Mathematical models for epidemic spreading on complex networks (original) (raw)

Epidemic dynamics on semi-directed complex networks

2013

In this paper an SIS model for epidemic spreading on semi-directed networks is established, which can be used to examine and compare the impact of undirected and directed contacts on disease spread. The model is analyzed for the case of uncorrelated semi-directed networks, and the basic reproduction number R 0 is obtained analytically. We verify that the R 0 contains the outbreak threshold on undirected networks and directed networks as special cases. It is proved that if R 0 < 1 then the disease-free equilibrium is globally asymptotically stable, otherwise the disease-free equilibrium is unstable and the unique endemic equilibrium exists, which is globally asymptotically stable. Finally the numerical simulations holds for these analytical results are given. Let pðk; l; mÞ be the probability that a randomly selected node has k incoming edges, l outgoing edges, and m undirected edges. One can then calculate the marginal distributions of in-degree, out-degree and undirected degree. Denote

Modeling Epidemic Spreading in Complex Networks: Concurrency and Traffic

2012

The study of complex networks sheds light on the relation between the structure and function of complex systems. One remarkable result is the absence of an epidemic threshold in infinite-size scale-free networks, which implies that any infection will perpetually propagate regardless of the spreading rate. However, realworld networks are finite and experience indicates that infections do have a finite lifetime. In this chapter, we will provide with two new approaches to cope with the problem of concurrency and traffic in the spread of epidemics. We show that the epidemic incidence is shaped by contact flow or traffic conditions. Contrary to the classical assumption that infections are transmitted as a diffusive process from nodes to all neighbors, we instead consider the scenario in which epidemic pathways are defined and driven by flows. Extensive numerical simulations and theoretical predictions show that whether a threshold exists or not depends directly on contact flow conditions. Two extreme cases are identified. In the case of low traffic, an epidemic threshold shows up, while for very intense flow, no epidemic threshold appears. In this way, the classical mean-field theory for epidemic spreading in scale free networks is recovered as a particular case of the proposed approach. Our results explain why some infections persist with low prevalence in scalefree networks, and provide a novel conceptual framework to understand dynamical processes on complex networks.

Spreading dynamics on complex networks: a general stochastic approach

Journal of Mathematical Biology, 2013

Dynamics on networks is considered from the perspective of Markov stochastic processes. We partially describe the state of the system through network motifs and infer any missing data using the available information. This versatile approach is especially well adapted for modelling spreading processes and/or population dynamics. In particular, the generality of our systematic framework and the fact that its assumptions are explicitly stated suggests that it could be used as a common ground for comparing existing epidemics models too complex for direct comparison, such as agentbased computer simulations. We provide many examples for the special cases of susceptible-infectious-susceptible (SIS) and susceptible-infectious-removed (SIR) dynamics (e.g., epidemics propagation) and we observe multiple situations where accurate results may be obtained at low computational cost. Our perspective reveals a subtle balance between the complex requirements of a realistic model and its basic assumptions.

Discrete-time Markov chain approach to contact-based disease spreading in complex networks

2010

Many epidemic processes in networks spread by stochastic contacts among their connected vertices. There are two limiting cases widely analyzed in the physics literature, the so-called contact process (CP) where the contagion is expanded at a certain rate from an infected vertex to one neighbor at a time, and the reactive process (RP) in which an infected individual effectively contacts all its neighbors to expand the epidemics. However, a more realistic scenario is obtained from the interpolation between these two cases, considering a certain number of stochastic contacts per unit time. Here we propose a discrete-time formulation of the problem of contact-based epidemic spreading. We resolve a family of models, parameterized by the number of stochastic contact trials per unit time, that range from the CP to the RP. In contrast to the common heterogeneous mean-field approach, we focus on the probability of infection of individual nodes. Using this formulation, we can construct the whole phase diagram of the different infection models and determine their critical properties.

