Review

Blackwell Publishing Ltd

Research review Modelling disease spread and control in networks: implications for plant sciences

Author for correspondence: Marco Pautasso Tel: +44 (0)20 759 42816 Fax: +44 (0)20 759 42669 Email: [email protected]

Mike J. Jeger1, Marco Pautasso1, Ottmar Holdenrieder2 and Mike W. Shaw3

Received: 13 December 2006 Accepted: 9 January 2007

Whiteknights, Reading RG6 6AS, UK

1

Division of Biology, Imperial College London, Wye Campus, Kent TN25 5AH, UK; 2Institute of

Integrative Biology, Department of Environmental Sciences, Eidgenössische Technische Hochschule, 8092 Zurich, Switzerland; 3The University of Reading, School of Biological Sciences, Lyle Tower,

Summary Key words: complex networks, disease management, epiphytotics, landscape pathology, network structure, plant pathogens, small-world, spatial dispersal.

Networks are ubiquitous in natural, technological and social systems. They are of increasing relevance for improved understanding and control of infectious diseases of plants, animals and humans, given the interconnectedness of today’s world. Recent modelling work on disease development in complex networks shows: the relative rapidity of pathogen spread in scale-free compared with random networks, unless there is high local clustering; the theoretical absence of an epidemic threshold in scale-free networks of infinite size, which implies that diseases with low infection rates can spread in them, but the emergence of a threshold when realistic features are added to networks (e.g. finite size, household structure or deactivation of links); and the influence on epidemic dynamics of asymmetrical interactions. Models suggest that control of pathogens spreading in scale-free networks should focus on highly connected individuals rather than on mass random immunization. A growing number of empirical applications of network theory in human medicine and animal disease ecology confirm the potential of the approach, and suggest that network thinking could also benefit plant epidemiology and forest pathology, particularly in human-modified pathosystems linked by commercial transport of plant and disease propagules. Potential consequences for the study and management of plant and tree diseases are discussed. New Phytologist (2007) doi: 10.1111/j.1469-8137.2007.02028.x © The Authors (2007). Journal compilation © New Phytologist (2007)

Introduction Much scientific work is currently focusing on the properties of networks (see Barabási & Albert, 1999; Newman, 2003; papers cited therein). From a physical point of view, networks may involve transport of energy, matter or information. From a mathematical standpoint, networks are sets of elements and

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of the relations between them. From both perspectives, networks are models applicable to a wide variety of natural, technological and social systems (e.g. Strogatz, 2001; Albert & Barabási, 2002; Table 1). Networks can be of interest to plant scientists when they are formed by a physical structure, when they refer to abstract relationships between connected entities (e.g. different species), and when they underline processes or flows in a structure.

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Table 1 Examples of natural, technological and social networks that have been the object of recent analyses Network

Type

Examples of references

Natural Prebiotic mutually catalytic pathways Microbial phylogenies (horizontal and vertical transfer of genes between microbes) Cellular and metabolic dynamics (interactions of cellular components and biochemical molecules)

V SF SF

The topology of food webs (who eats whom in ecological communities)

V

Neural networks (the connections between neurons of e.g. the nematode Caenorhabditis elegans) Ant nests (a set of chambers interconnected by galleries) Amphibian meta-populations in ponds Bats roosting in hollow trees Foraging walks of primates Spatially remote thunderstorms The Earth’s climate system (e.g. the correlations of monthly pressures on a 5 by 5 degrees grid) Earthquake networks (links between strongly correlated earthquake events) Syllable and word webs (the co-occurrence of syllables in words and of words in sentences)

SW N SF SF SF SW SW SF SF, SW

The decomposition of even numbers into two prime numbers (following the Goldbach conjecture)

SW

Shenhav et al. (2005) Kunin et al. (2005) Jeong et al. (2000); Albert (2005) Dunne et al. (2002); Woodward et al. (2005) Watts & Strogatz (1998); Humphries et al. (2006) Buhl et al. (2004) Fortuna et al. (2006) Rhodes et al. (2006) Boyer et al. (2006) Yair et al. (2006) Tsonis & Roebber (2004) Baiesi & Paczuski (2005) Cancho & Solé (2001); Soares et al. (2005) Chandra & Dasgupta (2005)

Technological Railways (stations and connecting trains) Urban street networks (streets and intersections) Electric power grids (power generators, substations, high-voltage transmission lines)

SW SW SF

Air transport (airports and connecting flights)

V

The Internet (transit backbone providers and their nodes)

V

Computing grids (a set of processors connected by some kind of communication network) Software maps (the topology of complex software systems) Electronic circuits in computers (logic gates connected by wires)

V V SW

Social Family networks (who is related to whom) Friendship groups (who knows whom)

SF SW

Bookworm contacts (book buyers spreading recommendations) Links between Wikipedia articles

SF SF

The World Wide Web (links to web pages)

V

Virtual learning communities (cultural transmission as contagion) Medieval heresies and inquisition Committees (who is in a meeting with whom) Telephone calls (sets of people with whom a telephone user communicates) Email patterns (electronic messages between email addresses) Co-authorship groups (who does research with whom)

? SF? ? SF SF SF

Citation webs of scientists (who cites whom) Terrorist groups (webs of perpetrators of terror attacks) Financial fluctuations (e.g. the cross-correlations of stock prices) Innovation flows (e.g. the flow of technological improvements from firm to firm) Human movements (tracked for instance following the dispersal of bank notes) The UK horse racing network The world trade web (trade relationships between different countries) Sexual partnerships

SF V SF SF SF SW V V

Sen et al. (2003) Jiang & Claramunt (2004) Amaral et al. (2000); Chassin & Posse (2005) Barrat et al. (2004a); Guimerá et al. (2005) Gorman & Malecki (2000); Pastor-Satorras et al. (2001) Costa et al. (2005) Valverde & Sole (2005) Cancho et al. (2001) Coelho et al. (2005) Milgram (1967); Amaral et al. (2000) Sornette et al. (2004) Capocci et al. (2006); Zlatic et al. (2006) Adamic (1999); Bornholdt & Ebel (2001) Giani et al. (2005) Ormerod & Roach (2004) Porter et al. (2005) Xia et al. (2005) Ebel et al. (2002) Newman (2001); Barber et al. (2006) Seglen (1992) Qin et al. (2005) Boginski et al. (2005) Di Matteo et al. (2005) Brockmann et al. (2006) Christley & French (2003) Serrano & Boguñá (2003) Kretzschmar (2000)

Network types (SF, scale-free; SW, small-world; V, various types; N, none of the previous types) are assigned on the basis of the references provided and do not rule out the possibility that other studies of networks of a similar nature may suggest a different structure. For other references the reader is referred to reviews by, for example, Albert & Barabási (2002), Dorogovtsev & Mendes (2002) and Newman (2003).

