One of the famous results of network science states that networks with heterogeneous connectivity are more susceptible to epidemic spreading than their more homogeneous counterparts. In particular, in networks of identical nodes it has been shown that heterogeneity can lower the epidemic threshold at which epidemics can invade the system. Network heterogeneity can thus allow diseases with lower transmission probabilities to persist and spread. Here, we point out that for real world applications, this result should not be regarded independently of the intra-individual heterogeneity between people. Our results show that, if heterogeneity among people is taken into account, networks that are more heterogeneous in connectivity can be more resistant to epidemic spreading. We study a susceptible-infected-susceptible model with adaptive disease avoidance. Results from this model suggest that this reversal of the effect of network heterogeneity is likely to occur in populations in which the individuals are aware of their subjective disease risk. For epidemiology, this implies that network heterogeneity should not be studied in isolation.
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 extens…
In this work, we aim to understand the influence of the heterogeneity of infection rates on the Susceptible-Infected-Susceptible (SIS) epidemic spreading. Employing the classic SIS model as the benchmark, we study the influence of the independently identically distributed infection rates on the average fraction of infected nodes in the metastable state. The log-normal, gamma and a newly designed distributions are considered for infection rates. We find that, when the recovery rate is small, i.e.\ the epidemic spreads out in both homogeneous and heterogeneous cases: 1) the heterogeneity of infe…
We show that the mean field equations for the SIR epidemic can be exactly solved for a network with arbitrary degree distribution. Our exact solution consists of reducing the dynamics to a lone first order differential equation, which has a solution in terms of an integral over functions dependent on the degree distribution of the network, and reconstructing all mean field functions of interest from this integral. Irreversibility of the SIR epidemic is crucial for the solution. We also find exact solutions to the sexually transmitted disease SI epidemic on bipartite graphs, to a simplified rum…
In this paper we investigate a susceptible-infected-susceptible (SIS) epidemic model describing data dissemination in opportunistic networks with heterogeneous setting of transmission parameters. We obtained the estimation of the final epidemic size assuming that amount of data transferred between network nodes possesses a Pareto distribution, implying scale-free properties. In this context, more heterogeneity in susceptibility means the less severe epidemic progression, and, on the contrary, more heterogeneity in infectivity leads to more severe epidemics -- assuming that the other parameter …
Infectious diseases outbreaks are often characterized by a spatial component induced by hosts' distribution, mobility, and interactions. Spatial models that incorporate hosts' movements are being used to describe these processes, to investigate the conditions for propagation, and to predict the spatial spread. Several assumptions are being considered to model hosts' movements, ranging from permanent movements to daily commuting, where the time spent at destination is either infinite or assumes a homogeneous fixed value, respectively. Prompted by empirical evidence, here we introduce a general …
Recent empirical observations suggest a heterogeneous nature of human activities. The heavy-tailed inter-event time distribution at population level is well accepted, while whether the individual acts in a heterogeneous way is still under debate. Motivated by the impact of temporal heterogeneity of human activities on epidemic spreading, this paper studies the susceptible-infected model on a fully mixed population, where each individual acts in a completely homogeneous way but different individuals have different mean activities. Extensive simulations show that the heterogeneity of activities …
We propose an efficient strategy to suppress epidemic explosion in heterogeneous metapopulation networks, wherein each node represents a subpopulation with any number of individuals and is assigned a curing rate that is proportional to $k^α$ with $k$ the node degree and $α$ an adjustable parameter. We have performed stochastic simulations of the dynamical reaction-diffusion processes associated with the susceptible-infected-susceptible model in scale-free networks. We found that the epidemic threshold reaches a maximum when the exponent $α$ is tuned to be $α_{opt}\simeq 1.3$. This nontrivial p…
We study a multi-type SIR epidemic process among a heterogeneous population that interacts through a network. When we base social contact on a random graph with given vertex degrees, we give limit theorems on the fraction of infected individuals. For a given social distancing individual strategies, we establish the epidemic reproduction number $R_0$ which can be used to identify network vulnerability and inform vaccination policies. In the second part of the paper we study the equilibrium of the social distancing game, in which individuals choose their social distancing level according to an a…
