ecological complexity 5 (2008) 1–8
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Epidemic threshold and network structure: The interplay of probability of transmission and of persistence in small-size directed networks Marco Pautasso *, Michael J. Jeger Division of Biology, Imperial College London, Wye Campus, High Street, Wye, Kent TN25 5AH, UK1
article info
abstract
Article history:
A growing number of studies are investigating the effect of contact structure on the
Received 25 June 2006
dynamics of epidemics in large-scale complex networks. Whether findings thus obtained
Accepted 5 July 2007
apply also to networks of small size, and thus to many real-world biological applications, is
Published on line 22 August 2007
still an open question. We use numerical simulations of disease spread in directed networks of 100 individual nodes with a constant number of links. We show that, no matter the type of
Keywords:
network structure (local, small-world, random and scale-free), there is a linear threshold
Basic reproduction number
determined by the probability of infection transmission between connected nodes and the
Infectious diseases
probability of infection persistence in an infected node. The threshold is significantly lower
Initial conditions
for scale-free networks compared to local, random and small-world ones only if super-
Phytophthora ramorum
connected nodes have a higher number of links both to and from other nodes. The starting
Small-size networks
point, the node at which the epidemic starts, does not affect the threshold conditions, but
Spread of ideas
has a marked influence on the final size of the epidemic in all kinds of network. There is evidence that contact structure has an influence on the average final size of an epidemic across all starting nodes, with significantly lower values in scale-free networks at equilibrium. Simulations in scale-free networks show a distinctive time-series pattern, which, if found in a real epidemic, can be used to infer the underlying network structure. The findings have relevance also for meta-population ecology and species conservation. # 2007 Elsevier B.V. All rights reserved.
1.
Introduction
A growing body of literature is investigating the influence of the structure of complex networks on the dynamics of epidemics (reviewed e.g. in Newman, 2002; Keeling, 2005a; Keeling and Eames, 2005; Shirley and Rushton, 2005; May, 2006). This line of inquiry has been made possible by the increased computational power and availability of data on the connectivity of very large networks (e.g. Albert and Baraba´si, 2002; Clauset et al., 2004). Network theory has thus found application not only in epidemiology, but also in several types
of complex systems at different scales (e.g. Green and Sadedin, 2005), from neuroscience (e.g. Sakata et al., 2005) to scientometrics (e.g. Borner et al., 2005), from business management (e.g. Caldarelli and Catanzaro, 2004) to social science (e.g. Butts, 2001) and English literature (Stiller et al., 2003) and from cell biology (e.g. Hatzimanikatis et al., 2004) to food web ecology (e.g. Jorda´n and Scheuring, 2004). There are a number of important findings from studies of disease spread in large-scale networks: (i) the restriction of infection spread from one individual to a subset of connected individuals rather than to the entire population slows the
* Corresponding author. E-mail address:
[email protected] (M. Pautasso). 1 From 1 August 2007: Silwood Park Campus, Buckhurst Road, Ascot, Berkshire SL5 7PY, UK. 1476-945X/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ecocom.2007.07.001
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ecological complexity 5 (2008) 1–8
spread of a disease (e.g. Keeling, 2005b), yet (ii) the presence of highly connected individuals (super-spreaders) makes it easier for an epidemic to develop compared to networks with a random structure (e.g. Keeling and Eames, 2005). Moreover, modeling work has shown (iii) the theoretical absence of an epidemic threshold in networks with super-spreaders (scalefree networks; e.g. Pastor-Satorras and Vespignani, 2001; Boguna et al., 2003), but the (iv) emergence of a non-null threshold for scale-free networks of finite size (Pastor-Satorras and Vespignani, 2002; May, 2006). Network structure has thus been shown to have an influence on epidemic development, other things being equal (e.g. Draief, 2006; Kiss et al., 2006), although differing initial conditions can result in variability of epidemic dynamics and final size (Cre´pey et al., 2006; White and Gilligan, 2006). Many real biological networks are not only of finite size but also of relatively small size. For instance, a study of flowerhead-feeding insects on Asteraceae at five localities in Southeast Brazil found 162 interactions between 35 insect species and their 81 host plants (Lewinsohn et al., 2006). The number of nodes and links in networks of host–parasite, plant–pollinator and plant–frugivore interactions recently object of study is similar (e.g. Va´zquez, 2005; Va´zquez et al., 2005). For example, 63 pollination interactions were recorded in a small-world network of 26 pollinator species and 17 plant species on a small island of Greenland (Lundgren and Olesen, 2005). Likewise, La´zaro et al. (2005) analyzed the network given by the co-occurrence of 18 fleshy-fruited plant species in 264, 25 m 25 m quadrates in a Denmark forest. This order of magnitude