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Table 2 Idealized network types discussed in the text (see Fig. 1 for a visual representation) Type

Description

Properties

Examples of references

Local

Neighbourhood connectivity (typically regular lattices) Nodes connected with probability P (Erdos–Renyi network) Local network rewired with shortcuts (Watts–Strogatz network) Nodes preferentially connecting to hubs (Barabási–Albert network)

High clustering, high path length Low clustering, low path length High clustering, low path length Proportion pk of vertices connected to k other vertices drops with increasing k as k–α for some constant α

Harris (1974); Keeling & Eames (2005); Shirley & Rushton (2005a) Erdos & Renyi (1960); Bollobás (1985); Roy & Pascual (2006) Watts & Strogatz (1998); Barrat & Weigt (2000); Moore & Newman (2000) Barabási & Albert (1999); Balthrop et al. (2004); Hwang et al. (2005); Newman (2005a)

Random Small-world Scale-free

Network theory (e.g. Bollobás, 1979; Chartrand, 1985; Chen, 1997) has many practical biological applications in plant sciences, for example biochemical networks (Aloy & Russell, 2004; Arita, 2005; De Silva & Stumpf, 2005; Green & Sadedin, 2005; Proulx et al., 2005; Sweetlove & Fernie, 2005; Uhrig, 2006), which form much of the focus of what is now termed systems biology. But epidemiological studies are also interesting dynamical problems in a system of connected entities (e.g. Matthews & Woolhouse, 2005; Zheng et al., 2005). A number of papers have summarized recent work modelling the spread of diseases in networks (to which we refer for more details, elaboration and perspectives; e.g. Wallinga et al., 1999; Watts, 1999; Koopman, 2004; Keeling & Eames, 2005; Keeling, 2005b,c; Shirley & Rushton, 2005a; May, 2006; Parham & Ferguson, 2006). However, most of this work has focused on human and animal diseases, thus raising the question of whether a similar approach may also be beneficial in plant and forest pathology. The relevance of network theory to the epidemiology of plant diseases is demonstrated by the growing literature on: how landscape patterns affect the spread and establishment of plant pathogens; large-scale site topographic, climatic and edaphic characters predisposing to plant disease risk; and host and pathogen genetic variation across their geographical distribution. New genetic tools enable a much more precise study of the dispersal of organisms on a geographic scale (e.g. Banke & McDonald, 2005; Garrett et al., 2006; Stukenbrock et al., 2006), and many questions in large-scale epidemiology can be conveniently framed in terms of network theory. In this review, we recapitulate important results of modelling work on disease spread and control in networks, present empirical studies applying network theory to a number of case studies of human and animal pathologies, and discuss potential consequences for plant disease epidemiology (e.g. Gilligan, 2002; Jeger, 2004; Burdon et al., 2006) and for landscape pathology, which is the study and management of tree diseases on a larger scale than previously common and making use of the tools of landscape ecology (e.g. Geils, 1992; Holdenrieder et al., 2004; Lundquist & Hamelin, 2005; Pautasso et al., 2005).

Modelling work Epidemiological approaches based on networks study individuals and their contacts as a set of vertices (also known as nodes, i.e. susceptible/infected entities) and connecting edges (links and infection events) (e.g. May & Lloyd, 2001; PastorSatorras & Vespignani, 2001a; Newman, 2002). The contact patterns between susceptible and/or infected individuals form a network, which can be classified into various types (Table 2; Fig. 1). In the approach most obviously related to network theory, disease has been modelled on regular lattices, where the probability of infection being passed to neighbouring cells on a grid can be constant (zero-dimensional models, or massaction mixing; e.g. Anderson & May, 1991; Filipe & Maule, 2004) or random (epidemics in random graphs; e.g. Barbour & Mollison, 1990), or can decrease as a certain function of the distance from an infected cell (typically producing travelling waves as in continuum models; e.g. Marchand et al., 1986; Zadoks & Van den Bosch, 1994; Van den Bosch et al., 1999; Russell et al., 2004). For plants, a further distinction can be drawn between models that operate in a landscape of a more or less fragmented natural environment and those studying the behaviour of trade-based networks. However, real populations rarely fall exactly into one of these idealizations, being neither completely artificial nor natural, and neither perfectly well-mixed, nor completely random, nor located on regular lattices (e.g. Mollison, 1977; Shaw, 1994; Kuperman & Abramson, 2001; Blyuss, 2005). Networks where connectivity is neither local, nor regular nor random but something in between these three extremes are dubbed ‘small-world’ networks, because the shortest path length between individuals increases only logarithmically with increasing size of the network (Table 2; Fig. 1). It is on these small-world networks, where global distances are low and local interconnectedness (clustering) is high (Petermann & De Los Rios, 2004; Roy & Pascual, 2006), that many modellers have focused their recent epidemiological work. Another type of network that has been the object of much investigation is the scale-free network. In scale-free networks, the probability that a given node has k connections follows an

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Fig. 1 Four basic kinds of network structure: (a) local, (b) random, (c) small-world, 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.

