Transboundary and Emerging Diseases
ORIGINAL ARTICLE
Epidemic Spreading in an Animal Trade Network – Comparison of Distance-Based and Network-Based Control Measures € ttner1,2, J. Krieter1, A. Traulsen2 and I. Traulsen1 K. Bu 1 2
Institute of Animal Breeding and Husbandry, Christian-Albrechts-University, Kiel, Germany €n, Germany Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, Plo
Keywords: SIR epidemic model; directed network; animal movements; network analysis; control measures Correspondence: €ttner. Institute of Animal Breeding and K. Bu Husbandry, Christian-Albrechts-University, Olshausenstr. 40, D-24098 Kiel, Germany. Tel.: +49 431 880 5457; Fax: +49 431 880 2588; E-mail: [email protected] Received for publication September 13, 2013 doi:10.1111/tbed.12245
Summary This study considered a simple SIR model for the spread of epidemics amongst holdings of a producer community in Northern Germany, based on the directed network of animal movements. The aim of this study was to evaluate the efficiency of different control measures to reduce the epidemic size substantially. The currently applied control measures based on the distance to an infected holding were compared with the control measures based on the specific network-based centrality parameters. We found that network-based measures led to a more efficient control of epidemics with a much smaller number of removed holdings. To assess the impact of different holding types, the analysed control measures were implemented by both including and excluding these holding types. The comparison revealed a crucial role of multipliers in the spread of an epidemic. The network-based control measures depending on the removal by out-degree, outgoing infection chain, betweenness centrality and outgoing closeness centrality showed the best results: In the three-year network, on average, 2.75, 4.15, 3.73 and 3.43 times more holdings had to be removed by the control measures based on the 1, 3, 5 and 10 km radius to reduce the epidemic to the same size compared with the network-based control measures. In an area with a higher holding density, the improvement of the network-based control measures may become even more obvious. The removal of holdings based on the above-mentioned centrality parameters did thus not only rapidly decompose the network into fragments, but also reduced the epidemic size most efficiently.
Introduction In addition to indirect disease transmission, for example, through the contact of persons or livestock trucks, shared equipment or the transmission due to the vicinity to another infected holding, one of the major transmission paths for highly contagious animal diseases is the direct contact with an infected animal. According to Fritzemeier et al. (2000), the majority (28%) of the secondary and follow-up outbreaks of classical swine fever in German domestic pig herds from 1993 to 1998 were caused by the trade with pigs. Also, Ribbens et al. (2004) stated that the direct virus transmission had the largest contribution to © 2014 Blackwell Verlag GmbH • Transboundary and Emerging Diseases.
the total between-herd transmission before the first infected holding was detected. During the classical swine fever outbreak in the Netherlands from 1997 to 1998, 17% of the between-herd transmission was due to direct contact of infected pigs. Additionally, in this time period, the highest transmission rate for the classical swine fever virus could be obtained for animal movements (Stegeman et al., 2002). Due to the high risk of disease spreading caused by animal movements, these transmission paths are subject to strict regulations during a disease outbreak. For example, in the EU directive 2001/89/EC (Anonymous, 2001), the control measures are written down, which are implemented in the case of a classic swine fever outbreak. Here, in an outbreak 1
€ttner et al. K. Bu
Disease Control in an Animal Trade Network
situation, the disease control policy focuses on movement restrictions and contact tracing, as well as culling of infected animals, respectively, holdings and preventive culling of holdings in a specific distance to the infected farms. Therefore, based on the information of animal movements, this transmission path can be well represented. These animal movements can be illustrated as a directed network and studied with the help of network analysis. In such a network, the holdings represent the nodes of the network and the animal movements are the directed links between the nodes. Each link has one specific supplier and one specific purchaser, as animals are only transported in one direction at a time. Centrality parameters identify highly central holdings within the trade network under investigation. In the recent years, a similar kind of network analysis has been used to characterize the