of the epidemic.To implement the two-phase flow model,we employ high-order numerical methods from computational fluid dynamics.We apply this model to simulate the COVID-19 outbreaks in the city of Wuhan in China and the state of Tennessee in the US.Our modeling and simulation framework allows us to conduct a detailed investigation into the complex spatiotemporal dynamics related to the transmission and spread of COVID-19.

Stochastic modeling,analysis,and simulation of the COVID-19 pandemic with explicit behavioral changes in Bogota:Acase study

In this paper,a stochastic epidemiological model is presented as an extension of a compartmental SEIR model with random perturbations to analyze the dynamics of the COVID-19 pandemic in the city of Bogota D.C.,Colombia.This model incorporates thespread of COVID-19 impacted by social behaviors in the population and allows for projecting the number of infected,recovered,and deceased individuals considering the mitigation measures,namely confinement and partial relaxed restrictions.Also,the role of randomness using the concept of Brownian motion is emphasized to explain the behavior of the population.Computational experiments for the stochastic model with random perturbations were performed,and the model is validated through numerical simulations for actual data from Bogota D.C.