inverse power-law distribution (Table 2; Fig. 1). This is called the degree distribution of the network. Illustrations of such a scale-free distribution in real-world networks can be found in many recent papers (e.g. Shirley & Rushton, 2005b; Barber et al., 2006; Montoya et al., 2006). Most scale-free networks have small-world properties (e.g. Amaral et al., 2000), but there are scale-free networks with low clustering, which are thus not small-worlds, and there are small-world networks that are not scale-free (see e.g. Jiang & Claramunt, 2004; Humphries et al., 2006). A particular class of networks with both scale-free and small-world properties is called Apollonian networks (e.g. Andrade et al., 2005), but there are many kinds of network with both scale-free and small-world properties that are not Apollonian (e.g. Matsuyama et al., 2005; Palotai et al., 2005). Each type of complex network described here can be embedded in space (e.g. Rozenfeld et al., 2002; Barthélemy, 2003; Morita, 2006). The distance between two nodes will be reflected in the strength of the connection between them, although it will not necessarily be the only determinant. This is of particular interest to plant scientists, as a plant lives in a single physical location. Epidemiological models in networks can be roughly subdivided into those pertaining to disease spread and those pertaining to disease control, although the interrelations are obvious. Throughout, for ease of readability, mathematical assumptions and notations of models are not reported from the original papers, to which interested readers are referred. Disease spread in networks Recent work on disease spread in networks shows that the probability of an epidemic occurring following an initial

infection is influenced by the contact structure in the first phases of the epidemic (Gallos & Argyrakis, 2003; Brauer, 2005; Saramäki & Kaski, 2005). Depending on where it originated, the first phase of an epidemic in a scale-free network is often marked by super-spreading events, in which a few infected entities with high numbers of connections are responsible for the vast majority of infections (Barthélemy et al., 2004; Brauer, 2005; Duan et al., 2005; James et al., 2007). An immediate finding is thus that infectious diseases can spread more easily in scale-free and small-world networks than in regular lattices (e.g. Watts & Strogatz, 1998; Kuperman & Abramson, 2001) and random networks (Kiss et al., 2006a; Matthäus, 2006), although spread is less predictable because of more stochasticity during the very early stages of the epidemic. High degrees of local clustering can lead to a less than exponential spread of diseases even at the very beginnings of epidemics (Szendrøi & Csányi, 2004; Verdasca et al., 2005). A second major result of models is that, in small-world networks, the threshold for an epidemic to occur decreases as a power-law with increasing number of shortcuts (longdistance infections) (Zekri & Clerc, 2001). As a result of the heterogeneity in the number of links per node, epidemics in scale-free networks of infinite size theoretically never die out, no matter how low the basic reproduction rate is (PastorSatorras & Vespignani, 2001b; Moreno et al., 2002b; Boguñá et al., 2003; Barthélemy et al., 2005). Therefore, in scale-free networks, in order to correctly estimate R0, some measure of the variance in the degree distribution of contacts is needed (Woolhouse et al., 2005; Ferrari et al., 2006; Green et al., 2006; May, 2006). The concept of a basic reproduction number R0 comes from studies of disease spread in homogenous landscapes (e.g. Heesterbeek, 2002; Brauer, 2005; Green et al., 2006; for

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plant epidemiology, e.g. Jeger, 1986; Jeger & van den Bosch, 1994). It is defined as the number of secondary infections caused by a single infective individual introduced into a wholly susceptible population, and depends normally on: the number of potentially infectious contacts per individual; the probability of infection per contact between infectious and susceptible individuals; and the duration of infection (e.g. Giesecke, 1994; Hethcote, 2000; Lloyd-Smith et al., 2005; May, 2006). If R0 is less than 1 in a homogeneous population, epidemics fail to establish. In networks in which the probability of a node having a certain number of contacts decreases exponentially as the number of contacts increases, which are therefore not scale-free, a threshold R0 exists even in case of an infinite size (Pastor-Satorras & Vespignani, 2001a). This is because the transmissibility scales in non-scale-free networks with the average degree (number of connections), whereas in scale-free networks it scales with the variance of the degree distribution. For an epidemic threshold to be absent (in the infinite size limit, and with the exponent of the power-law smaller than 3; see Table 2), the connectivity of the network must thus be scalefree. A scale-free connectivity (i.e. a linear decrease in log-log space of the number of links per node with increasing number of nodes in a network) implies the existence of hubs, or highly connected nodes. These hubs are largely responsible for the observed differences between processes (not only epidemiological) modelled on scale-free networks and other kinds of complex networks (e.g. Amaral et al., 2000). The key role of hubs is corroborated by the finding that not only is the epidemic threshold lower in scale-free networks, but the time needed for equilibrium levels of infection to be reached is shorter (e.g. Shirley & Rushton, 2005a). Work demonstrating the absence of an epidemic threshold in scale-free networks of infinite size is not a mathematical irrelevance, because, although scale-free networks of finite size (with cut-offs at the lower and higher ends of the distribution of connections) have thresholds, neglecting long-distance connectivity still leads to an overestimation of the epidemic threshold in finite populations (May & Lloyd, 2001; PastorSatorras & Vespignani, 2002a; Joo & Lebowitz, 2004; Hwang et al., 2005; Ying et al., 2005). Whether or not epidemic thresholds are really lower in real-world heterogeneous landscapes than in homogenous ones is an interesting question and requires field work for it to be tested also empirically. Whether or not scale-free networks are relevant for plant meta-populations will be discussed in detail later in the review (see ‘Potential implications for plant and forest pathology’). Suffice it to say for now that, although plant networks are finite (the world’s circumference has an order of magnitude of 107 m, and individual leaves of diseased plants are around 10−2 m, which would give potentially 10 orders of magnitude), if an epidemic were to spread around the world during a period substantially longer than its time of local development, then it might not be too far-fetched to apply results from models