The design of an efficient curing policy, able to stem an epidemic process at an affordable cost, has to account for the structure of the population contact network supporting the contagious process. Thus, we tackle the problem of allocating recovery resources among the population, at the lowest cost possible to prevent the epidemic from persisting indefinitely in the network. Specifically, we analyze a susceptible-infected-susceptible epidemic process spreading over a weighted graph, by means of a first-order mean-field approximation. First, we describe the influence of the contact network on…
We develop a theory for the susceptible-infected-susceptible (SIS) epidemic model on networks that incorporate both network structure and dynamic correlations. This theory can account for the multistage onset of the epidemic phase in scale-free networks. This phenomenon is characterized by multiple peaks in the susceptibility as a function of the infection rate. It can be explained by that, even under the global epidemic threshold, a hub can sustain the epidemics for an extended period. Moreover, our approach improves theoretical calculations of prevalence close to the threshold in heterogeneo…
We study SIS epidemic spreading processes unfolding on a recent generalisation of the activity-driven modelling framework. In this model of time-varying networks each node is described by two variables: activity and attractiveness. The first, describes the propensity to form connections. The second, defines the propensity to attract them. We derive analytically the epidemic threshold considering the timescale driving the evolution of contacts and the contagion as comparable. The solutions are general and hold for any joint distribution of activity and attractiveness. The theoretical picture is…
Modeling spreading processes in complex random networks plays an essential role in understanding and prediction of many real phenomena like epidemics or rumor spreading. The dynamics of such systems may be represented algorithmically by Monte-Carlo simulations on graphs or by ordinary differential equations (ODEs). Despite many results in the area of network modeling the selection of the best computational representation of the model dynamics remains a challenge. While a closed form description is often straightforward to derive, it generally cannot be solved analytically; as a consequence the…
The spread of disease on complex networks has attracted widely attention in the physics community. Recent works have demonstrated that heterogeneous degree and weight distributions have a significant influence on the epidemic dynamics. In this study, a novel edge-weight based compartmental approach is developed to estimate the epidemic threshold and epidemic size (final infected density) on networks with general degree and weight distributions, and a remarkable agreement with numerics is obtained. Even in complex network with the strong heterogeneous degree and weight distributions, this appro…
Realistic human contact networks capable of spreading infectious disease, for example studied in social contact surveys, exhibit both significant degree heterogeneity and clustering, both of which greatly affect epidemic dynamics. To understand the joint effects of these two network properties on epidemic dynamics, the effective degree model of Lindquist et al. is reformulated with a new moment closure to apply to highly clustered networks. A simulation study comparing alternative ODE models and stochastic simulations is performed for SIR (Susceptible-Infected-Removed) epidemic dynamics, inclu…
We study the impact of contact heterogeneity on epidemic dynamics. A system characterized by multiple susceptible populations is considered. The description of the spread of an infectious disease is obtained through the study of a system of Boltzmann-type equations for the number densities of social contacts of the introduced compartments. A macroscopic system of equations characterizing observable effects of the epidemic is then derived to assess the impact of contact heterogeneity.
Motivated by the importance of individual differences in risk perception and behavior change in people's responses to infectious disease outbreaks (particularly the ongoing COVID-19 pandemic), we propose a heterogeneous Disease-Behavior-Information (hDBI) transmission model, in which people's risk of getting infected is influenced by information diffusion, behavior change, and disease transmission. We use both a mean-field approximation and Monte Carlo simulations to analyze the dynamics of the model. Information diffusion influences behavior change by allowing people to be aware of the diseas…
Infection can spread easily on networks with heterogeneous degree distribution. Here, we considered targeted immunization on such networks, wherein a fraction of individuals with the highest connectivity are immunized. To quantify the effect of this targeted immunization approach on population immunity, we proposed a method using the type reproduction number. Consequently, we derived a precise and simple formula that can yield the immunization threshold, which, to the best of our knowledge, is the first such result presented in the literature.