is commonly used also in metapopulation ecology, for instance, Ovaskainen and Hanski (2003a) considered a subnetwork of 56 habitat patches in their analysis of metapopulations of the Glanville fritillary butterfly (Melitaea ˚ land Islands in Southwest Finland. cinxia) in the A A similar order of magnitude is found for the size of closely interacting human groups. Dunbar (1993) found a positive relationship between the size of neocortical brain and of primate group, with a predicted typical human group of roughly 150 individuals. This expectation fits well with available data of hunter-gatherer’s communities and business companies (Gladwell, 2000), patterns in cards sent for Christmas (Hill and Dunbar, 2003) and with theoretical analyses of the amount of information an individual can handle and the resulting number of its close acquaintances (Lo´pez and Sanjua´n, 2002; see also Margalit, 2002). Also, many networks of relevance to epidemiology may be of relatively small size, particularly if subsets of susceptible individuals are disconnected one from another, thus avoiding the creation of one giant component, i.e. a network where all individuals are connected to each other (e.g. Callaway et al., 2001). Leaving apart contacts to and from the external environment, the spread of computer viruses in a research group computer network, sexually transmitted diseases in a delimited town, infectious disease outbreaks in realistic urban social networks and networks of tree nurseries in a defined region, may all present similar small-size properties. For plant diseases, different types of structure in epidemics can arise through long distance connectivity resulting from the action of vectors (e.g. birds and insects, e.g. Leach, 1940; Tatem et al., 2006) or wind-borne dispersal (e.g. Aylor, 1978; Limpert et al., 1999). A
purely local spread would instead typically result in traveling waves (e.g. Johnson et al., 2004). The question arises of whether differences in the structure of networks of relatively small size can still influence the dynamics of epidemics, or whether such an influence is discernible only in the presence of large-scale networks. This is one of the 10 leading questions for further developments in network research highlighted at the conference ‘‘Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems’’ held in Rome in September 2003 (Amaral et al., 2004). This paper aims to investigate that question.
2.
Methods
Numerical simulations were based on networks of 100 individual nodes. Four basic scenarios with different network structure were used: local (nearest-neighbour transmission), small-world (nearest-neighbour transmission rewired with short-cuts), scale-free (absence of local connectivity, but presence of super-spreaders with high number of links) and random (absence of regular local links) connectivity. For each network structure, 20 replicate networks were built with randomly generated links. The number of replicates is high enough for the standard deviation of the threshold conditions (see below) for a given kind of network not to decline any more with additional replicates (e.g. Campbell and Madden, 1990). For each replicate, the overall number of connections was kept constant at an arbitrary value (369 out of the possible 9900 links). This means that in all cases the average number of links both to and from a node was 3.69. This value allows the construction of a lattice with local connectivity of nodes with their four nearest neighbours with some missing links. Since this average number of contacts between individuals (which, ceteris paribus, determines the basic reproduction number) is greater than one, an epidemic would theoretically develop (e.g. Anderson and May, 1991). 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; Schwartz et al., 2002; Boguna and Serrano, 2005). A graphical representation of one replicate for each network structure is given in Fig. 1 of Jeger et al. (2007). The probability for an infection to be transmitted from one node to another ( pt) was either zero (unconnected nodes) or a variable parameter, which is the equivalent of the decay constant z 1 modelling the corruption experienced by a piece of information when passed from one agent to another in sociological networks studies (e.g. Lo´pez and Sanjua´n, 2002). This pt was set 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 distinctive transmission probabilities according to whether links were to neighbouring cells or shortcuts). An additional variable parameter ( pp) was set to govern the persistence of an infection in a given node from time t to time t + 1. Also pp was set to be the same for all nodes. It combines in one single parameter the length of infectiousness, detection and control measures. As in Agliari et al. (2006), both pt and pp are real (i.e. not Boolean) variables, going from 0 to 1. This is a main difference of our model from the epidemiological
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ecological complexity 5 (2008) 1–8
Table 1a – Average (with standard deviation) threshold ( pt) for the 20 replicates of the four network structures at pp = 0 Network Local Small-world Random Scale-free (1) Scale-free (2)
Average 0.267 0.260 0.269 0.321 0.134
S.D. 0.011 0.009 0.007 0.048 0.021
For scale-free networks, values for networks where super-connected individuals to and from other nodes coincide (2) or do not coincide (1) are given.