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of scale-free networks of infinite size to plant networks in today’s globalized world. It would also be interesting to know how relevant the distinction established by these thresholds is for real epidemics. These might be theoretically unstable, but last sufficiently long to make the threshold irrelevant in practice. A third take-home message is that more complicated models also have an epidemic threshold. Thresholds are predicted in networks with high local clustering (Eguíluz & Klemm, 2002), in models taking into account differences in the rate of infection for individuals with different connectivity (Olinky & Stone, 2004), and, depending on the initial density of infection, in models of disease spread in scale-free networks created by the deactivation of links with probability inversely proportional to their number of connections (Moreno & Vázquez, 2003). Similarly, the combined effects of ageing (older individuals in a network differing in their susceptibility) and removal of links (dead individuals may no longer be connected to susceptible ones) on epidemic dynamics in scale-free networks lead to a critical value of effective links in the network below which only local spread of disease takes place (Chan et al., 2004; see also Amaral et al., 2000). The epidemic threshold in community networks (where there are groups of individuals with more connections between them than with individuals outside the group), however, is still lower than in a random network, other things being equal (Liu & Hu, 2005). This issue is of relevance for plant diseases, because plants in a field, forest or nursery may be at risk of infection from pathogens present on plants in the same location, but also from longer distance movements of disease propagules from other fields, forests or nurseries (e.g. Parnell et al., 2006; Shaw et al., 2006). However, in scale-free networks with household structure (which enables a distinction between infection among and within households (Bian, 2004); for plant meta-populations, the concept of household may be conveniently thought of in terms of farms, pathosystems, or pathoregions (see Holdenrieder et al., 2004)), models predict that a disease can spread through the network even if the recovery rate in single households is greater than the local infection rate (Liu et al., 2004). However, as for homogenous networks, Grabowski & KosiNski (2004) have found that in household networks disease spread is slower when there is much local clustering. The same authors have reported that, in these household network models, diseases with a lower rate of spread have a greater probability of surviving endemically. This finding is in agreement with the general theory of disease spread when there is a high variance in contacts between individuals (Anderson & May, 1991). Household structure also introduces into models an asymmetry between individuals inside groups and individuals connecting groups, with a higher probability of infection from connecting individuals to those within a household than vice versa (Meyers et al., 2003). A fourth general finding is thus that asymmetries can have an influence on disease spread in networks. It is known that

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models with symmetrical interactions are often unrealistic approximations of real systems (e.g. Bauch & Galvani, 2003; Bascompte et al., 2006; Chavez et al., 2006). Many real pathosystems present heterogeneities or asymmetries in the flow of disease propagules carried by different links (e.g. Dall’Asta, 2005; Yan et al., 2005). A number of studies have started investigating the spread of pathogens in directed networks of various kinds (e.g. Newman et al., 2001; Schwartz et al., 2002; Boguñá & Serrano, 2005; Meyers et al., 2006). For plants, directed networks may be of relevance, for instance, in the case of the trade among nurseries, garden centres and retail centres, where the probability of movement from a certain nursery to a given garden centre will tend to differ greatly from the probability of movement in the reverse direction. In many real-world examples, not only the number and direction of connections of different nodes may be heterogeneous, but also their strength. Recent work has started to take into account the variation in intensity of different links (e.g. Yook et al., 2001; Barrat et al., 2004b; Jezewski, 2005), but investigations of the implications of including weight in models of disease spread and control in complex networks are still in their infancy (e.g. Wu et al., 2005; Dall’Asta et al., 2006). Intuitively, when higher connectivity strength per link is assigned to highly connected individuals, the findings discussed for nonweighted scale-free networks will tend to be present even more strongly. But it would be interesting to know whether or not (and under which conditions) strong links for the many nodes with few connections can cancel out the effect of hubs if these have weak links. For plant diseases, hubs with weak links may be a realistic model of control targeted exclusively at plant movements from and to highly connected traders, while strong links to many other nodes would be an appropriate model if commercial growers with many customers also tended to sell significantly more plants than small-scale retailers with few connections. Disease control in networks A number of consequences for disease management can be inferred from modelling work on disease spread in networks. First, models show that random immunization of even a high proportion of individuals is not an effective strategy to control an epidemic operating in scale-free networks. This result follows from the theoretical absence of an epidemic threshold in scale-free networks with infinite variance in connectivity: in this case, immunity is not conferred even by high densities of randomly immunized individuals (e.g. PastorSatorras & Vespignani, 2002b; Zanette & Kuperman, 2002; Takeuchi & Yamamoto, 2005). In networks with small-world properties, as a result of the effect of shortcuts, the influence of an untargeted immunization protocol is generally lower than when only local infection is possible (KosiNski & Adamowski, 2004). In a botanical context, protection may be conferred by biological control or fungicides at the level of the

individual field, and by the intermixing of resistant varieties and species at the landscape level. Rather than random immunization, when disease spreads on a scale-free network, an effective control strategy should immunize highly connected nodes (e.g. Pastor-Satorras & Vespignani, 2002b; Zanette & Kuperman, 2002; Liu et al., 2003; Chang & Young, 2005; Hwang et al., 2005; LloydSmith et al., 2005). Models suggest that the more strategies focus on immunization of highly connected individuals, the more likely they are to bring under control an epidemic spreading on a scale-free network, and the cheaper a successful strategy will be (Dezsø & Barabási, 2002). These models predict that in a finite population, even with small-world properties, above some critical immunization level the disease is confined locally (Zanette & Kuperman, 2002). Moreover, just as local clustering slows down the spread of disease in networks (see ‘Disease spread in networks’), a lower efficiency in contact tracing is required to control disease in a clustered network, other things being equal (Eames & Keeling, 2003). For plant diseases, contact tracing often translates into removal of infected plants and the containment of further pathogen spread across a dispersal network through quarantine measures. Even without human intervention, it is a common observation – for example, in the saprotrophic invasion of the soil-borne pathogen Rhizoctonia solani – that patches of susceptible plants can remain uninfected because they are surrounded by immune individuals (e.g. Jeger, 1989; Bailey et al., 2000; Sander et al., 2002). A number of parameters have been analysed to enable the identification of highly connected individuals in small-world and random networks: degree (number of contacts), betweenness (a measure of the probability of an individual being on the path between other individuals), shortest-path betweenness (the same, but for the shortest path), and farness (the sum of the number of steps between an individual and all other individuals). Degree, the network parameter most easily measured, was found to be at least as good as the other metrics in identifying highly connected individuals and thus in predicting risk of infection (Christley et al., 2005). This result is to be expected from the common finding in real-world networks that the betweenness of a node is positively correlated with the degree of the node (e.g. Lee et al., 2006). From a practical point of view, however, there is often only limited knowledge at the beginning of an epidemic outbreak about the number of connections single individuals have (Dybiec et al., 2004). For many airborne diseases, a substantial fraction of contacts may be untraceable (Eames & Keeling, 2003). When contact tracing is possible in theory, if latent periods are short there may not be time in practice to trace the contacts of connected and infected individuals (Huerta & Tsimring, 2002; Kiss et al., 2005). But in plant epidemiology, where long latency periods are common, it may also be difficult to trace contacts. Researchers have thus tried to identify control strategies that do not require pathologists to know the

