Ecological Complexity 7 (2010) 424–432

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Ecological Complexity journal homepage: www.elsevier.com/locate/ecocom

The number of links to and from the starting node as a predictor of epidemic size in small-size directed networks Marco Pautasso a,*, Mathieu Moslonka-Lefebvre a,b,c, Michael J. Jeger a a

Division of Biology, Imperial College London, Silwood Park Campus, Ascot, SL5 7PY, UK Department of Biology, Universite´ Paris-Sud XI, 91405 Orsay, France c Biochemistry and Bio-engineering Department, ENS Cachan, 61 av President Wilson, 94230 Cachan, France b

A R T I C L E I N F O

A B S T R A C T

Article history: Received 18 September 2008 Received in revised form 2 March 2009 Accepted 20 October 2009

Much recent modelling is focusing on epidemics in large-scale complex networks. Whether or not findings of these investigations also apply to networks of small size is still an open question. This is an important gap for many biological applications, including the spread of the oomycete pathogen Phytophthora ramorum in networks of plant nurseries. We use numerical simulations of disease spread and establishment in directed networks of 100 individual nodes at four levels of connectivity. Factors governing epidemic spread are network structure (local, small-world, random, scale-free) and the probabilities of infection persistence in a node and of infection transmission between connected nodes. Epidemic final size at equilibrium varies widely depending on the starting node of infection, although the latter does not affect the threshold condition for spread. The number of links from (out-degree) but not the number of links to (in-degree) the starting node of the epidemic explains a substantial amount of variation in final epidemic size at equilibrium irrespective of the structure of the network. The proportion of variance in epidemic size explained by the out-degree of the starting node increases with the level of connectivity. Targeting highly connected nodes is thus likely to make disease control more effective also in case of small-size populations, a result of relevance not just for the horticultural trade, but for epidemiology in general. ß 2009 Elsevier B.V. All rights reserved.

Keywords: Initial conditions Meta-population Network epidemiology Plant sciences Susceptible-infected-susceptible (SIS) Uncorrelated scale-free networks

1. Introduction Plant pathology has a long tradition of investigating disease spread, establishment and persistence in meta-populations (e.g. Thrall and Antonovics, 1995; Thrall and Burdon, 2002; Laine, 2004; Koslow and DeAngelis, 2006; Jeger et al., 2007; Brooks et al., 2008). Epidemic models have recently moved from the assumption of perfect mixing or of regularly and randomly connected individuals to scenarios involving networks of more complex structure (e.g. Sander et al., 2002; Franc, 2004; Shirley and Rushton, 2005; Roy and Pascual, 2006; Zhou et al., 2006; Trapman, 2007; Pellis et al., 2009). Researchers have thus investigated the properties of epidemic dynamics in small-world (e.g. Moore and Newman, 2000; Xu et al., 2004; Verdasca et al., 2005) and scale-free networks (e.g. PastorSatorras and Vespignani, 2003; Latora et al., 2006; Colizza et al., 2007a). Small-world networks are commonly obtained by rewiring a varying proportion of the local connections of a regular lattice

* Corresponding author. Tel.: +44 020 759 42533. E-mail address: [email protected] (M. Pautasso). 1476-945X/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecocom.2009.10.003

into shortcuts or long-distance links (e.g. Watts and Strogatz, 1998; Amaral et al., 2000; Zhao and Gao, 2007). Small-world (Watts–Strogatz) networks show low average path distance between nodes but high clustering (e.g. da Gama and Nunes, 2006; May, 2006; Aparicio and Pascual, 2007). Small-world properties have been documented in a variety of complex networks (Bassett and Bullmore, 2006; Boccaletti et al., 2006; Costa et al., 2007) and have recently also been shown to be fundamental for the functionality of the human brain (Stam, 2004; Sporns and Kotter, 2004; Liu et al., 2008; Supekar et al., 2008; Wang et al., 2009). Small-world properties can be found also in scale-free networks (e.g. Eguiluz et al., 2005). Scale-free networks are defined by a degree distribution (the frequency distribution of the number links of the nodes) which can be fitted by a power-law (e.g. Barabasi and Albert, 1999; Wang, 2002; Silva et al., 2007). This follows from the presence in scale-free networks of a small number of superconnected individuals and of a high proportion of nodes with only a few connections (e.g. Albert et al., 2000; May and Lloyd, 2001; Barthe´lemy et al., 2005). For a visualization of small-world vs. scalefree networks we refer to e.g. Watts and Strogatz (1998), Albert et al. (2000), Jeger et al. (2007) and Dorogovtsev et al. (2008).

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Modelling of epidemic spread in complex networks so far has mostly relied on asymptotic results for graphs of large size (e.g. Albert and Barabasi, 2002; Boccaletti et al., 2006; Riley, 2007). Also, empirical work has mainly dealt with large-size networks. For instance, Bigras-Poulin et al. (2006) investigated the properties of the Danish cattle industry network over a six-month period. This turned out to be a directed scale-free network of ca. 30,000 nodes. Similar analyses exist on the livestock movement networks in other countries (e.g. Webb, 2005; Kiss et al., 2006b; Lentz et al., 2009; Ribbens et al., 2009). Similar orders of magnitudes are found in studies of the degree and weight distributions of small-world and scale-free networks (Masuda et al., 2005), in investigations of the effectiveness of vaccination strategies in various kinds of contact networks (Takeuchi and Yamamoto, 2006), and in analyses of dynamical reaction-diffusion processes in heterogeneous networks (Colizza et al., 2007b). Nevertheless, many networks of biological significance can be of markedly smaller size. Examples include closely interacting primate groups (Dunbar, 1993), meta-populations (e.g. Hanski and Ovaskainen, 2000; Brooks, 2006; Lobel et al., 2006; Burns, 2007), plant-pollinator interactions (e.g. Olesen et al., 2006; Nielsen and Bascompte, 2007), mycorrhiza and rhizomorphs (Southworth et al., 2005; Selosse et al., 2006; Lamour et al., 2007), and food webs (e.g. Montoya and Sole´, 2002; Estrada, 2007; Neutel et al., 2007). Networks of small size (of the order of magnitude of hundreds of nodes) can also be formed by trade movements among plant nurseries, an issue of growing relevance in plant epidemiology given the increasing trade in horticultural material (e.g. Bandyopadhyay and Frederiksen, 1999; Anderson et al., 2004; Slippers et al., 2005). A recent example is given by the spread in regional networks of plant nurseries of Phytophthora ramorum, the causal agent of Sudden Oak Death and of leaf blight and dieback in many ornamental shrubs (Werres et al., 2001; Meentemeyer et al., 2008; Prospero et al., 2009). In spite of the relevance of small-size networks for many issues in ecology, it is not clear whether theoretical results derived from analyses of large-scale complex networks apply also to small-size networks (e.g. Amaral et al., 2004; Guimara˜es et al., 2007).

