Long-range correlations improve understanding of the influence of network structure on contact dynamics

Mean-field models of dynamics on networks via moment closure: An automated procedure

In the study of dynamics on networks, moment closure is a commonly used method to obtain low-dimensional evolution equations amenable to analysis. The variables in the evolution equations are mean counts of subgraph states and are referred to as moments. Due to interaction between neighbors, each moment equation is a function of higher-order moments, such that an infinite hierarchy of equations arises. Hence, the derivation requires truncation at a given order and an approximation of the highest-order moments in terms of lower-order ones, known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equations for SIR epidemic spreading or to unclosed moment equations for arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equations for dynamics with nearest-neighbor interactions on undirected networks. Automation of the closure step was made possible by our generalized closure scheme, which systematically decomposes the largest subgraphs into their smaller components. We show that this decomposition is exact if these components form a tree, there is independence at distances beyond their graph diameter, and there is spatial homogeneity. Testing our method for SIS epidemic spreading on lattices and random networks confirms that biases are larger for networks with many short cycles in regimes with long-range dependence. A Mathematica package that automates the moment closure is available for download.

Understanding the temporal pattern of spreading in heterogeneous networks: Theory of the mean infection time

For a reliable prediction of an epidemic or information spreading pattern in complex systems, well-defined measures are essential. In the susceptible-infected model on heterogeneous networks, the cluster of infected nodes in the intermediate-time regime exhibits too large fluctuation in size to use its mean size as a representative value. The cluster size follows quite a broad distribution, which is shown to be derived from the variation of the cluster size with the time when a hub node was first infected. On the contrary, the distribution of the time taken to infect a given number of nodes is well concentrated at its mean, suggesting the mean infection time is a better measure. We show that the mean infection time can be evaluated by using the scaling behaviors of the boundary area of the infected cluster and use it to find a nonexponential but algebraic spreading phase in the intermediate stage on strongly heterogeneous networks. Such slow spreading originates in only small-degree nodes left susceptible, while most hub nodes are already infected in the early exponential-spreading stage. Our results offer a way to detour around large statistical fluctuations and quantify reliably the temporal pattern of spread under structural heterogeneity.

Network impact on persistence in a finite population dynamic diffusion model: application to an emergent seed exchange network

Dynamic extinction colonisation models (also called contact processes) are widely studied in epidemiology and in metapopulation theory. Contacts are usually assumed to be possible only through a network of connected patches. This network accounts for a spatial landscape or a social organization of interactions. Thanks to social network literature, heterogeneous networks of contacts can be considered. A major issue is to assess the influence of the network in the dynamic model. Most work with this common purpose uses deterministic models or an approximation of a stochastic Extinction-Colonisation model (sEC) which are relevant only for large networks. When working with a limited size network, the induced stochasticity is essential and has to be taken into account in the conclusions. Here, a rigorous framework is proposed for limited size networks and the limitations of the deterministic approximation are exhibited. This framework allows exact computations when the number of patches is small. Otherwise, simulations are used and enhanced by adapted simulation techniques when necessary. A sensitivity analysis was conducted to compare four main topologies of networks in contrasting settings to determine the role of the network. A challenging case was studied in this context: seed exchange of crop species in the Réseau Semences Paysannes (RSP), an emergent French farmers׳ organisation. A stochastic Extinction-Colonisation model was used to characterize the consequences of substantial changes in terms of RSP׳s social organization on the ability of the system to maintain crop varieties.

Predicting epidemic thresholds on complex networks: limitations of mean-field approaches

Over the last decade considerable research effort has been invested in an attempt to understand the dynamics of viruses as they spread through complex networks, be they the networks in human population, computers or otherwise. The efforts have contributed to an understanding of epidemic behavior in random networks, but were generally unable to accommodate specific nonrandom features of the network's actual topology. Recently, though still in the context of the mean field theory, Chakrabarti et al. (2008) proposed a model that intended to take into account the graph's specific topology and solve a longstanding problem regarding epidemic thresholds in both random and nonrandom networks. Here we review previous theoretical work dealing with this problem (usually based on mean field approximations) and show with several relevant and concrete counter examples that results to date breakdown for nonrandom topologies.

The consequences of spatial structure for the evolution of pathogen transmission rate and virulence

The distribution of organisms in space can be an important mediator of species interactions, but its evolutionary effects on those interactions are only beginning to be explored. These effects may be especially relevant to pathogen-host interactions. A detailed understanding of how and when spatial structure will affect the evolution of pathogen traits is likely to aid our ability to control rapidly emerging infectious diseases. Here we review a growing body of theoretical studies suggesting that spatial structure can lead to the evolution of an intermediate pathogen transmission rate and virulence. We explain the results of these studies in terms of a competition-persistence trade-off. These studies strongly suggest that local host interactions, local host dispersal, and relatively low host reproduction rates create a host population spatial structure that enforces this trade-off and leads to the evolution of lower pathogen transmission rates and virulence. They also suggest that when spatial structure exists, it can dominate over the shape of the transmission-virulence trade-off in determining pathogen traits. We also identify important areas of future research, including quantifying pathogen fitness in a spatial context in order to gain a more mechanistic understanding of the effects of spatial structure and observationally and experimentally testing theoretical predictions.