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complete structure of the network at risk. One of these is the immunization of a small fraction of random acquaintances of randomly selected individuals (acquaintance immunization). As hubs have by definition a large number of links, the probability that a random neighbour of a random node is a hub is very large. This is thus a simple way to identify and remove highly connected individuals even without knowing who they are in advance (Cohen et al., 2003; Holme, 2004; Madar et al., 2004; May, 2006). Alternatively, ring vaccination of individuals at less than a certain radius from infected cases has been modelled (e.g. Ahmed et al., 2002; Pourbohloul et al., 2005), which is both more effective and more costly the larger the radius chosen (Dybiec et al., 2004). This is essentially a local strategy and has long been studied for regular lattices and carried out in homogenous pathosystems. The same authors report that the effect of including long-distance interactions in models (thus moving from a regular network to a small-world one) is that the radius of the local control strategy has to be greatly increased, with proportionally poorer cost-effectiveness. Similar implications for disease control are obtained from models aiming to determine the best strategy for protecting computer networks. In this case, of course, the finding that random removal of links does not affect a scale-free network is a good rather than a bad thing because it makes networks more robust (e.g. Vázquez & Moreno, 2003). Similarly, in computer networks highly connected individuals are not to be removed to prevent disease spread, but protected to decrease network vulnerability (e.g. Crucitti et al., 2003). This is because, in networks with a heterogeneous distribution of connections, when highly connected individuals are disconnected a global cascade of failures is likely to follow (e.g. Moreno et al., 2002a; Motter, 2004; Zhao et al., 2004). Further examples of this kind include studies of the structural vulnerability of electric power grids, which were found to be robust to most random perturbations, but very sensitive to disturbances affecting key power stations (e.g. Albert et al., 2004; Crucitti et al., 2004; Chassin & Posse, 2005; Kinney et al., 2005). Similar conclusions about the general robustness of scale-free networks to random disruptions of their components have been drawn from studies of metabolic networks (e.g. Albert et al., 2000; Dorogovtsev & Mendes, 2002), although for plant cells the picture may be more complicated (Sweetlove & Fernie, 2005).

Case Studies Not only has the impact of network structure on disease development been modelled, but the tools of network theory have been applied to a number of case studies. In this section, we review some recent empirical applications, drawing conclusions for epidemiology whenever possible. An exemplary application of network theory to an epidemiologic case study is the investigation of how computer network structure affects so-called epidemic algorithms. These

Review

are mechanisms that allow data dissemination (e.g. software updates, peer-to-peer networks and database maintenance) to computers connected in a network (e.g. Acosta-Elias et al., 2004). Large-scale numerical simulations of epidemic algorithms suggest that in scale-free networks data transfer is more efficient but less reliable than in homogenous topologies (Moreno et al., 2004). This finding corroborates the higher speed of disease spread in scale-free networks pointed out in the section ‘Disease spread in network’. The lower reliability emphasizes that disease development in scale-free networks is stochastically affected by the number of connections of the first individuals infected (e.g. Keeling, 1999; Verdasca et al., 2005). A similar suite of studies is related to the spread of memes through social networks. By analogy with the susceptible/ infected/removed (SIR) model in epidemiology, individuals of a population can be subdivided into those not having heard an idea yet; those aware of the concept and communicating it to others; and those having become uninterested and not disseminating it any longer (e.g. Zanette, 2002). Also in this case, models show that one strategy for a successful dissemination of memes is to target hubs (e.g. Duan et al., 2005). Translated into terms of disease control, this finding suggests again that disease spread can best be constrained in scale-free networks by removing from the network individuals with the highest number of connections. For plant diseases that are spread through the nursery trade, the susceptible/infected/ susceptible (SIS) model may be more realistic, as infected nurseries, unless under complete quarantine, may continue to operate even if under surveillance or if plants within a certain distance from the infected material are quarantined. The implications of such models will be addressed in the last section of the review. It is already clear, however, that there are many examples of spatially structured host–pathogen systems where the identification of highly connected nodes in the network underlying the long-distance spread of disease might have been an effective way to delay plant disease expansion (e.g. chestnut blight, Dutch elm disease, black sigatoka of banana and potato blight). The spread of viruses via email messages in computer networks is a further instance that has been analysed from a network theory perspective. A remarkable finding of some models is that the whole network of computers can be made immune from infection by the targeted immunization of a selected 10% of connected computers (Newman et al., 2002). But further analyses have shown that the way in which a virus replicates itself can affect the topology of the computer network, thus making it difficult to control an epidemic (Balthrop et al., 2004). In this case, epidemic control is also made difficult by the increasing disparity between the speed of automated disease spread and that of manual eradication. Apart from epidemic algorithms and the spread of ideas and of computer viruses, work applying network theory to empirical cases can be subdivided into that pertaining to

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human and animal diseases. Few applications have referred to plant diseases. Network theory applied to human diseases A whole series of case studies involves human pathologies. Here the motivation is the greater threat posed by human pathogens in a more and more interconnected world (e.g. Eubank et al., 2004; Hufnagel et al., 2004; Brockmann et al., 2006; Colizza et al., 2006b; Tatem et al., 2006). Recent studies have been motivated by the threat of pandemic influenza. Detailed network models of this and of other globally relevant infectious diseases need accurate estimation of model parameters (Ferguson et al., 2005; Longini et al., 2005; Arino et al., 2006; Colizza et al., 2006a; Germann et al., 2006). However, for the recent severe acute respiratory syndrome (SARS) outbreak, modellers found that random networks did not satisfactorily catch the observed dynamics of the epidemic, and that only the addition of small-world properties allowed realistic description of disease development. In particular, small-world networks are able to account for the otherwise puzzling disparities between the markedly different developments of outbreaks that started simultaneously in different regions (e.g. Masuda et al., 2004; Small et al., 2004, 2006; Bauch et al., 2005; Meyers et al., 2005). Incorporating the heterogeneity in the contact structure into models also allows an accurate matching of predictions with observed dynamics at relatively small scales, as shown by an analysis of a dengue outbreak on Easter Island (Favier et al., 2005). A study of childhood infections dynamics in Canada showed that it is possible to reconstruct the probable network structure for a disease given the time-series data of the epidemics. The epidemic size distribution follows an inverse power-law for rubella and mumps, implying heterogeneous individual contacts and thus a scale-free network; whereas for pertussis a homogenous transmission network is suggested by the exponential distribution of epidemic sizes (Trottier & Philippe, 2005). When the basic structure of a network is known, as in the case of an outbreak of bacterial pneumonia in a residential institution where a household structure was clearly present in the different wards, models can help the management of the disease by pointing out that nurses are the super-spreaders who need to be immunized (Meyers et al., 2003). A similar result, showing that preventive measures need to be applied to individuals with many partners, was found in analyses of an outbreak of gonorrhoea in Alberta, Canada (De et al., 2004). The last example is part of the research using network theory to improve forecasts of the spread of sexually transmitted diseases (STDs; e.g. Liljeros et al., 2001, 2003; Eames & Keeling, 2002; Jolly et al., 2005). Modelling work on STDs illustrates that for real populations of finite size, even though organized in a scale-free network, there exists a non-null epidemic threshold, so the spread of STDs can be stopped