Fig. 1 – Threshold between epidemic development and control in the four basic kinds of network structures (middle lines are average values, outer lines represent the standard deviation of the 20 replicates for each kind of network). Only scale-free networks where superconnected individuals to and from other nodes coincide are shown.
literature, which typically assumes a binary status for individuals (either susceptible or infected). Also in our case, the outcome is that nodes are either infected or not infected, but this happens not on a binary base but along a continuum. This can be realistic for plant diseases in the context of plant nurseries, which can be completely devoid of a pathogen, with all susceptible plants infected, or everything in between. All nodes were assumed to be of equal size and kind. For each iteration of the simulation, the probability of a given node x to be infected ( pi) was obtained by: piðxÞ ¼
X
ptðx;yÞ piðyÞ
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 pi(y) is the infection status of the node y at the previous iteration. For the connection of a node with itself, pp was used instead of pt. Two conditions were thus necessary for a node to become infected: (i) the node had to have a connection from (ii) an infected node. Models of stochastic epidemics usually model a contact process together with transmission (Diekmann et al., 1998). We decided to keep the two things separate in order to unravel the influences of, respectively, contact structure and infectivity on disease spread in different kinds of networks. At each iteration, the contact structure of the network was maintained exactly the same. The development of the epidemic 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 (1%). The epidemic was started with a single infection of a single node (as e.g. in Ben-Naim and Krapivsky, 2004), but results are robust to multiple starting infections. The effect on the epidemic development of differing starting points of the infection was assessed by making the epidemic start in each of the 100 different nodes of the networks. Simulations were run for 100 iterations, or for longer (up to 400 in some kinds of local networks) if this was necessary in
order to reach equilibrium. In many cases, equilibrium at the threshold conditions was reached long before the 100th iteration.
3.
Results
Irrespective of the structure of a network, whether the epidemic developed or not was governed by the interplay between pt and pp. A linear threshold was recovered in all types of networks on a graph of pp as a function of pt (Fig. 1). The generality of the linearity of the threshold can be assumed given that the proportion of variance explained by linear regressions of five randomly chosen data points on the pp as a function of pt graph for all the 20 different replicates of the four kinds of network structure was 1.00. All these linear thresholds have the same intercept on the y-axis (0, 1), because when pt is null, equilibrium is given by pp = 1. The slopes of the linear thresholds were thus determined by the second intercepts on the x-axis, which varied between different replicates and were determined by the structure of the network. The average intercept on the x-axis (Tables 1a and 1b) was not significantly different only between networks of local and random structure. Small-world networks showed a significantly lower threshold than local and random networks. In turn, scale-free networks showed a significantly lower threshold compared to all other types. It is important to note that scale-free networks showed a lower threshold (at pp = 0) than other types of networks only if super-connected individuals exhibited both a high number of connections to other nodes
Table 1b – ANOVA comparisons between the average threshold values (see Table 1a) of the 20 replicates of the different kinds of networks Comparison Local–random Local–small-world Random–small-world Scale-free (1)–random Scale-free (1)–local Scale-free (1)–small-world Small-world–scale-free (2) Local–scale-free (2) Random–scale-free (2) Scale-free (1)–scale-free (2)
r2 0.01 0.10 0.20 0.38 0.39 0.45 0.94 0.94 0.95 0.97
p 0.59 0.05 0.003 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001 <0.0001
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ecological complexity 5 (2008) 1–8
Table 2a – Average (with standard deviation) epidemic final size (assessed on the basis of (i) the number of nodes with pi greater than 1%, and (ii) the sum of pi across all nodes) for the 100 possible starting nodes of a replicate and 20 replicates of the four network structures at the threshold conditions Network
Local Small-world Random Scale-free (1) Scale-free (2)
Average 1% (i)
Average sum (ii)
Mean
S.D.