Epidemic spreading and cascading failure are two important dynamical processes over complex networks. They have been investigated separately for a long history. But in the real world, these two dynamics sometimes may interact with each other. In this paper, we explore a model combined with SIR epidemic spreading model and local loads sharing cascading failure model. There exists a critical value of tolerance parameter that whether the epidemic with high infection probability can spread out and infect a fraction of the network in this model. When the tolerance parameter is smaller than the crit…
It is often useful to represent the infectious dynamics of mobile agents by metapopulation models. In such a model, metapopulations form a static network, and individuals migrate from one metapopulation to another. It is known that heterogeneous degree distributions of metapopulation networks decrease the epidemic threshold above which epidemic spreads can occur. We investigate the combined effect of heterogeneous degree distributions and diffusion on epidemics in metapopulation networks. We show that for arbitrary heterogeneous networks, diffusion suppresses epidemics in the sense of an incre…
Since roughly a decade ago, network science has focused among others on the problem of how the spreading of diseases depends on structural patterns. Here, we contribute to further advance our understanding of epidemic spreading processes by proposing a non-perturbative formulation of the heterogeneous mean field approach that has been commonly used in the physics literature to deal with this kind of spreading phenomena. The non-perturbative equations we propose have no assumption about the proximity of the system to the epidemic threshold, nor any linear approximation of the dynamics. In parti…
Contact tracing via digital tracking applications installed on mobile phones is an important tool for controlling epidemic spreading. Its effectivity can be quantified by modifying the standard methodology for analyzing percolation and connectivity of contact networks. We apply this framework to networks with varying degree distributions, numbers of application users, and probabilities of quarantine failures. Further, we study structured populations with homophily and heterophily and the possibility of degree-targeted application distribution. Our results are based on a combination of explicit…
Drug resistance and strong contacts actually play crucial roles in epidemic spread in complex systems. Nevertheless, neither theoretical model or methodology is proposed to address this. We thus consider an edge-based epidemic spread model considering the two key ingredients, in which the contacts are grouped into two classes: strong contacts and normal ones. Next, we present a unified edge-based compartmental approach to the spread dynamics on Erdös-Rényi (ER) networks and validate its results by extensive numerical simulations. In case that epidemic is totally drug-resistant, we both numeric…
We consider epidemic extinction in finite networks with broad variation in local connectivity. Generalizing the theory of large fluctuations to random networks with a given degree distribution, we are able to predict the most probable, or optimal, paths to extinction in various configurations, including truncated power-laws. We find that paths for heterogeneous networks follow a limiting form in which infection first decreases in low-degree nodes, which triggers a rapid extinction in high- degree nodes, and finishes with a residual low-degree extinction. The usefulness of the approach is furth…
Numerical simulation of continuous-time Markovian processes is an essential and widely applied tool in the investigation of epidemic spreading on complex networks. Due to the high heterogeneity of the connectivity structure through which epidemics is transmitted, efficient and accurate implementations of generic epidemic processes are not trivial and deviations from statistically exact prescriptions can lead to uncontrolled biases. Based on the Gillespie algorithm (GA), in which only steps that change the state are considered, we develop numerical recipes and describe their computer implementa…
The dynamics of ecological as well as chemical systems may depend on heterogeneous configurations. Heterogeneity in reaction-diffusion systems often increase modelling and simulating difficulties when non-linear effects are present. One synthetic epidemic system with short range heterogeneous composition is modelled and its space-time evolution studied using maximum heterogeneity details. Two other modelling alternatives are applied, one of them using elementary mean-field variables, one other using non-localized geometrical parameters, so avoiding the limitations of the used mean-field model,…
Empirical studies suggest that contact patterns follow heterogeneous inter-event times, meaning that intervals of high activity are followed by periods of inactivity. Combined with birth and death of individuals, these temporal constraints affect the spread of infections in a non-trivial way and are dependent on the particular contact dynamics. We propose a stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event times, and leave and enter the system at fixed rates. We study how these temporal properties affect the prevalence…
Network-based epidemic models have been extensively employed to understand the spread of infectious diseases, but have generally overlooked the fact that most realistic networks are dynamical rather than static. In this paper, we study a susceptible-infected-susceptible epidemic model with vaccination in a dynamical contact network of moving individuals, where we regard mobile individuals as random walkers that are allowed to perform long-range jumps. Different from previous studies of epidemics in a random walk network with a constant interaction radius, we consider the scenario where the ind…
In the present work the spread of epidemic is studied over complex networks which are characterized by power law degree distribution of links and heterogeneous rate of disease transmission. The random allocation of epidemic transmission rates to the nodes results in the heterogeneity, which in turn causes the segregation of nodes in terms of various sub populations. The aim of the study is to gain microscopic insight into the effect of interactions among various sub populations in the spreading processes of disease over such networks. The discrete time Markov chain method based upon the suscep…