(Gonçalves & Kuperman, 2003; Jones & Handcock, 2003). Whether the dynamic nature (sexual partnerships may evolve through time) of these networks will tend to facilitate (by creating new connections) or hamper (by disrupting the structure of the network) the spread of STDs deserves further investigation, also in the context of venereal diseases of plants (Antonovics, 2005). However, sexual partnership networks tend to be scale-free, as the distribution of the number of sexual partners cumulated over time typically follows an inverse power-law. In this case, only targeted action (aimed at individuals connecting subgroups of the population) can be effective in preventing further spread of STDs (e.g. Liljeros, 2004; Schneeberger et al., 2004). An example is the use of network data to predict the development of an AIDS outbreak in Houston, Texas, USA, where data on social network structure were assessed as the most important requirement for more effective management (Bell et al., 2002). Network theory applied to animal diseases A prime example of the application of network analysis to the study of disease spread in animals is foot and mouth disease (FMD). Much modelling work has been done following the 2001 outbreak in the UK (e.g. Woolhouse, 2003; Keeling, 2005a). But in this case too, it has been advocated that models need to use aspects of network theory (e.g. Haydon et al., 2003; Shirley & Rushton, 2005b; Woolhouse et al., 2005). This is because of the long-distance dispersal exhibited by the viral pathogen (via farm management, commercial exchanges, and possibly airborne dispersal), and by the scale-free contact structure of the farm network, including hubs such as markets and animal shows (e.g. Keeling et al., 2003; Shirley & Rushton, 2005b; Webb, 2005, 2006; Kiss et al., 2006b). However, Woolhouse et al. (2005) argue that, even if the network among livestock farms has a scale-free distribution of contacts, the basic reproduction number is not increased by this because the probability of one farm infecting another was not significantly related to the probability of the first farm becoming infected itself. For the FMD outbreak of 2001 in the UK, a reconstruction of epidemic trees (from putative sources of infection for infected premises) revealed that, if the national ban on movement of cattle had been declared 2 d earlier, the size of the epidemic would probably have been reduced by half (Haydon et al., 2003). Even more effective would have been the removal from the network of the three hubs from which nearly 80% of subsequent infections are thought to have originated. Unfortunately, livestock can spread the disease without showing clinical signs of it for up to 10 d (Shirley & Rushton, 2005b), so the rapid identification of the markets that caused the long-range spread of the disease was not possible. FMD is only one example of the many wildlife diseases potentially spread by movements of animals (Woolhouse et al., 2005). Network analyses related to wildlife diseases

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include a study of bovine tuberculosis of African buffalo in Kruger National Park, South Africa (Cross et al., 2004). This showed that buffalo herds were less tightly clustered in years of dry weather, and that this mixing of the overall population could lead to faster spread of the disease. A somewhat different example is an assessment of the role of long-distance dispersal for the spread of raccoon rabies in Connecticut, USA (Smith et al., 2005). In this case, establishment of disease foci from small-world shortcuts was rare, and the disease can be managed with a local containment strategy. A different finding in relation to Lyme disease is that in northern Spain there are critical stepping-stone habitats with high tick densities, whose removal can markedly alter the connectivity of the landscape (Estrada-Peña, 2003). Network theory applied to plant diseases The applications of network theory discussed above prompt the question of whether network theory can also improve our understanding of the spread of plant and tree diseases. Of course, plants are not as mobile as humans and animals (although their pathogens are not static, and plants themselves can cross long distances from one generation to the next through seeds, pollen and, in some cases, vegetative material). In fact, at a first glance plants do not seem to form social scalefree networks of highly mobile interactions such as, for instance, fish are classically able to materialize, thus potentially facilitating the spread of their epidemics (e.g. Croft et al., 2004, 2005). This may explain why there has so far been relatively little use of network theory in plant epidemiology. Models of epidemics on networks, rather than in continuous space or on lattices, might work better for animal or human than for plant diseases (Bolker, 1999). But there exist threads of plant epidemiological research that use meta-population theory, and in many cases follow similar lines of reasoning to modelling work on networks (e.g. Hanski, 1994; Park et al., 2001, 2002; Gilligan, 2002; Franc, 2004; Otten et al., 2004, 2005; Vuorinen et al., 2004; Watts et al., 2005; Brooks, 2006).

Potential Implications for Plant and Forest Pathology Network theory may be relevant to plant diseases, but not yet have been applied. If so, one explanation of the delayed application may be that the theory is not yet sufficiently mathematically developed to apply to epiphytotics, as plant– pathogen networks in the real world are not only complex, but transient and dynamic. Another explanation is that plant epidemiologists need the development of appropriate software tools to exploit the potential of network theory (see Garrett et al., 2004). It may also be that data on the network structure of plant communities are frequently harder to obtain than the often meticulous records for human and animal epidemics.