topology of various animal trade networks. Several studies have used static network analysis to describe the network properties of a specific observation period (Webb, 2005, Bigras-Poulin et al., 2006; 2007, Lentz et al., 2009; Volkova et al., 2010a,b; B€ uttner et al., 2013b), whereas other studies have compared the results obtained by network analysis of different time periods (Vernon and Keeling, 2009; Bajardi et al., 2011; N€ oremark et al., 2011; Rautureau et al., 2011; B€ uttner et al., 2013c). In this way, it has been possible to detect structural changes over time or stable characteristics. The studies of B€ uttner et al. (2013b, c) showed that the pork supply chain had stable values for the centrality parameters based on the outgoing trade contacts between different time periods, meaning that the pyramidal structure of this kind of network did not show temporal fluctuations. The knowledge of the network topology can be utilized to control processes, which take place in the pork supply chain, such as disease transmission or percolation. By applying percolation theory, the fragmentation of a network can be analysed (Sahimi, 1994; Stauffer and Aharony, 1994; ben-Avraham and Havlin, 2000; Newman, 2010; B€ uttner et al., 2013a). It has been shown that networks with a right-skewed distribution of the centrality parameters are highly vulnerable to the targeted removal of holdings (Albert et al., 2000; Cohen et al., 2000, 2001; May and Lloyd, 2001; Kiss et al., 2006; Dube et al., 2009; Natale et al., 2009). Thus, these kinds of networks can be decomposed into fragments by removing only a small amount of nodes. In a network that is disjointed, that is, which has many unconnected components, disease transmission is less likely than in a totally connected network. To validate this effect, a simple epidemiological SIR (susceptible–infected– recovered) model was implemented in the present study based on the trade network under investigation (Kermack and McKendrick, 1927). This SIR model made it possible to illustrate not only the efficiency of the network-based con2
trol measures but also the differences to the control measures based on the radius around an infected holding. Therefore, the objective of this study was to use simulation modelling to compare the application of distance-based control measures to control measures based on network characteristics. Thus, the most appropriate control measure to reduce the epidemic size efficiently could be identified. Materials and Methods Data and network construction Pig movement data from a producer community in Northern Germany were analysed in an observation period between 1 June 2006 and 31 May 2009. The date of the movements, the supplier, the purchaser as well as the batch size and the type and age group of the delivered livestock were recorded. For data protection reasons, the exact coordinates of the holdings were not available. To approximate the geographical location (longitude and latitude) of the holdings, information of the town and the zip code was used. The geographic distances were calculated as the Euclidean distance between the holdings. In total, the data contained 15 372 directed animal movements between 658 holdings, meaning that each of the animal movements had one specific supplier and one specific purchaser. For the analysis of different time periods, the data were separated into the three-year network, three yearly networks and 36 monthly networks. Multiple trade contacts throughout the analysed time periods were aggregated into a single one. The production type of the different holdings was classified into five categories by means of number, type and age group of the delivered livestock: multipliers, farrowing farms, finishing farms, farrow-to-finishing farms and abattoirs. Due to the dead-end characteristic of abattoirs for the transport of live animals, this holding type was excluded from the analysis. Thus, in the total three-year network, the number of holdings reduced to 483 and 4635 animal movements, which were aggregated to 926 group movements (B€ uttner et al., 2013b). In the yearly networks, the number of holdings was on average 322 (319–323), which were connected by 1545 (1522–1571) animal movements, that is, 449 (431–468) aggregated group movements. The monthly networks had on average 129 (107 to 148) holdings and 427 (359 to 479) animal movements, which were aggregated to 114 (93–134) group movements (B€ uttner et al., 2013c). The proportions of the four holding types in the specific time intervals are reported in B€ uttner et al. (2013b,c). Disease simulation and implemented control measures In this study, the disease spread in the three observed time periods was modelled using a simple SIR model. Each © 2014 Blackwell Verlag GmbH • Transboundary and Emerging Diseases.