Mean
37.7 33.1 41.2 22.6 24.2
16.7 9.9 6.0 4.9 8.3
1.05 0.83 0.98 1.24 0.69
S.D. 1.02 0.15 0.05 0.44 0.04
For scale-free networks, values for networks where super-connected individuals to and from other nodes coincide (2) or do not coincide (1) are given.
and from other nodes (both-ways scale-free networks). In the case that super-connected nodes only showed a higher number of connections to other nodes, but a low number of connections from other nodes, or vice versa (one-way scalefree networks), the threshold for the epidemic to develop was significantly higher than the one for all the other types of networks. However, even if the super-connected individuals used are super-connected in both directions, the scale-free networks arising are still directed, because some nodes connected to these super-connected individuals may still not be connected from the same super-connected individuals, although most will be. Interestingly, in all kinds of networks, the threshold was not influenced by the starting point of the epidemic. In other words, no matter from which node the epidemic started, the combination of pt and pp in a given network for equilibrium to be reached was constant. Nonetheless, the node from which the epidemic started had a marked influence on the size of the epidemic at equilibrium, both in terms of the sum of pi across all nodes and of the number of nodes with pi higher than 1%. In spite of this high variability in the final size of the epidemic depending on the starting node, there was evidence that the epidemic final size averaged across all starting nodes is influenced by the network structure (Tables 2a and 2b). The average final size of the epidemic at the threshold conditions
was not significantly different between networks of local and random, and of local and small-world types. However, random networks showed a significantly higher average final size than small-world ones. Random networks (as well as local and small-world ones) also showed a significantly higher proportion of nodes with pi higher than 1% at equilibrium than scalefree networks (whether these were one-way or both-ways scale-free networks). But for one-way scale-free networks, this result was reversed when considering the sum of pi across all nodes at equilibrium, for in this case they showed a significantly higher final size of the epidemic than random and small-world networks (there were no significant differences with local networks). This was not the case for bothways scale-free networks: in this case, also when the final size of the epidemic was expressed as sum of pi, it was significantly lower than for random and small-world networks (again, there were no significant differences with local networks). These findings followed from the non-significantly different proportion of nodes with pi higher than 1% between one-way and both-ways scale-free networks and the significantly higher sum of pi across all nodes for one-way scale-free networks compared to both-ways ones.
4.
Discussion
The basic reproduction number R0 is now well known among pathologists, as it a fundamental quantity in the study of epidemics (Heesterbeek, 2002). It is defined as the expected number of new cases of infection caused by a typical infected individual in a population consisting of susceptible individuals only (e.g. Hethcote, 2000; May, 2006). It has been shown for networks of large size that a contact structure with high variability in the number of links between individuals can have an impact on epidemic development and threshold properties (e.g. Lloyd and May, 2001; Koopman, 2004). This analysis shows that also small-size networks with constant average number of connections per node can exhibit markedly different epidemic developments depending on the network structure. The interplay between pp and pt in determining the epidemic threshold in networks of small size is consistent with previous work by Tuckwell et al. (1998), where diseases
Table 2b – ANOVA comparisons between the average epidemic final size values at equilibrium of the 20 replicates of the different kinds of networks. Same conventions as in Table 2a Comparison
Average 1% (i) r
Local–random Local–small-world Random–small-world Scale-free (1)–random Scale-free (1)–local Scale-free (1)–small-world Small-world–scale-free (2) Local–scale-free (2) Random–scale-free (2) Scale-free (1)–scale-free (2)
2
0.02 0.03 0.20 0.75 0.28 0.32 0.20 0.21 0.59 0.01
Average sum (ii) 2
p
r
0.38 0.30 0.003 <0.0001 0.0004 <0.0001 0.004 0.003 <0.0001 0.45
0.00 0.02 0.31 0.15 0.01 0.28 0.31 0.06 0.92 0.44
p 0.76 0.35 0.0002 0.01 0.46 0.0004 0.0002 0.12 <0.0001 <0.0001
ecological complexity 5 (2008) 1–8
with low establishment ability did not spread in the population modeled unless this population was highly mobile. This corresponds to the bottom-right region of Fig. 1, with epidemic development even if pp is low because of a high pt. A similar finding by Tuckwell et al. (1998) is that a pathogen will pervade an entire population even if the individuals in the population interact very little among themselves if the pathogen readily establishes itself within individuals. This corresponds to the top-left region of Fig. 1, with epidemic development even if pt is low because of a high pp. Of novelty is here that in various kinds of complex networks of small size, when a pathogen spreads under the assumptions described in Section 2, the threshold on a graph of pp as a function of pt is linear. Fig. 1 can provide guidance in attempting to control a disease spreading in a network when the actual and threshold pt and pp are known. The disease can be brought under control either by lowering pp with immunization and removal strategies, or by lowering the pt with quarantine measures and attempting in some other way to diminish the spread of a pathogen across links between nodes. Theoretically, the graph suggests that while it is always possible to reach the threshold conditions by only lowering pt (unless pp is one), for values of pt greater than the x-axis intercept, it is not sufficient to lower pp, but it is also necessary to lower pt below the intercept in order to stop the spread of the disease. The finding that only in scale-free networks, where nodes with a high number of links to and from other nodes coincide, there is a significantly lower threshold compared to other kinds of networks is in agreement with work by Woolhouse