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But studies are beginning to show that plant communities can be part of scale-free networks, at least when considered in conjunction with other interacting species (on their own, plants seem to depart from a true fractal spatial distribution; e.g. Lennon et al., 2002; see also Erickson, 1945; Kunin, 1998). An intriguing related question is whether such an absence of a fractal spatial distribution for plants would preclude the existence of a scale-free network (e.g. Berntson & Stoll, 1997). For example, a network of plants and their pollinator species in Greenland was found to show small-world properties (Lundgren & Olesen, 2005). High clustering and small path length between plant species were also reported from a study of the network of frugivorous birds and fleshyfruited plant species in Denmark (Lázaro et al., 2005). These studies are only a few of many investigations of plant–pollinator and plant–frugivore networks. However, these webs may not generally show scale-free properties as, in an analysis of 53 such networks in natural communities, only roughly one-fifth exhibited scale invariance in the connectivity distribution (Jordano et al., 2003). The reasons for such a finding are being debated, but the range of scales involved in each case may be of relevance here (see also Khanin & Wit, 2006). Mycorrhizal networks, when seen from a mycocentric point of view (i.e. considering individual trees as connecting fungal morphotypes and not vice versa), can be scale-free (Southworth et al., 2005). There is much research potential in investigating whether this scale invariance is present more generally in microbial, mycelial and host–parasitoid networks (e.g. Davidson et al., 1996; Klein & Paschke, 2004; Cairney, 2005; Károlyi, 2005; Killingback et al., 2006; see also Friesen et al., 2006). Of course, scale-free networks may also be relevant in plant sciences in relation to food webs (e.g. Dunne et al., 2002) and from a metabolic point of view (e.g. Sweetlove & Fernie, 2005; Uhrig, 2006), for instance in the context of the pathways controlling stomata at different scales (Hetherington & Woodward, 2003). Provided that it is feasible to obtain network data from plant and forest ecosystems, there are a number of reasons to think that network theory may be a convenient tool when dealing with the health of plant and forest pathosystems. In a modern landscape, it may be relatively easy to recognize a scale of description at which one should switch to a network. For crop plants, the obvious unit are fields, farms and trading units on a production chain; for trees, nodes may be forest stands, plantations, urban parks and tree nurseries. The great promise of network theory is that it can help in investigating how disease development parameters vary within and across individual meta-populations (Heesterbeek & Zadoks, 1987; Parnell et al., 2006). Moreover, shipments of plants across continents are a matter of routine nowadays, with often unpredictable consequences for the introduction and spread of exotic plants and their pathogens (e.g. Reichard & White, 2001; Stokstad, 2004; Dumroese & James, 2005; Perrings et al., 2005; Dehnen-Schmutz et al., 2007).

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Fig. 2 Findings of Phytophthora ramorum on plants at retail (crosses), nursery (diamonds), estates/environment (squares) and other (triangles) sites in England and Wales in 2003–2005. Data source: Department for Environment, Food and Rural Affairs, UK.

There is thus a need to assess the properties of plant nursery networks in a number of representative regions and for various pathogens and endophytes (e.g. Stanosz et al., 2005; GiménezJaime et al., 2006; Menkis et al., 2006; Pinto et al., 2006; Stepniewska-Jarosz et al., 2006). This need is immediate wherever nurseries have been tested positive for the presence of Phytophthora ramorum, the causal agent of sudden oak death (e.g. Parke et al., 2004; Daughtrey & Benson, 2005; Rizzo et al., 2005; Fig. 2). Figure 2 suggests that contact tracing information about ongoing and eradicated outbreaks of P. ramorum in the UK may enable the reconstruction of the network underlying the spread of the pathogen, which in turn might enable a more effective control strategy. Nurseries may also be contributing to the spread of Phytophthora alni, as a study from Bavaria suggests ( Jung & Blaschke, 2004), of the western flower thrips Frankliniella occidentalis, which is the vector of tospoviruses, both in North America and in Europe (Kirk & Terry, 2003; Jones et al., 2005), and of Ralstonia solanacearum, the bacterium causing potato brown rot, which has been the object of individual-based modelling in the Netherlands (Breukers et al., 2006). A network approach seems sensible also in relation to botanical gardens, which acted historically as hubs in the introduction of plants outside their natural geographic range (e.g. Mamaev & Andreev, 1996; Ingram, 1999; He, 2002). It remains true that understanding the network, especially its topology, is useful in

devising effective disease management policies even if there is probably no single immunization strategy that can be effective for all types of scale-free networks (Volchenkov et al., 2002). Network thinking may also be relevant for natural plant communities because, although individuals cannot move, seeds can, and provide a means of long-distance dispersal and invasion for species with appropriate life history traits (e.g. Grotkopp et al., 2002; Pysek et al., 2004; Richardson & Rejmanek, 2004; Zedler & Kercher, 2004; Hamilton et al., 2005). There is evidence that the level of long-distance dispersal in tree recolonization after glaciations can determine the genetic pool of newly founded populations (Le Corre et al., 1997; Petit et al., 2004; Bialozyt et al., 2006). It could be useful to describe invasion processes with flows along the network of locations where a plant species establishes itself; it is possible that this would reveal differences between fast and slow invaders. Network theory might be combined with landscape genetics to improve our understanding of the consequences of rapid climate change, as now predicted, for plant (and associated pathogen) distributions (e.g. Manel et al., 2003; Bacles et al., 2004; Neilson et al., 2005; Simberloff et al., 2005; Webber & Brasier, 2005). Although contact tracing of disease is often impractical both in plantations and in more pristine forests, recent studies demonstrate that it is possible to track long-range spore dispersal of wood-decaying fungal organisms (e.g. Edman et al.,