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More in detail, previous studies have shown the importance of the network structure and of the initial conditions in determining invasion thresholds in complex networks of large size (e.g. Barthe´lemy et al., 2005; Cre´pey et al., 2006; Vespignani, 2008), but it is unclear whether a similar effect is still observed for smallsize networks. In a previous investigation of disease spread in directed, complex networks of small size (Pautasso and Jeger, 2008), we found that the epidemic threshold (the boundary between no epidemic development and epidemic development) was significantly lower for scale-free networks with a positive correlation between links in and out of nodes compared to local, random and small-world networks. We also observed marked variations in the final size of an epidemic at the threshold conditions depending on the starting node. This result prompts the question of whether such variations can be predicted by some characteristic of the starting node. In this paper, we focus again on networks of small size, model disease spread in directed networks of various types, and study in more depth the effect of the starting node on the epidemic final size. We test whether characteristics of the starting node of the epidemic such as the in- and out-degree can be predictors of the final size of the epidemic, and whether there are any differences in this respect among different network structures. In this analysis, we also vary the level of connectivity, thus testing the effect of the in- and out-degree of the starting node on epidemic final size not only for different network structures, but also over a range of connectivity levels. 2. Materials and methods Networks have 100 individual nodes and four different kinds of network structure (Fig. 1): local (nearest-neighbour transmission), random (absence of regular local links), small-world (nearestneighbour transmission rewired with shortcuts), and scale-free (presence of super-connected nodes with high number of connections) connectivity. For scale-free networks, we considered separately networks with the presence of super-connected nodes in terms both of links

Fig. 1. The four basic kinds of network structure used: (a) local, (b) small-world, (c) random, and (d) scale-free. Graphs are networks of 100 individuals with a constant number of links. The circular layout does not reflect the spatial arrangement of nodes.

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from and links to these nodes (two-way scale-free networks) and networks with nodes super-connected either in terms of links from these nodes or to these nodes (one-way scale-free networks, see Pautasso and Jeger, 2008). We also considered uncorrelated scalefree networks, where, in spite of the presence of super-connected nodes, the in- and the out-degree of the nodes are not significantly correlated. The networks were directed, i.e. the connectivity from node a to node b could be different from the one from node b to node a (as e.g. in Newman et al., 2001; Allesina and Bodini, 2005; Boguna and Serrano, 2005; Meyers et al., 2006; Park and Kim, 2006; Wang and Liu, 2009). Directed networks are realistic approximations of the horticultural trade, where the probability that a plant nursery will be connected to a distribution or retail centre will tend to be different from the reverse connection. For each network structure, 100 replicate networks were built in MATLAB at each level of connectivity (100, 200, 400, and 1000 links) with different algorithms for the different network structures. Local networks were built starting from a regular ring with 100 links more than the target number of links and with a procedure randomly generating 100 gaps in the regular ring. For the lowest level of connectivity, variability among network realizations was created with a function generating some variability among a node and its four clockwise neighbours. Random digraphs were generated using the G(N,M) model where M directed links are placed randomly and independently between the N nodes of the graph. Small-world networks were built with the Watts and Strogatz algorithm (Watts and Strogatz, 1998) starting from local networks and with a rewiring coefficient of 0.25. Scale-free networks (with in- and out-degree of nodes positively, uncorrelated, and negatively correlated) were built with a preferential attachment algorithm, starting with a simple seed network and based on five parameters adding nodes and/or links depending on the in-, out-, and total degree of existing nodes. Given the six kinds of network structure, the four levels of connectivity, and the 100 replicate networks per network structure and level of connectivity, the 100 starting nodes resulted in 240,000 individual epidemic simulations. At each iteration, the contact structure of the network realization was maintained exactly the same. Networks were not necessarily fully connected, so it is possible that at the lower levels of connectivity not all nodes could be reached from all nodes. Epidemic development was deterministic and governed by two parameters: the probability of infection transmission between nodes and the probability of infection persistence in a node. The probability for an infection to be transmitted from one node to another (pt) was either zero (unconnected nodes) or a parameter constant for different nodes of a network but variable among network replicates (in order to work at the threshold conditions in each replicate). The probability of infection transmission among nodes is a key feature of the trade among plant nurseries and garden centres. Plant shipments are made of several individual plants, and a varying amount of these plants will tend to carry a given pathogen from nursery a to retail centre b. We set this pt to be the same for each connection (fixed-edge networks; in contrast to Sarama¨ki and Kaski (2005), who modelled epidemic dynamics in small-world networks with two different transmission probabilities according to whether links were to neighbouring cells or shortcuts). We set an additional parameter (pp) for the probability of persistence of an infection in a given node from time t to time t + 1. This pp combined in one single parameter the length of infectiousness, detection and control measures. We also set pp to be the same for all nodes. By definition, both pt and pp are real variables, going from 0 to 1 (as e.g. in Agliari et al., 2006). This can be a realistic assumption for example for plant nurseries and shipments of plants among nurseries, which contain a certain

