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