5
et al. (2005) on the epidemiological implications of the contact network structure for cattle farms in the UK, where it was possible to ascertain that the risk of acquiring infection and the risk of transmitting infection were not positively correlated. This means that the high degree of heterogeneity in the number of contacts, if not coupled with a high correlation coefficient between links to and from highly connected nodes, does not result in an increase in the basic reproduction number R0. It is to be noted that the lower threshold for scale-free networks does not mean that this threshold is null, as it has been reported for scale-free networks of infinite size (e.g. Peltoma¨ki et al., 2005). In networks of small size, assuming a random or a local distribution of contacts will overestimate the threshold if the underlying structure is scale-free. But also in scale-free networks of small size, diseases with very low transmission rate will not spread. A further important point is that for a network of 100 individual nodes, and when disease spreads according to the assumptions of the model, there are no significant differences in the epidemic threshold only when comparing local and random networks. The threshold for small-world networks, albeit significantly higher than the one for scale-free networks, is significantly lower than the one for local and random networks. This finding confirms the potential of different network structures in determining epidemic development (e.g. Shirley and Rushton, 2005; Draief, 2006; May, 2006) in small-size networks. If significant differences in the epidemic threshold for small-world versus local and random networks,
Fig. 2 – Examples of simulations at the threshold conditions for each of the four basic kinds of network structure used in analyses: (a) local, (b) small-world, (c) random and (d) scale-free. Epidemic development can be assessed on the basis of the sum of the probability of infection across all nodes (grey line, left-hand y-axis) and of the proportion of nodes with probability of infection higher than 1% (black line, right-hand y-axis).
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ecological complexity 5 (2008) 1–8
and for scale-free networks compared to all other types, can be observed already in networks of 100 individual nodes, then it makes sense also at this order of magnitude inquiring in epidemiological studies of human, animal and plant diseases whether a particular network has a scale-free, random, smallworld or local connectivity (e.g. Southworth et al., 2005). Regardless from which particular node an infection is started, the threshold for a network remains the same. But the final size of an epidemic at the threshold is markedly influenced by the starting point of the epidemic. This result is consistent with previous work documenting an effect of the initial conditions on epidemic dynamics and variability in large-scale homogeneous, scale-free and random networks (Cre´pey et al., 2006; White and Gilligan, 2006). Similarly, Gallos and Argyrakis (2003) found that disease spreading in scale-free networks could either die off or develop in an epidemic depending on the initial starting point. However, in our case, if the set of pp and pt is below, at or above the threshold, it does not matter where the epidemic starts, the epidemic will, respectively, be quenched, reach equilibrium or develop. Although the final size of the epidemic is primarily influenced by its starting point, there is evidence that network structure can affect the average final epidemic size across all starting nodes. Both-ways scale-free networks show lower final epidemic size at threshold conditions than other kinds of networks, possibly in relation to their lower threshold. Similarly, random networks show a higher final epidemic size than small-world ones, again possibly due to their higher threshold. The discrepancies between results assessing the final size of the epidemic in terms of the sum of pi across all nodes and the number of nodes with pi higher than 1% deserve further investigation. The finding that it is possible to obtain different time-series curves with the same network structure, pp and pt but differing starting point cautions against trying to infer from a timeseries curve, the contact structure of a network of small size. This is all the more so given that similar time-series curves can arise with different types of network structure (Fig. 2a–c). But scale-free networks consistently exhibit a peculiar development, i.e. high initial variations in the proportion of nodes with a certain pi from one iteration to the following one (Fig. 2d). This pattern is not found in the other types of network structure. A similar issue is whether it is possible to trace back the starting point of an epidemic spreading in a network by analyzing its time-series, as Kosin´ski and Adamowski (2004) successfully do in a two-dimensional lattice. The problem here is that in many cases similar time-series curves and final epidemic sizes can be obtained by starting the epidemic in different nodes.
