Epidemic processes in complex networks (1408.2701v2)

Abstract: 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 analytical frameworks, and it has produced results of conceptual and practical relevance. A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, mathematicians, epidemiologists, computer, and social scientists share a common interest in studying epidemic spreading and rely on similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while focusing on the main results and the paradigmatic models in infectious disease modeling, the major results concerning generalized social contagion processes are also presented. Finally, the research activity at the forefront in the study of epidemic spreading in coevolving, coupled, and time-varying networks is reported.

Epidemic Processes in Complex Networks

The paper "Epidemic Processes in Complex Networks" presents an extensive review of the theoretical and numerical findings on epidemic spreading within the context of complex networks. The work underscores the critical role that the connectivity structure of networks plays in the diffusion of diseases and broader contagion phenomena.

Theoretical Frameworks and Models

The paper extensively discusses network characteristics and how they influence epidemic processes. Initially, the focus is on defining fundamental network metrics such as the degree distribution $P(k)$, clustering coefficients, path lengths, and assortativity. These metrics form the basis for understanding the heterogeneity present in real-world networks.

Two primary models to capture disease spreading among individuals are discussed: the Susceptible-Infected-Susceptible (SIS) and Susceptible-Infected-Removed (SIR) models. Each model presents distinct challenges and behaviors depending on the network's structure:

1. SIS Model: This model, where recovered individuals can be re-infected, is discussed in great detail. The degree-based mean-field (DBMF) and individual-based mean-field (IBMF) approaches are primarily used to analyze the model on heterogeneous networks. For scale-free networks with degree exponent $2 < \gamma \leq 3$, the paper states that the epidemic threshold tends to zero, highlighting the ease with which diseases can spread in these settings.
2. SIR Model: This model considers permanent recovery, offering a mapping to percolation theory for analyzing network spreading phenomena. This approach sheds light on outbreak sizes and the critical transmission probability for epidemic outbreaks.

Impact of Network Structures on Epidemics

Heterogeneous Networks: The networks' degree distribution critically affects both SIS and SIR models. In power-law networks, where node degrees can be highly heterogeneous, highly connected nodes (hubs) significantly enhance the spread of diseases. However, in SIS models, this effect results in a vanishing epidemic threshold as network size approaches infinity.

Clustering Coefficients: Clustering in networks can curtail the spread of diseases by limiting the effective network connectivity. This results in smaller epidemic sizes as the disease's path is often constrained within clusters.

Degree Correlations: Network assortativity (the tendency of nodes to connect with similar degree nodes) can either elevate or suppress the likelihood of a network-wide epidemic depending on whether the network is assortative or disassortative.

Temporal Networks

The review also explores studies of temporal networks, where the structure evolves in time, capturing more realistic interaction patterns. The temporal dimension introduces additional complexity, such as burstiness and periodicity in contacts, which significant impact on spreading dynamics is:

• Slow Dynamics: For slow-evolving networks, static network approximations (integrated networks over time windows) might suffice.
• Fast Dynamics: Networks with rapid structural changes require precise modeling of time-ordered interactions.

Metapopulation and Reaction-Diffusion Frameworks

The analysis extends to metapopulation models, where nodes represent subpopulations with many individuals. Diffusion among subpopulations can transition between localized outbreaks and global pandemics. The threshold for global invasion, $R_*$, depends on parameters including the reproductive number $R_0$ and the mobility rate $p$.

Practical Implications

From a practical perspective, the paper emphasizes the potential of network-targeted interventions, such as:

• Targeted Immunization: Strategies that prioritize higher-degree nodes or central nodes optimize immunization efforts by leveraging network topology to enhance herd immunity effects, vastly outperforming random immunization.
• Behavioral Interventions: The consideration of human behavioral adaptation to epidemic states adds realism but also complexity, necessitating co-evolution models that treat network evolution in tandem with spreading dynamics.

Future Directions

The paper speculates on several future directions:

• Multiplex Networks: Studying epidemic processes in layered or interdependent networks could provide insights into complex real-world scenarios where different types of interactions co-exist and influence each other.
• Predictive Models: Advancing data-driven, agent-based models and enhancing their transparency will help synthesize detailed empirical data into understandable and actionable predictions.
• Control Strategies: Developing strategies for controlling or maximizing the spread of information or diseases efficiently within complex, adaptive networks remains a critical and open challenge.

In summary, this comprehensive review reveals that understanding epidemic dynamics in complex networks requires a confluence of theoretical models, robust computational simulations, and real-world data. Such integrated approaches will be pivotal in addressing global health challenges and optimizing information dissemination strategies in an increasingly connected world.