number of plants, and will have a varying proportion of these individual plants infected by a certain pathogen at any point in time. We assumed all nodes to be of equal size and kind. For each iteration of the simulation, we obtained the infection status of a given node Pi(x) in the following way: P iðxÞ ¼ S ptðx;yÞ P iðyÞ for y going from 1 to 100, where pt is the probability of infection transmission for the connection of the node x from a node y, and Py is the infection status of the nodes y at the previous iteration. This infection status is not a probability and can thus assume values greater than 1. Pi is essentially a metric of infection severity in plant nurseries, which can be infected to varying degrees depending on how many individual plants are affected by the pathogen. This assumption distinguishes our model from much of the plant epidemiology literature, where the infection status of nodes is typically either devoid of infection (susceptible) or completely infected (e.g. Madden, 2006). The summation for ‘‘y’’ goes across all nodes that are directly connected to ‘‘x’’ and not across all nodes. At the beginning of the epidemic Pi(x) was equal zero for all nodes except for the node from which the epidemic was started, with P(i) = 1. For the connection of a node with itself, pp was used instead of pt. Hence, the networks had self-loops. The biological motivation for self-loops is that plant nurseries which have become infected by a given pathogen have a certain probability to remain infected due to the persistence of inoculum through time. The model was thus a SIS (susceptibleinfected-susceptible) model (e.g. Joo and Lebowitz, 2004; Grabowski and Kosin´ski, 2005; Hiebeler, 2006). This can be a realistic assumption for plant nurseries, which are still at risk even after eradication of a disease outbreak if they continue trading susceptible material (Jeger et al., 2007). As in Pautasso and Jeger (2008), the development of the epidemic (i.e. whether the epidemic died away, reached equilibrium affecting a certain proportion of nodes, or affected all nodes) was assessed on the basis of the sum of Pi across all nodes at a given iteration or on the basis of the number of nodes with Pi higher than an arbitrary value (0.01). The epidemic was started with a single infection of a single node (as e.g. in Ben-Naim and Krapivsky, 2004). The effect on the epidemic development of differing starting nodes of the infection was assessed by making the epidemic start in each of the 100 different nodes of the networks. Making the epidemic start from different nodes did not affect the combination of pp and pt producing a threshold between no epidemic and epidemic development. Moreover, the particular combination of pp and pt at the threshold did not affect epidemic final size (Pautasso and Jeger, 2008). However, the starting node had a marked influence on the epidemic size at equilibrium (Pautasso and Jeger, 2008). Analysis of variance (ANOVA) of the correlation coefficient between epidemic final size and in- and out-degree of the starting node of the epidemic for the different levels of connectivity (within a network structure) and for the different types of network structure (at a given level of connectivity) was carried out in SAS 9.1. For scalefree networks (two-way, uncorrelated, and one-way) this correlation coefficient was calculated between log-transformed epidemic final size and log-transformed number of links from and to a node to better approach a normal distribution of the two variables. 3. Results Irrespective of the structure of the network, there was a wide variation in final epidemic size at the threshold conditions depending on the node from which the epidemic was started. This wide variation was observed both in terms of the sum of infection status across all nodes (Fig. 2) and of the proportion of nodes with infection status higher than 0.01 (Fig. 3).

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Fig. 2. Epidemic final size at equilibrium (sum of infection status across all nodes) depending on the starting node for one out of the 100 replicates and the six network structures: (a) local, (b) small-world, (c) random, (d) two-way scale-free, (e) uncorrelated scale-free, and (f) one-way scale-free networks.

Fig. 3. Epidemic final size at equilibrium (proportion of nodes with infection status higher than 0.01) depending on the starting node for one out of the 100 replicates and the six network structures: (a) local, (b) small-world, (c) random, (d) two-way scale-free, (e) uncorrelated scale-free, and (f) one-way scale-free networks.

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Fig. 4. Correlation between epidemic final size and out-degree of the starting node for 100 network replicates and 100 starting nodes. The correlation between the sum of the infection status of all nodes and the number of links from the starting node for the different (a) levels of connectivity and (b) network structures. Error bars are standard deviations. Different letters show significant differences (ANOVA, p < 0.05) within (a) levels of connectivity for a given network structure, and (b) network structures for a given level of connectivity.

The out-degree of the node from which the epidemic started explained a substantial amount of the variation in the final size of the epidemic, both in terms of the sum of infection status across all nodes (Fig. 4) and of the proportion of nodes with infection status higher than 0.01 (Fig. 5). The proportion of variance in epidemic final size explained by the out-degree of the starting node increased for all network structures with increasing level of connectivity (Figs. 4a and 5a). Exception to this trend was made by two-way scale-free networks in the case of the proportion of nodes with infection status higher than 0.01. There was no consistent influence of network structure on the proportion of variance in epidemic final size explained by the out-degree of the starting node at a given level of connectivity (Figs. 4b and 5b), but in all cases this proportion was substantial. The in-degree of the node from which the epidemic started did not explain any substantial variation in the final size of the epidemic, both in terms of the sum of infection status across all nodes (Fig. 6) and of the proportion of nodes with infection status higher than 0.01 (Fig. 7). An exception to this pattern was observed with two-way scale-free networks, at least at high levels of connectivity, where the in-degree of the starting node was positively correlated with epidemic final size, and with one-way scale-free networks, where the in-degree of the starting node was negatively correlated with epidemic final size. Both results are reasonable: for two-way scale-free networks in- and out-degree

Fig. 5. Correlation between epidemic final size and out-degree of the starting node for 100 network replicates and 100 starting nodes. The correlation between the proportion of individuals with infection status higher than 0.01 and the number of links from the starting node for the different (a) levels of connectivity and (b) network structures. Same conventions as in Fig. 4.

are positively correlated and for one-way scale-free networks inand out-degree are negatively correlated. 4. Discussion Network approaches are increasingly used in ecology (Lugo and McKane, 2008; Muneepeerakul et al., 2008; Namba et al., 2008; Bascompte, 2009; de Santana et al., 2009) and epidemiology (Goldstein et al., 2009; Meloni et al., 2009; Tildesley and Keeling, 2009; Wylie and Getz, 2009). In spite of the relevance of small-size networks for a range of such applications (Blick and Burns, 2009; Fontaine et al., 2009; Green et al., 2009; Moslonka-Lefebvre et al., 2009; Westgarth et al., 2009; see also Section 1), it is still unclear whether results obtained from analyses of large-size networks also apply to networks of small size. In particular, we need to know whether epidemic development is still affected by the heterogeneity in the contact structure and by differing initial conditions even in the presence of a relatively small number of individuals in a network. Our analysis extends the results reported in Pautasso and Jeger (2008), where we investigated variations among network structures in the epidemic final size averaged across all starting nodes. Here, we studied again the effect of the network structure on the epidemic final size at the threshold conditions, but at different levels of connectivity, and without losing the information provided by the initial node of infection, given the marked variations in epidemic final size in a given network replicate depending on the starting node (Figs. 2 and 3).

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Fig. 6. Correlation between epidemic final size and in-degree of the starting node for 100 network replicates and 100 starting nodes. The correlation between the sum of the infection status of all nodes and the number of links to the starting node for the different (a) levels of connectivity and (b) network structures. Same conventions as in Fig. 4.

Fig. 7. Correlation between epidemic final size and in-degree of the starting node for 100 network replicates and 100 starting nodes. The correlation between the proportion of individuals with infection status higher than 0.01 and the number of links to the starting node for the different (a) levels of connectivity and (b) network structures. Same conventions as in Fig. 4.