Recent Covid-19 pandemic has demonstrated the need of efficient epidemic outbreak management. We study the optimal control problem of minimizing the fraction of infected population by applying vaccination and treatment control strategies, while at the same time minimizing the cost of implementing them. We model the epidemic using the degree based Susceptible-Infected-Recovered (SIR) compartmental model. We study the impact of varying network topologies on the optimal epidemic management strategies and present results for the Erdos-Renyi, scale free, and real world networks. For efficient compu…
We study the dynamics of epidemic and reaction-diffusion processes in metapopulation models with heterogeneous connectivity pattern. In SIR-like processes, along with the standard local epidemic threshold, the system exhibits a global invasion threshold. We provide an explicit expression of the threshold that sets a critical value of the diffusion/mobility rate below which the epidemic is not able to spread to a macroscopic fraction of subpopulations. The invasion threshold is found to be affected by the topological fluctuations of the metapopulation network. The presented results provide a ge…
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 symbolising 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 metaopoluation in the long time limit, as well as the relevant parameters characterising the epidemic threshold that separates an epidemic from a non-epidemic phase. W…
We explore how heterogeneity in the intensity of interactions between people affects epidemic spreading. For that, we study the susceptible-infected-susceptible model on a complex network, where a link connecting individuals $i$ and $j$ is endowed with an infection rate $β_{ij} = λw_{ij}$ proportional to the intensity of their contact $w_{ij}$, with a distribution $P(w_{ij})$ taken from face-to-face experiments analyzed in Cattuto $et\;al.$ (PLoS ONE 5, e11596, 2010). We find an extremely slow decay of the fraction of infected individuals, for a wide range of the control parameter $λ$. Using a…
Most epidemic models assume equal mixing among members of a population. An alternative approach is to model a population as random network in which individuals may have heterogeneous connectivity. This paper builds on previous research by describing the exact dynamical behavior of epidemics as they occur in random networks. A system of nonlinear differential equations is presented which describes the behavior of epidemics spreading through random networks with arbitrary degree distributions. The degree distribution is observed to have significant impact on both the final size and time scale of…
A social (sexual) network is modeled by an extension of the configuration model to the situation where edges have weights, e.g. reflecting the number of sex-contacts between the individuals. An epidemic model is defined on the network such that individuals are heterogeneous in terms of how susceptible and infectious they are. The basic reproduction number R_0 is derived and studied for various examples, but also the size and probability of a major outbreak. The qualitative conclusion is that R_0 gets larger as the community becomes more heterogeneous but that different heterogeneities (degree …
The epidemic spreading has been widely studied when each node may get infected by an infected neighbor with the same rate. However, the infection rate between a pair of nodes is usually heterogeneous and even correlated with their nodal degrees in the contact network. We aim to understand how such correlated heterogeneous infection rates influence the spreading on different network topologies. Motivated by real-world datasets, we propose a correlated heterogeneous Susceptible-Infected-Susceptible model which assumes that the infection rate $β_{ij}(=β_{ji})$ between node $i$ and $j$ is correlat…
This study is concerned with the dynamical behaviors of epidemic spreading over a two-layered interconnected network. Three models in different levels are proposed to describe cooperative spreading processes over the interconnected network, wherein the disease in one network can spread to the other. Theoretical analysis is provided for each model to reveal that the global epidemic threshold in the interconnected network is not larger than the epidemic thresholds for the two isolated layered networks. In particular, in an interconnected homogenous network, detailed theoretical analysis is prese…
The evolution of network structure and the spreading of epidemic are common coexistent dynamical processes. In most cases, network structure is treated either static or time-varying, supposing the whole network is observed in a same time window. In this paper, we consider the epidemic spreading on a network consisting of both static and time-varying structures. At meanwhile, the time-varying part and the epidemic spreading are supposed to be of the same time scale. We introduce a static and activity driven coupling (SADC) network model to characterize the coupling between static (strong) struc…
Networks of contacts capable of spreading infectious diseases are often observed to be highly heterogeneous, with the majority of individuals having fewer contacts than the mean, and a significant minority having relatively very many contacts. We derive a two-dimensional diffusion model for the full temporal behavior of the stochastic susceptible-infectious-recovered (SIR) model on such a network, by making use of a time-scale separation in the deterministic limit of the dynamics. This low-dimensional process is an accurate approximation to the full model in the limit of large populations, eve…
We propose a deterministic compartmental model of infectious disease which considers the test-kits as an important ingredient for the suppression and mitigation of epidemics. A rigorous simulation (with analytical argument) is provided to reveal the effective reduction of final outbreak size and peak of infection as a function of basic reproduction number in a single patch. Further, to study the impact of long and short-distance human migration among the patches, we have considered heterogeneous networks where the linear diffusive connectivity is determined by the network link structure. We nu…
The nodes in communication networks are possibly and most likely equipped with different recovery resources, which allow them to recover from a virus with different rates. In this paper, we aim to understand know how to allocate the limited recovery resources to efficiently prevent the spreading of epidemics. We study the susceptible-infected-susceptible (SIS) epidemic model on directed scale-free networks. In the classic SIS model, a susceptible node can be infected by an infected neighbor with the infection rate $β$ and an infected node can be recovered to be susceptible again with the recov…
