Disease spread in small-size directed networks: Epidemic threshold, correlation between links to and from nodes, and clustering

https://doi.org/10.1016/j.jtbi.2009.06.015Get rights and content

Abstract

Network epidemiology has mainly focused on large-scale complex networks. It is unclear whether findings of these investigations also apply to networks of small size. This knowledge gap is of relevance for many biological applications, including meta-communities, plant–pollinator interactions and the spread of the oomycete pathogen Phytophthora ramorum in networks of plant nurseries. Moreover, many small-size biological networks are inherently asymmetrical and thus cannot be realistically modelled with undirected networks. We modelled disease spread and establishment in directed networks of 100 and 500 nodes at four levels of connectance in six network structures (local, small-world, random, one-way, uncorrelated, and two-way scale-free networks). The model was based on the probability of infection persistence in a node and of infection transmission between connected nodes. Regardless of the size of the network, the epidemic threshold did not depend on the starting node of infection but was negatively related to the correlation coefficient between in- and out-degree for all structures, unless networks were sparsely connected. In this case clustering played a significant role. For small-size scale-free directed networks to have a lower epidemic threshold than other network structures, there needs to be a positive correlation between number of links to and from nodes. When this correlation is negative (one-way scale-free networks), the epidemic threshold for small-size networks can be higher than in non-scale-free networks. Clustering does not necessarily have an influence on the epidemic threshold if connectance is kept constant. Analyses of the influence of the clustering on the epidemic threshold in directed networks can also be spurious if they do not consider simultaneously the effect of the correlation coefficient between in- and out-degree.

Section snippets

1. Introduction

Epidemic models assuming regularly or randomly connected individuals are now involving more complex networks (Keeling, 2005; May, 2006; Jeger et al., 2007). Compared to regular lattices, epidemics in small-world networks are facilitated by long-distance connections (Moore and Newman, 2000). In scale-free networks of infinite size, epidemics lack a threshold, which implies that even pathogens with a low probability of transmission will persist (Pastor-Satorras and Vespignani, 2001). Whether

Materials and Methods

We simulated disease spread and establishment in networks of 100 and 500 nodes. For both network sizes, we used six kinds of structure: (1) local (nearest-neighbour transmission), (2) random (nodes connected with probability p), (3) small-world (local networks rewired with short-cuts), and scale-free structure (see Jeger et al., 2007 for a visualization). For scale-free networks, we considered separately networks with in- and out-degree of nodes (4) positively, (5) not, and (6) negatively

Results

The threshold pt* significantly decreased with increasing connectance for all structures and with both network sizes (Fig. 1a and b). With the exception of the lowest connectance level for both network sizes, two-way scale-free networks showed a significantly lower and one-way scale-free networks a significantly higher threshold than all other structures (Fig. 1c and d). For network size of 100 nodes, random networks showed a significantly lower threshold than local networks, but not at the

Discussion

Networks of small size have biological significance in a variety of ecological fields. Examples include meta-populations, mutualistic, and antagonistic interactions (Dunne et al., 2002; Lundgren and Olesen, 2005; Brooks, 2006; Pautasso et al., 2008; Thebault and Fontaine, 2008). In spite of the relevance of small-size networks for many issues in natural sciences, it is not clear whether theoretical results derived from analyses of large-scale complex networks apply also to small-size networks (

Acknowledgements

Many thanks to T. Harwood, O. Holdenrieder, J. Parke, M. Shaw, J. Tufto, F. Van den Bosch, and X. Xu for discussions and insights, and to T. Matoni and anonymous reviewers for helpful comments on a previous version of the ms. This study was funded by the Department for Environment, Food and Rural Affairs, the Rural Economy and Land Use Programme, UK, and the French Ministry of National Education and Research.

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