The result that the number of links from the starting node of the epidemic is consistently associated with the final size of an epidemic in small-size directed networks at threshold conditions, irrespective of the network structure (Figs. 4b and 5b), is evidence of the importance of the initial conditions in epidemic development (e.g. Lung-Escarmant and Guyon, 2004; Cre´pey et al., 2006; White and Gilligan, 2006; Kleczkowski and Gilligan, 2007; Riley, 2007). The directed nature of the network contact structure makes it likely that if a node has a high number of connections from it, a pathogen can move from it to most other nodes in a network, whereas if a node has a low number of connections from it, the outcomponent (i.e. the set of nodes reachable from that node) will tend to be of low size (Meyers et al., 2006). In our model, the number of links to a node explains only negligible variation in epidemic final size (Figs. 6 and 7), whereas it is the number of links from a node which appears to play an important role in determining the epidemic outcome. Other things being equal, the out-degree of the starting node of an infection event, when unknown, can thus add to the stochastic nature and unpredictability of diseases (e.g. Shaw, 1994; Madden, 2006; James et al., 2007). This result underlines the need for reliable trace-forward information in plant epidemiology. In spite of the small size of the networks studied, there were significant differences in the correlation coefficient between the out-degree of the starting node of the epidemic and the corresponding final epidemic size among the different network

types. At a given level of connectivity, small-world networks showed a smaller average correlation coefficient between outdegree and epidemic final size compared to random networks (Figs. 4b and 5b). This suggests that in small-world networks the number of connections from the starting point of an epidemic is less important than whether there are long-distance connections (e.g. Petit et al., 2004; Vuorinen et al., 2004; Trakhtenbrot et al., 2005). Two-way scale-free networks showed a higher average correlation coefficient between the out-degree of the starting node and epidemic final size than the other types of scale-free networks, but only in the case of the sum of the infection status across all nodes (Fig. 4b). This discrepancy, possibly caused by a different partitioning of disease presence at equilibrium among nodes in two-way scale-free networks, implies that the number of links from the starting node is a better predictor of the final amount of disease in these networks than of the total number of nodes affected. These two aspects of a plant epidemic are likely to be positively correlated but do not represent the same thing, as a given amount of pathogen inoculum in a horticultural trade system can be concentrated in some trade pathways or be spread among many of them. Apart from this discrepancy, and irrespective of the structure of the network, the out-degree of the starting node of the epidemic is a reliable predictor of the outcome of an epidemic under the assumptions of the model. The out-degree of the initially infected node explained between 29 and 92% of the variability in the

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proportion of nodes with infectious status higher than 0.01, and between 33 and 94% of the variability in the sum of infectious status across all nodes in the different types of networks with different levels of connectivity. The variability in epidemic final size explained by the out-degree of the starting tended to increase with the connectivity level (Figs. 4a and 5a). These findings underline the importance of the number of out-connections as a diagnostic feature of the epidemic risk arising from an infection event at a single node in a system of connected entities with heterogeneous contact structure (Christley et al., 2005; Ferrari et al., 2006), also in the case of small-size networks. Targeting highly connected individuals is thus likely to be an effective strategy to control the development of epidemics (e.g. Keeling and Eames, 2005; Kiss et al., 2006a; Kleczkowski et al., 2006) also in metapopulations of limited size. Even if in our analyses the in-degree of the starting node had a negligible role in explaining variations in epidemic final size (Figs. 6 and 7), the in-degree of a node can still be important in real-world epidemics, as it will affect the likelihood that a node will become infected in the first place. Plant nurseries with a high in-degree will be likely to receive plant material from a range of sources and will thus be more likely to become infected. In real life, when disease is first noted at the ‘starting node’ it may have been present before that node was reached. Trace-back investigations are thus of key importance in epidemics in directed networks. Moreover, for two-way scale-free networks, in- and out-degree are positively correlated, so one can be predicted from the other. Whenever the structure of a network of relatively small size is not or only patchily known, as e.g. in the case of the network of plant nurseries trading material potentially infected by Phytophthora ramorum (Parke et al., 2004; Stokstad, 2004; Rizzo et al., 2005), preferential control of commercial players with a higher number of links is potentially an effective way to reduce the likelihood that a pathogen will spread through the system. Hence, regardless of whether a network of plant nurseries in a certain country or region has a local, small-world, random or scale-free degree distribution, the number of links from a plant nursery may be used as a straightforward indicator of the risk of further spread posed by the introduction in that nursery of a given plant pathogen or exotic invasive organism. The issue of biological invasions and the problem of finding ways to prevent new introductions and establishments are much debated in today’s globalized world (e.g. Perrings et al., 2005; Hulme, 2006; Desprez-Loustau et al., 2007; Kowalski and Holdenrieder, 2008). For the current epidemic of Phytophthora ramorum in Britain (e.g. Lane et al., 2003; Appiah et al., 2004; Brasier et al., 2004; Kluza et al., 2007), marked variations in the number of plants affected are reported from different outbreak sites, both in the natural environment and in the horticultural trade (Xu et al., 2009). This phenomenon has obvious consequences for the likelihood that eradication measures at different sites will be successful and for the potential risk for further spread of the pathogen. Similar widely diverging outcomes can be observed for introduced species depending on a variety of starting conditions. Godfray and Crawley (1998) note that of the 20,000 to 200,000 alien plant species introduced into Britain, only ca. 1200 have become naturalized, and about 70 have become widespread in natural habitats. Jones and Baker (2007) show that a large proportion of the plant pathogens introduced into Britain from 1970 to 2004 were first reported in the South-East, where climatic conditions may be more suitable for establishment and connectivity from the continent (as well as to the rest of the country) is better represented than in other regions. In these and other examples, the connectivity of different nodes in a network is likely to vary widely and has the potential to influence the proportion of the system affected.