5.
Conclusion
The model presented applies to pathosystems where individual nodes are not removed following the onset of disease (either because of death or prophylaxis). This is the case wherever individual nodes contain multiple susceptible units and carry on being in contact with other nodes even if some units are infected or removed. Also in this case, it is true that when individuals systematically contact the same individuals,
the probability that an infectious individual contacts already infected individuals is higher than it is otherwise (Diekmann et al., 1998). But the effect of the infectious output is not reduced because infected individual nodes are still able to receive more infected units. This applies for instance in networks of plant nurseries, which may contain thousands of individual plants susceptible, for example, to Phytophthora ramorum, the oomycete responsible for Sudden Oak Death (Jeger et al., 2007), and may go on trading with other nurseries even if they already have some infected plants or have destroyed them (e.g. Tubajika et al., 2006). This kind of S-I model can represent also the spread of ideas (e.g. Nakamaru and Levin, 2004; Agliari et al., 2006; Kuperman, 2006). In this case, individuals having become fascinated by a certain thought will tend to spread it to people with whom they are connected, unless this thought causes people to withdraw from interactions with other people. Further interactions with people who are convinced by the same idea are possible and will tend to make it even more likely that the idea will become established. Further investigations of epidemic development in networks of small size could extend to situations where infection is likely to develop in removal of infected individuals. Furthermore, it would be interesting to study the effect of a dynamic contact structure on the threshold properties (e.g. Agliari et al., 2006) or of the number of links on the final size of epidemics in networks of various types (on the lines of Diekmann et al., 1998). Also, the effect of variations in the probability of infection transmission between different nodes could be explored (Olinky and Stone, 2004). Additional open questions are which factors can explain the marked variations in the final size of the epidemic with differing starting nodes, and whether the time scale for outbreaks to reach equilibrium at the threshold conditions is different in different types of networks (see Barthe´lemy et al., 2004). The framework developed can be used to assess the importance of single patches for meta-population persistence (Ovaskainen and Hanski, 2003b) and has thus potential applicability in conservation biology, where population viability analyses often deal with networks of individuals of threatened species of similar size to the ones used in the simulations presented. Of course, in meta-population ecology, the aim is reversed, for it is not an issue of keeping a disease under the threshold conditions, but of maintaining the viability of a species of interest in a conservation network (e.g. Hanski and Ovaskainen, 2003; Kindlmann et al., 2005; Fortuna et al., 2006; Gaston et al., 2006; Moilanen and Wintle, 2007). Mathematical analysis and modeling of plant diseases have gone a long way since the seminal contribution of Kranz (1974), and there is now much scope for going further in integrating biological heterogeneities and network complexity in the study and management of epidemics (Franc, 2004; Roy and Pascual, 2006; Jeger et al., 2007).
Acknowledgements Many thanks to T. Harwood and O. Holdenrieder for comments on a previous version of the draft and to F. Matthaeus, J. Parke, M. Shaw, F. Van den Bosch and X. Xu for discussions on the
ecological complexity 5 (2008) 1–8
analyses. This study was funded by Department for Environment, Food, and Rural Affairs, UK.
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