Unification of theoretical approaches for epidemic spreading on complex networks

Reports on progress in physics. Physical Society (Great Britain), 2017

Models of epidemic spreading on complex networks have attracted great attention among researchers in physics, mathematics, and epidemiology due to their success in predicting and controlling scenarios of epidemic spreading in real-world scenarios. To understand the interplay between epidemic spreading and the topology of a contact network, several outstanding theoretical approaches have been developed. An accurate theoretical approach describing the spreading dynamics must take both the network topology and dynamical correlations into consideration at the expense of increasing the complexity of the equations. In this short survey we unify the most widely used theoretical approaches for epidemic spreading on complex networks in terms of increasing complexity, including the mean-field, the heterogeneous mean-field, the quench mean-field, dynamical message-passing, link percolation, and pairwise approximation. We build connections among these approaches to provide new insights into dev...

Different epidemic models on complex networks

2009

Models for diseases spreading are not just limited to SIS or SIR. For instance, for the spreading of AIDS/HIV, the susceptible individuals can be classified into different cases according to their immunity, and similarly, the infected individuals can be sorted into different classes according to their infectivity. Moreover, some diseases may develop through several stages. Many authors have shown that the individuals' relation can be viewed as a complex network. So in this paper, in order to better explain the dynamical behavior of epidemics, we consider different epidemic models on complex networks, and obtain the epidemic threshold for each case. Finally, we present numerical simulations for each case to verify our results.

Effect of Heterogeneous Transmission Rate on Epidemic Spreading Over Scale Free Networks

arXiv: Physics and Society, 2016

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 susceptible infected susceptible (SIS) model of diseases transmission has been used to describe the spreading of epidemic over the networks. The study is parameterized in terms of variable lambda\lambdalambda, defined as the number of contacts a node makes with the fraction of its neighboring nodes. From the simulation results, it is found that the spread of epidemic on such networks is critical in terms of nu...

Modelling Spreading Process Induced by Agent Mobility in Complex Networks

IEEE Transactions on Network Science and Engineering, 2018

Most conventional epidemic models assume contact-based contagion process. We depart from this assumption and study epidemic spreading process in networks caused by agents acting as carrier of infection. These agents traverse from origins to destinations following specific paths in a network and in the process, infecting the sites they travel across. We focus our work on the Susceptible-Infected-Removed (SIR) epidemic model and use continuous-time Markov chain analysis to model the impact of such agent mobility induced contagion mechanics by taking into account the state transitions of each node individually, as oppose to most conventional epidemic approaches which usually consider the mean aggregated behavior of all nodes. Our approach makes one mean field approximation to reduce complexity from exponential to polynomial. We study both network-wide properties such as epidemic threshold as well as individual node vulnerability under such agent assisted infection spreading process. Furthermore, we provide a first order approximation on the agents' vulnerability since infection is bi-directional. We compare our analysis of spreading process induced by agent mobility against contact-based epidemic model via a case study on London Underground network, the second busiest metro system in Europe, with real dataset recording commuters' activities in the system. We highlight the key differences in the spreading patterns between the contact-based vs. agent assisted spreading models. Specifically, we show that our model predicts greater spreading radius than conventional contact-based model due to agents' movements. Another interesting finding is that, in contrast to contact-based model where nodes located more centrally in a network are proportionally more prone to infection, our model shows no such strict correlation as in our model, nodes may not be highly susceptible even located at the heart of the network and vice versa.

Epidemics spreading in interconnected complex networks

Physics Letters A, 2012

We study epidemic spreading in two interconnected complex networks. It is found that in our model the epidemic threshold is always lower than that in any of the two standalone networks. Detailed theoretical analysis is proposed which allows quick and accurate calculations of epidemic threshold and average outbreak/epidemic size. Theoretical analysis and simulation results show that, generally speaking, the epidemic size is not significantly affected by the inter-network correlation. In interdependent networks which can be viewed as a special case of interconnected networks, however, impacts of inter-network correlation on the epidemic threshold and outbreak size are more significant.