Our results are of relevance for a number of applications in meta-population ecology, conservation biology and plant epidemiology. Efforts at reintroducing species in parts of their historic range can be successful, or not, depending not only on the quality of the habitat chosen for the reintroduction, but also on the number of other suitable habitat patches to which the introduced organisms can disperse from the first stepping stone (e.g. LeonCortes et al., 2003; Gardner and Gustafson, 2004; Kramer-Schadt et al., 2004). A similar argument can be made for conservation activities aiming to ensure meta-population persistence, where connectivity among habitat patches has been shown to be a key for species survival (e.g. Burel, 1996; Swart and Lawes, 1996; Sork and Smouse, 2006). Conversely, for plant diseases, and provided that the aim is to reduce their incidence, the implication of this modelling work is that landscape connectivity needs to be reduced by targeting nodes with more connections than others in order to decrease the magnitude of epidemics (e.g. Holdenrieder et al., 2004; Plantegenest et al., 2007; Margosian et al., 2009; Harwood et al., 2009). How to manage the connectivity of landscapes to achieve sufficient dispersal of threatened species but also adequate containment of pathogen spread remains an outstanding challenge of applied ecology (Hess, 1994; Chetkiewicz et al., 2006; Brenn et al., 2008; Urban et al., 2009). Acknowledgments Many thanks to M. Bertaglia, T. Harwood, M. Keeling, A. Inman, O. Holdenrieder, J. Parke, M. Shaw, F. Van den Bosch and X. Xu for discussions on these analyses, and to T. Matoni and anonymous reviewers for helpful comments on a previous draft. This study was funded by the Department for Environment, Food and Rural Affairs and the Rural Economy and Land Use Research Program, UK. M. M.L. was funded by a grant of the French Ministry of National Education and Research. References Agliari, E., Burioni, R., Cassi, D., Neri, F.M., 2006. Efficiency of information spreading in a population of diffusing agents. Phys. Rev. E 73, 046138. Albert, R., Barabasi, A.L., 2002. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97. Albert, R., Jeong, H., Barabasi, A.L., 2000. Error and attack tolerance of complex networks. Nature 406, 378–382. Allesina, S., Bodini, A., 2005. Food web networks: scaling relation revisited. Ecol. Compl. 2, 323–338. Amaral, L.A.N., Scala, A., Barthe´lemy, M., Stanley, H.E., 2000. Classes of small-world networks. Proc. Natl. Acad. Sci. U.S.A. 97, 11149–11152. Amaral, L.A.N., Barrat, A., Barabasi, A.L., Caldarelli, G., De los Rios, P., Erzan, A., Kahng, B., Mantegna, R., Mendes, J.F.F., Pastor-Satorras, R., Vespignani, A., 2004. Virtual round table on ten leading questions for network research. Eur. Phys. J. B 38, 143–145. Anderson, P.K., Cunningham, A.A., Patel, N.G., Morales, F.J., Epstein, P.R., Daszak, P., 2004. Emerging infectious diseases of plants: pathogen pollution, climate change and agrotechnology drivers. Trends Ecol. Evol. 19, 535–544. Aparicio, J.P., Pascual, M., 2007. Building epidemiological models from R0: an implicit treatment of transmission in networks. Proc. R. Soc. Lond. B 274, 505–512. Appiah, A.A., Jennings, P., Turner, J.A., 2004. Phytophthora ramorum: one pathogen and many diseases, an emerging threat to forest ecosystems and ornamental plant life. Mycologist 18, 145–150. Bandyopadhyay, B., Frederiksen, R.A., 1999. Contemporary global movement of emerging plant diseases. Ann. N. Y. Acad. Sci. 894, 28–36. Barabasi, A.L., Albert, R., 1999. Emergence of scaling in random networks. Science 286, 509–512. Barthe´lemy, M., Barrat, A., Pastor-Satorras, R., Vespignani, A., 2005. Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275–288. Bascompte, J., 2009. Mutualistic networks. Front. Ecol. Environ. 7, 429–436. Bassett, D.S., Bullmore, E., 2006. Small-world brain networks. Neuroscientist 12, 512–523. Ben-Naim, E., Krapivsky, P.L., 2004. Size of outbreaks near the epidemic threshold. Phys. Rev. E 69, 050901. Bigras-Poulin, M., Thompson, R.A., Chriel, M., Mortensen, S., Greiner, M., 2006. Network analysis of Danish cattle industry trade patterns as an evaluation of risk potential for disease spread. Prev. Vet. Med. 76, 11–39.

M. Pautasso et al. / Ecological Complexity 7 (2010) 424–432 Blick, R., Burns, K.C., 2009. Network properties of arboreal plants: are epiphytes, mistletoes and lianas structured similarly? Persp. Plant Ecol. Evol. Syst. 11, 41–52. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U., 2006. Complex networks: structure and dynamics. Phys. Rep. 424, 175–308. Boguna, M., Serrano, M.A., 2005. Generalized percolation in random directed networks. Phys. Rev. E 72, 016106. Brasier, C.M., Denman, S., Rose, J., Kirk, S.A., Hughes, K.J.D., Griffin, R.L., Lane, C.R., Inman, A.J., Webber, J.F., 2004. First report of ramorum bleeding canker on Quercus falcata, caused by Phytophthora ramorum. Plant Pathol. 53, 804–1804. Brenn, N., Menkis, A., Gru¨nig, C.R., Sieber, T.N., Holdenrieder, O., 2008. Community structure of Phialocephala fortinii s. lat. in European tree nurseries, and assessment of the potential of the seedlings as dissemination vehicles. Mycol. Res. 112, 650–662. Brooks, C.P., 2006. Quantifying population substructure: extending the graphtheoretic approach. Ecology 87, 864–872. Brooks, C.P., Antonovics, J., Keitt, T.H., 2008. Spatial and temporal heterogeneity explain disease dynamics in a spatially explicit network model. Am. Nat. 172, 149–159. Burel, F., 1996. Hedgerows and their role in agricultural landscapes. Crit. Rev. Plant Sci. 15, 169–190. Burns, K.C., 2007. Network properties of an epiphyte metacommunity. J. Ecol. 95, 1142–1151. Chetkiewicz, C.L.B., Clair, C.C.S., Boyce, M.S., 2006. Corridors for conservation: integrating pattern and process. Ann. Rev. Ecol., Evol. Syst. 37, 314–342. Christley, R.M., Pinchbeck, G.L., Bowers, R.G., Clancy, D., French, N.P., Bennett, R., Turner, J., 2005. Infection in social networks: using network analysis to identify high-risk individuals. Am. J. Epidemiol. 162, 1024–1031. Colizza, V., Barthe´lemy, M., Barrat, A., Vespignani, A., 2007. Epidemic modeling in complex realities. C. R. Biol. 330, 364–374. Colizza, V., Pastor-Satorras, R., Vespignani, A., 2007. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys. 3, 276–282. Costa, L.D., Rodrigues, F.A., Travieso, G., Boas, P.R.V., 2007. Characterization of complex networks: a survey of measurements. Adv. Phys. 56, 167–242. Cre´pey, P., Alvarez, F.P., Barthe´lemy, M., 2006. Epidemic variability in complex networks. Phys. Rev. E 73, 046131. da Gama, M.M.T., Nunes, A., 2006. Epidemics in small world networks. Eur. Phys. J. B 50, 205–208. de Santana, C.N., Fontes, A.S., Cidreira, M.A.D., Almeida, R., Gonzalez, A.P., Andrade, R.F.S., Miranda, J.G.V., 2009. Graph theory defining non-local dependency of rainfall in Northeast Brazil. Ecol. Compl. 6, 272–277. Desprez-Loustau, M.-L., Robin, C., Bue´e, M., Courtecuisse, R., Garbaye, J., Suffert, F., Sache, I., Rizzo, D.M., 2007. The fungal dimension of biological invasions. Trends Ecol. Evol. 22, 472–480. Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F., 2008. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335. Dunbar, R.I.M., 1993. Coevolution of neocortical size, group-size and language in humans. Behav. Brain Sci. 16, 681–694. Eguiluz, V.M., Chialvo, D.R., Cecchi, G.A., Baliki, M., Apkarian, A.V., 2005. Scale-free brain functional networks. Phys. Rev. Lett. 94, 018102. Estrada, E., 2007. Food webs robustness to biodiversity loss: the roles of connectance, expansibility and degree distribution. J. Theor. Biol. 244, 296–307. Ferrari, M.J., Bansal, S., Meyers, L.A., Bjornstad, O.N., 2006. Network frailty and the geometry of herd immunity. Proc. R. Soc. Lond. B 273, 2743–2748. Fontaine, C., Thebault, E., Dajoz, I., 2009. Are insect pollinators more generalist than insect herbivores? Proc. R. Soc. B 276, 3027–3033. Franc, A., 2004. Metapopulation dynamics as a contact process on a graph. Ecol. Compl. 1, 49–63. Gardner, R.H., Gustafson, E.J., 2004. Simulating dispersal of reintroduced species within heterogeneous landscapes. Ecol. Model. 71, 339–358. Godfray, H.C.J., Crawley, M.J., 1998. Introductions. In: Sutherland, W.J. (Ed.), Conservation Science and Action. Blackwell Science, Oxford, pp. 39–65. Goldstein, E., Paur, K., Fraser, C., Kenah, E., Wallinga, J., Lipsitch, M., 2009. Reproductive numbers, epidemic spread and control in a community of households. Math. Biosci. 221, 11–25. Grabowski, A., Kosin´ski, R.A., 2005. The sis model of epidemic spreading in a hierarchical social network. Acta Phys. Polo. B 36, 1579–1593. Green, D.M., Gregory, A., Munro, L.A., 2009. Small- and large-scale network structure of live fish movements in Scotland. Prev. Vet. Med. 91, 261–269. Guimara˜es, P.R., Bonaldo, R.M., Krajewski, J.P., Guimara˜es, P., Pinheiro, A., Powers, J., dos Reis, S.F., 2007. Investigating small fish schools: selection of school— formation models by means of general linear models and numerical simulations. J. Theor. Biol. 245, 784–789. Hanski, I., Ovaskainen, O., 2000. The metapopulation capacity of a fragmented landscape. Nature 404, 755–758. Harwood, T.D., Xu, X.M., Pautasso, M., Jeger, M.J., Shaw, M.W., 2009. Epidemiological risk assessment using linked network and grid based modelling: Phytophthora ramorum and P. kernoviae in the UK. Ecol. Model. 220, 3353–3361. Hess, G.R., 1994. Conservation corridors and contagious disease—a cautionary note. Cons. Biol. 8, 256–262. Hiebeler, D., 2006. Moment equations and dynamics of a household SIS epidemiological model. Bull. Math. Biol. 68, 1315–1333. Holdenrieder, O., Pautasso, M., Weisberg, P.J., Lonsdale, D., 2004. Tree diseases and landscape processes: the challenge of landscape pathology. Trends Ecol. Evol. 19, 446–452. Hulme, P.E., 2006. Beyond control: wider implications for the management of biological invasions. J. Appl. Ecol. 43, 835–847.