The spatial structure of populations is a key element in the understanding of the large scale spreading of epidemics. Motivated by the recent empirical evidence on the heterogeneous properties of transportation and commuting patterns among urban areas, we present a thorough analysis of the behavior of infectious diseases in metapopulation models characterized by heterogeneous connectivity and mobility patterns. We derive the basic reaction-diffusion equation describing the metapopulation system at the mechanistic level and derive an early stage dynamics approximation for the subpopulation inva…
We study numerically how the structures of distinct networks influence the epidemic dynamics in contact process. We first find that the variability difference between homogeneous and heterogeneous networks is very narrow, although the heterogeneous structures can induce the lighter prevalence. Contrary to non-community networks, strong community structures can cause the secondary outbreak of prevalence and two peaks of variability appeared. Especially in the local community, the extraordinarily large variability in early stage of the outbreak makes the prediction of epidemic spreading hard. Im…
Epidemics in large complete networks is well established. In contrast, we consider epidemics in non-complete networks. We establish the fluid limit macroscopic dynamics of a multi-virus spread over a multipartite network as the number of nodes at each partite or island grows large. The virus spread follows a peer-to-peer random rule of infection in line with the Harris contact process. The model conforms to an SIS (susceptible-infected-susceptible) type, where a node is either infected or it is healthy and prone to be infected. The local (at node level) random infection model induces the emerg…
Single virus epidemics over complete networks are widely explored in the literature as the fraction of infected nodes is, under appropriate microscopic modeling of the virus infection, a Markov process. With non-complete networks, this macroscopic variable is no longer Markov. In this paper, we study virus diffusion, in particular, multi-virus epidemics, over non-complete stochastic networks. We focus on multipartite networks. In companying work http://arxiv.org/abs/1306.6198, we show that the peer-to-peer local random rules of virus infection lead, in the limit of large multipartite networks,…
The adoption of prophylaxis attitudes, such as social isolation and use of face masks, to mitigate epidemic outbreaks strongly depends on the support of the population. In this work, we investigate a susceptible-infected-recovered (SIR) epidemic model, in which the epidemiological perception of the environment can adapt the behavior of susceptible individuals towards preventive behavior. {Two compartments of susceptible individuals are considered, to distinguish those that adopt or not prophylaxis attitudes.} Two rules, depending on local and global epidemic prevalence, for the spread of the e…
Many real networks are not isolated from each other but form networks of networks, often interrelated in non trivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately, and even when the number of coupling connections is small. Our analytic results are successf…
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 …
Recently, some studies have revealed that non-Poissonian statistics of human behaviors stem from the hierarchical geographical network structure. On this view, we focus on epidemic spreading in the hierarchical geographical networks, and study how two distinct contact patterns (i. e., homogeneous time delay (HOTD) and heterogeneous time delay (HETD) associated with geographical distance) influence the spreading speed and the variability of outbreaks. We find that, compared with HOTD and null model, correlations between time delay and network hierarchy in HETD remarkably slow down epidemic spre…
In this paper, we propose a modified susceptible-infected-recovered (SIR) model, in which each node is assigned with an identical capability of active contacts, $A$, at each time step. In contrast to the previous studies, we find that on scale-free networks, the density of the recovered individuals in the present model shows a threshold behavior. We obtain the analytical results using the mean-field theory and find that the threshold value equals 1/A, indicating that the threshold value is independent of the topology of the underlying network. The simulations agree well with the analytic resul…
We study the SIR ("susceptible, infected, removed/recovered") model on directed graphs with heterogeneous transmission probabilities within the message-passing approximation. We characterize the percolation transition, predict cluster size distributions and suggest vaccination strategies. All predictions are compared to numerical simulations on real networks. The percolation threshold which we predict is a rigorous lower bound to the threshold on real networks. For large, locally tree-like networks, our predictions agree very well with the numerical data.
In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and sociotechnical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and nonequilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel a…
We study the effect of heterogeneous temporal activations on epidemic spreading in temporal networks. We focus on the susceptible-infected-susceptible (SIS) model on activity-driven networks with burstiness. By using an activity-based mean-field approach, we derive a closed analytical form for the epidemic threshold for arbitrary activity and inter-event time distributions. We show that, as expected, burstiness lowers the epidemic threshold while its effect on prevalence is twofold. In low-infective systems burstiness raises the average infection probability, while it weakens epidemic spreadin…
We show how one can trace in a systematic way the coarse-grained solutions of individual-based stochastic epidemic models evolving on heterogeneous complex networks with respect to their topological characteristics. In particular, we have developed algorithms that allow the tuning of the transitivity (clustering coefficient) and the average mean-path length allowing the investigation of the "pure" impacts of the two characteristics on the emergent behavior of detailed epidemic models. The framework could be used to shed more light into the influence of weak and strong social ties on epidemic s…