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2004). Will this enable the reconstruction of the colonization history of these endangered species, so as to allow a rough understanding of the type of network involved? It is known that models predicting travelling waves of pathogen spread at constant speed are unrealistic if spore dispersal of plant pathogens is best fitted by a fat-tailed and not an exponential distribution (e.g. Ferrandino, 1993; Shaw, 1995; see also Scherm, 1996; Gibson, 1997; Jeger, 1999; Brown & Hovmøller, 2002; Bicout & Sache, 2003; Filipe & Maule, 2004; Shaw et al., 2006). This issue is also relevant for investigations of the effect of landscape structure on the spread of P. ramorum in California (e.g. Meentemeyer et al., 2004; Rizzo et al., 2005). In the landscape of forest and grassland patches where the pathogen is currently spreading, dispersal gradients will be affected by relatively efficient impaction and slow wind speeds within forest patches but by inefficient deposition and faster winds through open areas, with potentially different spore dispersal functions in the two cases. Insect vectors are another way for fungal pathogens of plants to jump from patch to patch of potential hosts in a landscape without other dispersal pathways (e.g. Geils, 1992). Elm bark beetles (Coleoptera: Scolytidae), the vectors of Dutch elm disease, attack clusters of debilitated trees, and in many cases avoid local dispersal to neighbouring healthy trees by flying long distances to different forest patches. There is evidence that the presence of elm trees (Ulmus spp.) in the landscape is made more manifest to elm bark beetles by sesquiterpene emissions induced on infected trees by the fungus responsible for Dutch elm disease (McLeod et al., 2005). This implies that the underlying network structure of the pathosystem is dynamic, and can differ from that deduced from the distribution of trees in the landscape. In the case of infections spread by vectors, managers may profitably make use of models incorporating mobile agents in the study of disease spread in complex networks (e.g. Miramontes & Luque, 2002; González & Herrmann, 2004; Frasca et al., 2006; see also Rvachev & Longini, 1985). Network modelling including mobile agents may, for instance, help in understanding and predicting the progress of invasions such as that of the horse chestnut leaf miner, Cameraria ohridella, which is facilitated by car movements and therefore follows major roads (e.g. Gilbert et al., 2005). Other potential applications of network theory in plant and forest pathology include the spread of fire blight (Erwinia amylovora; e.g. Jock et al., 2002) and of chestnut blight (Cryphonectria parasitica) hypovirulence (e.g. Milgroom & Cortesi, 2004). A network description of a tree pathosystem from a phytocentric point of view would specify nodes (host trees) and the links between them by whatever transport mechanism is responsible for spreading the disease, with or without any explicit geography. If data were available at a sufficiently large scale, it would be interesting to compare the networks of the population structure of newly introduced, aggressive tree pathogens (e.g. Phytophthora cinnamomi; e.g. Hardham, 2005) with those of endemic, long established ones (e.g. Heteroba-

Review

sidion annosum s.l.; e.g. Asiegbu et al., 2005). Unfortunately, to the best of our knowledge, only data from the local foraging behaviour of pathogens are available (for e.g. Armillaria spp.; Prospero et al., 2003a,b; Mihail & Bruhn, 2005), although large-scale information is accumulating, for example for the tree root endophyte and opportunistic pathogen Phialocephala fortinii s.l. (e.g. Queloz et al., 2005; Grünig et al., 2006). Network models might help in predicting the outcome of the dynamic interactions between pathogens of tree stumps and saprotrophic fungi used as biological control agents (Holdenrieder & Greig, 1998; Boddy, 2000). Models suggest that by manipulating the network structure it may be possible to diminish the incidence of one of two competing species, to the benefit of the other (Newman, 2005b). But other modelling work on the spread of two social norms in a network suggests that the contact structure may not be the only factor determining the outcome of the interactions between two competing species (Nakamaru & Levin, 2004). Heterogeneous forested landscapes are hard to represent as regular grids and contain multiple layers of evolving interactions. Elaborations of basic models of disease spread in networks may allow us to use insights provided by network theory in these ecosystems. Models making use of small-world networks in the simulation of the spread of forest fires are already established (e.g. Moukarzel, 1999; Graham & Matthai, 2003; Porterie et al., 2005) and have much more potential. It would be most fascinating to use these models to integrate the combined effects of sudden oak death and fire (Moritz & Odion, 2005). The patchiness of the host distribution and of environmental conditions may contribute to the heterogeneous spread of epidemics in forests (e.g. Sander et al., 2003; Vannucchi & Boccaletti, 2004). Models trying to include in their structure the variable susceptibility of hosts have certainly much scope for application in real forests (e.g. Sander et al., 2002). Models investigating the effect of the contact structure of networks on pathogen diversity (e.g. Buckee et al., 2004; Nunes et al., 2006) also offer new insights and possibly practical applications. Given the often long timescale of disease evolution in real forests, models can provide a rapid forecast of the direction towards which a pathosystem may evolve, given a certain network structure (e.g. Read & Keeling, 2003; Ferrari et al., 2006; Kao, 2006). A network perspective can also help in the selection of protected areas, because of the importance for the success of conservation efforts of understanding the connectivity within metapopulations (e.g. Brito & Grelle, 2004; Frank, 2004; Cerdeira et al., 2005).

Conclusion Network theory will be a useful addition to the set of concepts and tools available to understand and manage disease in plant populations. It may have its most obvious uses, as in our examples, in human-modified pathosystems where discrete,

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separated units can be identified, and where commercial transport may make distance a poor guide to the strength of a link between two units. It is less likely to be useful where the capacity of channels depends simply on proximity, and homogeneity is a reasonable approximation to the plant spatial distribution. But given the interconnectedness of today’s world, it may matter less and less that the long-distance jumps characteristic of wind-blown pathogens are hard to describe in a network model. The key factor in an increasing number of epiphytotics today is the transport by humans of disease propagules. Anything involving human transport of plants or their pathogens may be usefully modelled as a network, at least at some stage. We argue that more use of network theory in plant epidemiology would ensure that the critical features of many real epidemics would be clarified and studied earlier than if the normal task is regarded as modelling dispersal on a grid or even growth in a homogeneous population. Modelling work arising from issues in plant epidemiology may also motivate the investigation of questions of basic mathematical interest.

Acknowledgements Many thanks to A. Ramsay, N. Russell and R. Smith for comments on a previous version of this draft and to R. Baker, S. Hardy, T. Harwood, A. Inman, N. Künzli, J. Parke, L. Paul, V. Queloz, N. Salama, C. Sansford, T. Sieber, P. Todeschini, J. Webber and X. Xu for insights and discussions. The comments of two anonymous reviewers were particularly helpful. Work on this review was funded by the Department for Environment, Food and Rural Affairs, UK and is partly based on a talk on Epidemiology and Networks at the APS, CPS & MSA joint meeting in Quebec City, July 2006.

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