431

James, A., Pitchford, J.W., Plank, M.J., 2007. An event-based model of superspreading in epidemics. Proc. R. Soc. Lond. B 274, 741–747. Jeger, M.J., Pautasso, M., Holdenrieder, O., Shaw, M.W., 2007. Modelling disease spread and control in networks: implications for plant sciences. New Phytol. 174, 279–297. Jones, D.R., Baker, R.H.A., 2007. Introductions of non-native plant pathogens into Great Britain, 1970–2004. Plant Pathol. 56, 891–910. Joo, J., Lebowitz, J.L., 2004. Behavior of susceptible-infected-susceptible epidemics on heterogeneous networks with saturation. Phys. Rev. E 69, 066105. Keeling, M.J., Eames, K.T.D., 2005. Networks and epidemic models. J. R. Soc. Interf. 2, 295–307. Kiss, I.Z., Green, D.M., Kao, R.R., 2006. Infectious disease control using contact tracing in random and scale-free networks. J. R. Soc. Interf. 3, 55–62. Kiss, I.Z., Green, D.M., Kao, R.R., 2006. The network of sheep movements within Great Britain: network properties and their implications for infectious disease spread. J. R. Soc. Interf. 3, 669–677. Kleczkowski, A., Gilligan, C.A., 2007. Parameter estimation and prediction for the course of a single epidemic outbreak of a plant disease. J. R. Soc. Interf. 4, 865–877. Kleczkowski, A., Dybiec, B., Gilligan, C.A., 2006. Economic and social factors in designing disease control strategies for epidemics on networks. Acta Phys. Polo. B 37, 3017–3026. Kluza, D.A., Vieglais, D.A., Andreasen, J.K., Peterson, A.T., 2007. Sudden oak death: geographic risk estimates and predictions of origins. Plant Pathol. 56, 580–587. Koslow, J.M., DeAngelis, D.L., 2006. Host mating system and the prevalence of disease in a plant population. Proc. R. Soc. Lond. B 273, 1825–1831. Kowalski, T., Holdenrieder, O., 2008. Eine neue Pilzkrankheit an Esche in Europa. Schweiz. Z. Forstw. 159, 45–50. Kramer-Schadt, S., Revilla, E., Wiegand, T., Breitenmoser, U., 2004. Fragmented landscapes, road mortality and patch connectivity: modelling influences on the dispersal of Eurasian lynx. J. Appl. Ecol. 41, 711–723. Laine, A.L., 2004. Resistance variation within and among host populations in a plantpathogen metapopulation: implications for regional pathogen dynamics. J. Ecol. 92, 990–1000. Lamour, A., Termorshuizen, A.J., Volker, D., Jeger, M.J., 2007. Network formation by rhizomorphs of Armillaria lutea in natural soil: their description and ecological significance. FEMS Microb. Ecol. 62, 222–232. Lane, C.R., Beales, P.A., Hughes, K.J.D., Griffin, R.L., Munro, D., Brasier, C.M., Webber, J.F., 2003. First outbreak of Phytophthora ramorum in England, on Viburnum tinus. Plant Pathol. 52, 414. Latora, V., Nyamba, A., Simpore, J., Sylvette, B., Diane, S., Sylvere, B., Musumeci, S., 2006. Network of sexual contacts and sexually transmitted HIV infection in Burkina Faso. J. Med. Virol. 78, 724–729. Lentz, H., Kasper, M., Selhorst, T., 2009. Beschreibung des Handels mit Rindern in Deutschland mittels Netzwerkanalyse–Ergebnisse von Voruntersuchungen. Berl. Muench. Tierarzt. Wochenschr. 122, 193–198. Leon-Cortes, J.L., Lennon, J.J., Thomas, C.D., 2003. Ecological dynamics of extinct species in empty habitat networks. 1. The role of habitat pattern and quantity, stochasticity and dispersal. Oikos 102, 449–464. Liu, Y., Liang, M., Zhou, Y., He, Y., Hao, Y.H., Song, M., Yu, C.S., Liu, H.H., Liu, Z.N., Jiang, T.Z., 2008. Disrupted small-world networks in schizophrenia. Brain 131, 945–961. Lobel, S., Snall, T., Rydin, H., 2006. Metapopulation processes in epiphytes inferred from patterns of regional distribution and local abundance in fragmented forest landscapes. J. Ecol. 94, 856–868. Lugo, C.A., McKane, A.J., 2008. The robustness of the Webworld model to changes in its structure. Ecol. Compl. 5, 106–120. Lung-Escarmant, B., Guyon, D., 2004. Temporal and spatial dynamics of primary and secondary infection by Armillaria ostoyae in a Pinus pinaster plantation. Phytopathology 94, 125–131. Madden, L.V., 2006. Botanical epidemiology: some key advances and its continuing role in disease management. Eur. J. Plant Pathol. 115, 3–23. Margosian, M.L., Garrett, K.A., Hutchinson, J.M.S., With, K.A., 2009. Connectivity of the American agricultural landscape: assessing the national risk of crop pest and disease spread. Bioscience 59, 141–151. Masuda, N., Miwa, H., Konno, N., 2005. Geographical threshold graphs with smallworld and scale-free properties. Phys. Rev. E 71, 036108. May, R.M., 2006. Network structure and the biology of populations. Trends Ecol. Evol. 21, 394–399. May, R.M., Lloyd, A.L., 2001. Infection dynamics on scale-free networks. Phys. Rev. E 64, 066112. Meentemeyer, R.K., Rank, N.E., Anacker, B.L., Rizzo, D.M., Cushman, J.H., 2008. Influence of land-cover change on the spread of an invasive forest pathogen. Ecol. Appl. 18, 159–171. Meloni, S., Arenas, A., Moreno, Y., 2009. Traffic-driven epidemic spreading in finitesize scale-free networks. Proc. Natl. Acad. Sci. U.S.A. 106, 16897–16902. Meyers, L.A., Newman, M.E.J., Pourbohloul, B., 2006. Predicting epidemics on directed contact networks. J. Theor. Biol. 240, 400–418. Montoya, J.M., Sole´, R.V., 2002. Small world patterns in food webs. J. Theor. Biol. 214, 405–412. Moore, C., Newman, M.E.J., 2000. Epidemics and percolation in small-world networks. Phys. Rev. E 61, 5678–5682. Moslonka-Lefebvre, M., Pautasso, M., Jeger, M.J., 2009. Disease spread in small-size directed networks: epidemic threshold, correlation between links to and from nodes, and clustering. J. Theor. Biol. 260, 402–411. Muneepeerakul, R., Bertuzzo, E., Rinaldo, A., Rodriguez-Iturbe, I., 2008. Patterns of vegetation biodiversity: the roles of dispersal directionality and river network structure. J. Theor. Biol. 252, 221–229.

432

M. Pautasso et al. / Ecological Complexity 7 (2010) 424–432

Namba, T., Tanabe, K., Maeda, N., 2008. Omnivory and stability of food webs. Ecol. Compl. 5, 73–85. Neutel, A.M., Heesterbeek, J.A.P., van de Koppel, J., Hoenderboom, G., Vos, A., Kaldeway, C., Berendse, F., de Ruiter, P.C., 2007. Reconciling complexity with stability in naturally assembling food webs. Nature 449, 599–602. Newman, M.E.J., Strogatz, S.H., Watts, D.J., 2001. Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E 64, 026118. Nielsen, A., Bascompte, J., 2007. Ecological networks, nestedness and sampling effort. J. Ecol. 95, 1134–1141. Olesen, J.M., Bascompte, J., Dupont, Y.L., Jordano, P., 2006. The smallest of all worlds: pollination networks. J. Theor. Biol. 240, 270–276. Park, S.M., Kim, B.J., 2006. Dynamic behaviors in directed networks. Phys. Rev. E 74, 026114. Parke, J.L., Linderman, R.G., Osterbauer, N.K., Griesbach, J.A., 2004. Detection of Phytophthora ramorum blight in Oregon nurseries and completion of Koch’s postulates on Pieris, Rhododendron, Viburnum and Camellia. Plant Dis. 88, 87– 187. Pastor-Satorras, R., Vespignani, A., 2003. Epidemics and immunization in scale-free networks. In: Bornholdt, S., Schuster, H.G. (Eds.), Handbook of Graphs and Networks. From the Genome to the Internet. Wiley, Weinheim, pp. 111–130. Pautasso, M., Jeger, M.J., 2008. Epidemic threshold and network structure: the interplay of probability of transmission and of persistence in small-size directed networks. Ecol. Complex. 5, 1–8. Pellis, L., Ferguson, N.M., Fraser, C., 2009. Threshold parameters for a model of epidemic spread among households and workplaces. J. R. Soc. Interf. 6, 979–987. Perrings, C., Dehnen-Schmutz, K., Touza, J., Williamson, M., 2005. How to manage biological invasions under globalization. Trends Ecol. Evol. 20, 212–215. Petit, R.J., Bialozyt, R., Garnier-Gere, P., Hampe, A., 2004. Ecology and genetics of tree invasions: from recent introductions to Quaternary migrations. For. Ecol. Manage. 197, 117–137. Plantegenest, M., Le May, C., Fabre, F., 2007. Landscape epidemiology of plant diseases. J. R. Soc. Interf. 4, 963–972. Prospero, S., Gru¨nwald, N.J., Winton, L.M., Hansen, E.M., 2009. Migration patterns of the emerging plant pathogen Phytophthora ramorum on the West coast of the United States of America. Phytopathology 99, 739–749. Ribbens, S., Dewulf, J., Koenen, F., Mintiens, K., de Kruif, A., Maes, D., 2009. Type and frequency of contacts between Belgian pig herds. Prev. Vet. Med. 88, 57–66. Riley, S., 2007. Large-scale spatial-transmission models of infectious disease. Science 316, 1298–1301. Rizzo, D.M., Garbelotto, M., Hansen, E.A., 2005. Phytophthora ramorum: integrative research and management of an emerging pathogen in California and Oregon forests. Ann. Rev. Phytopathol. 43, 309–335. Roy, M., Pascual, M., 2006. On representing network heterogeneities in the incidence rate of simple epidemic models. Ecol. Compl. 3, 80–90. Sander, L.M., Warren, C.P., Sokolov, I.M., Simon, C., Koopman, J., 2002. Percolation on heterogeneous networks as a model for epidemics. Math. Biosci. 180, 293–305. Sarama¨ki, J., Kaski, K., 2005. Modelling development of epidemics with dynamic small-world networks. J. Theor. Biol. 234, 413–421. Selosse, M.A., Richard, F., He, X., Simard, S.W., 2006. Mycorrhizal networks: des liaisons dangereuses? Trends Ecol. Evol. 21, 621–628. Shaw, M.W., 1994. Modelling stochastic processes in plant pathology. Ann. Rev. Phytopathol. 32, 523–544. Shirley, M.D.F., Rushton, S.P., 2005. The impacts of network topology on disease spread. Ecol. Compl. 2, 287–299. Silva, S.L., Ferreira, J.A., Martins, M.L., 2007. Epidemic spreading in a scale-free network of regular lattices. Physica A 377, 689–697. Slippers, B., Stenlid, J., Wingfield, M.J., 2005. Emerging pathogens: fungal host jumps following anthropogenic introduction. Trends Ecol. Evol. 20, 420–421. Sork, V.L., Smouse, P.E., 2006. Genetic analysis of landscape connectivity in tree populations. Landsc. Ecol. 21, 821–836. Southworth, D., He, X.H., Swenson, W., Bledsoe, C.S., Horwath, W.R., 2005. Application of network theory to potential mycorrhizal networks. Mycorrhiza 15, 589– 595. Sporns, O., Kotter, R., 2004. Motifs in brain networks. PLoS Biol. 2, 1910–1918.

Stam, C.J., 2004. Functional connectivity patterns of human magnetoencephalographic recordings: a ‘small-world’ network? Neurosci. Lett. 355, 25–28. Stokstad, E., 2004. Nurseries may have shipped sudden oak death pathogen nationwide. Science 303, 1959–11959. Supekar, K., Menon, V., Rubin, D., Musen, M., Greicius, M.D., 2008. Network analysis of intrinsich functional brain connectivity in Alzheimer’s disease. PLoS Comp. Biol. 4, e1000100. Swart, J., Lawes, M.J., 1996. The effect of habitat patch connectivity on samango monkey (Cercopithecus mitis) metapopulation persistence. Ecol. Model. 93, 57– 74. Takeuchi, F., Yamamoto, K., 2006. Effectiveness of realistic vaccination strategies for contact networks of various degree distributions. J. Theor. Biol. 243, 39–47. Thrall, P.H., Antonovics, J., 1995. Theoretical and empirical studies of metapopulations—population and genetic dynamics of the Silene-Ustilago system. Can. J. Bot. 73, S1249–S1258. Thrall, P.H., Burdon, J.J., 2002. Evolution of gene-for-gene systems in metapopulations: the effect of spatial scale of host and pathogen dispersal. Plant Pathol. 51, 169–184. Tildesley, M.J., Keeling, M.J., 2009. Is R0 a good predictor of final epidemic size: footand-mouth disease in the UK. J. Theor. Biol. 258, 623–639. Trakhtenbrot, A., Nathan, R., Perry, G., Richardson, D.M., 2005. The importance of long-distance dispersal in biodiversity conservation. Div. Dist. 11, 173–181. Trapman, P., 2007. On analytical approaches to epidemics on networks. Theor. Pop. Biol. 71, 160–173. Urban, D.L., Minor, E.S., Treml, E.A., Schick, R.S., 2009. Graph models of habitat mosaics. Ecol. Lett. 12, 260–273. Verdasca, J., da Gama, M.M.T., Nunes, A., Bernardino, N.R., Pacheco, J.M., Gomes, M.C., 2005. Recurrent epidemics in small world networks. J. Theor. Biol. 233, 553–561. Vespignani, A., 2008. Reaction-diffusion processes and epidemic metapopulation models in complex networks. Eur. Phys. J. B 64, 349–353. Vuorinen, V., Peltomaki, M., Rost, M., Alava, M.J., 2004. Networks in metapopulation dynamics. Eur. Phys. J. B 38, 261–268. Wang, X.F., 2002. Complex networks: topology, dynamics and synchronization. Int. J. Bifurc. Chaos 12, 885–916. Wang, J.Z., Liu, Z.R., 2009. Mean-field level analysis of epidemics in directed networks. J. Phys. A 42, 355001. Wang, L., Zhu, C.Z., He, Y., Zang, Y.F., Cao, Q.J., Zhang, H., Zhong, Q.H., Wang, Y.F., 2009. Altered small-world brain functional networks in children with attention-deficit/hyperactivity disorder. Human Brain Map. 30, 638–649. Watts, D.J., Strogatz, S.H., 1998. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442. Webb, C.R., 2005. Farm animal networks: unraveling the contact structure of the British sheep population. Prev. Vet. Med. 68, 3–17. Werres, S., Marwitz, R., Veld, W.A.M.I., De Cock, A.W.A.M., Bonants, P.J.M., De Weerdt, M., Themann, K., Ilieva, E., Baayen, R.P., 2001. Phytophthora ramorum sp nov., a new pathogen on Rhododendron and Viburnum. Mycol. Res. 105, 1155– 1165. Westgarth, C., Gaskell, R.M., Pinchbeck, G.L., Bradshaw, J.W.S., Dawson, S., Christley, R.M., 2009. Walking the dog: exploration of the contact networks between dogs in a community. Epidemiol. Infect. 137, 1169–1178. White, K.A.J., Gilligan, C.A., 2006. The role of initial inoculum on epidemic dynamics. J. Theor. Biol. 242, 670–682. Wylie, D.C., Getz, W.M., 2009. Sick and edgy: walk-counting as a metric of epidemic spreading on networks. J. R. Soc. Interf. 6, 897–907. Xu, X.J., Wu, Z.X., Chen, Y., Wang, Y.H., 2004. Steady states of epidemic spreading in small-world networks. Int. J. Mod. Phys. C 15, 1471–1477. Xu, X.M., Harwood, T.D., Pautasso, M., Jeger, M.J., 2009. Spatio-temporal analysis of Phytophthora ramorum cases in England and Wales (2003–2006). Ecography 32, 504–516. Zhao, H., Gao, Z.Y., 2007. Modular epidemic spreading in small-world networks. Chin. Phys. Lett. 24, 1114–1117. Zhou, T., Fu, Z.Q., Wang, B.H., 2006. Epidemic dynamics on complex networks. Prog. Nat. Sci. 16, 452–457.