€ttner et al. K. Bu
Disease Control in an Animal Trade Network
holding represented a single epidemiological unit, which was connected to its trade partners by animal movements. Each holding could be in three basic states: susceptible, infected or recovered, respectively, removed. All holdings had the state susceptible at the beginning of the simulation. The epidemic started with a single infected holding chosen uniformly at random from the whole number of holdings in the network. The successors, that is, the holdings that received animals from the infected holding, became themselves infected by different transmission probabilities T = 0.1, 0.2 . . . 0.9, 0.95. For each of the new infected holdings, their successors were determined and became again infected with the transmission probability T. A holding immediately becomes recovered, respectively, removed after its successors had been determined and it played no further part in the course of the epidemic. Removal in this context does not necessarily mean depopulation, but can also stand for trade restrictions. No uncertainty or variability in the detection or the control was implemented, and the detection of infected holdings was assumed to be instantaneous. The epidemic terminated when there were no more successors remaining in the network which could be infected. This model was implemented in Python (van Rossum and Drake, 2001). Temporal aspects were not included in the model as the network-based control measures were based on the centrality parameters calculated for the aggregated networks. Control measures based on the removal of holdings in a specific radius around the infected holdings The first control measure implemented in the SIR model was the removal of holdings in a specific radius around the infected holding, that is, 1, 3, 5 and 10 km. When the simulation started and the first holding was infected, the successors of this holding were detected. The next step was to identify the holdings, which were located in one of the specified radii (e.g. 5 km) around the infected holding. These holdings were removed from the network and played therefore no further role in the course of the epidemic. All
other successors became infected with a specific transmission probability (e.g. 0.95). The epidemic terminated when there were no more successors remaining in the network, which could be infected. For each transmission probability and radius, 10 000 independent simulations were performed. To evaluate the impact of the four holding types on the implementation of the control strategies, each control measure was performed additionally excluding one specific holding type. Control measures based on the removal of holdings using the rank of specific centrality parameters The second control measure implemented in the SIR model was the removal of holdings using the rank of the centrality parameters in-degree and out-degree, ingoing and outgoing infection chain, betweenness centrality and ingoing and outgoing closeness centrality (Table 1). For each time period, the analysis of the centrality parameters was performed separately, see B€ uttner et al. (2013b,c). The centrality parameters as well as the general network properties were calculated using the Python module NetworkX (Hagberg et al., 2008). Holdings were removed from the network in descending order of the centrality parameters. More precisely, for each holding in the network, the centrality parameters were calculated and then sorted in descending order. According to this order, the most central holdings were removed from the networks, meaning the holdings with the highest centrality parameter. Then, the simulation started as described above. For each value of the centrality parameters obtained, the simulation was performed 10 000 times. The control measures based on the removal of holdings using the rank of specific centrality parameters were also carried out excluding the different holding types, as mentioned above. Comparison of control methods The control measures based on the radius around an infected holding were compared with the control measures based on the targeted removal of holdings according to
Table 1. Definition of parameters used in network analysis for the characterization of animal movements Parameter
Definition
References
In-degree Out-degree Ingoing infection chain
Number of trade partners which deliver animals to a specific holding Number of trade partners which receive animals from a specific holding Number of direct and indirect trade contacts which lead to a specific holding taking the chronological order of the contacts into account Number of direct and indirect trade contacts which originate at a specific holding taking the chronological order of the contacts into account The betweenness centrality measures the extent to which a holding lies on paths between other holdings Mean distance from all other reachable holdings to one specific holding Mean distance from one holding to all other reachable holdings
Newman (2010) Newman (2010) €remark et al. (2011) No
Outgoing infection chain Betweenness Ingoing closeness Outgoing closeness
© 2014 Blackwell Verlag GmbH • Transboundary and Emerging Diseases.
Webb (2006); Dube et al. (2008) Wasserman and Faust (1994) Wasserman and Faust (1994) Wasserman and Faust (1994)
3
Disease Control in an Animal Trade Network
specific centrality parameters. The efficiencies of both approaches based on the resulting epidemic size were analysed. In contrast to the distance-based control measures, the network-based control measures are independent from the course of the epidemic, meaning the first infected holding. Furthermore, for each rank of the centrality parameters, a mean epidemic size was obtained, whereas the distance-based control measures had only one mean epidemic size for each analysed transmission probability and radius. To facilitate a comparison of these different results, the average values for the distance-based control measures were approximated with the help of the slope obtained for the network-based control measures based on the same epidemic size. By means of this approximation, the difference between the network-based and the distance-based control measures could be obtained represented as a factor. As reference value the fraction of removed holdings in the distance-based control measures was set to 1. The approximation was performed using SAS statistical software package (SASâ Institute Inc., 2008).
€ttner et al. K. Bu
61.8% (572) of the contacts originated from a multiplier and 25.6% (237) from a farrowing farm. The yearly and monthly networks showed similar results with 65.3% (293) and 68.4% (78) for the contacts starting at a multiplier and 24.1% (108) and 23.7% (27) for the farrowing farms as supplier. Otherwise, the majority of the direct trade contacts in the three-year network ended at finishing farms with 40.1% (371) and farrow-to-finishing farms with 49.8% (461). In the yearly and monthly networks, 37.4% (168) and 36.0% (41) of the trade contacts ended at a finishing farm and 51.9% (233) and 52.6% (60) ended at a farrow-to-finishing farm. Furthermore, almost 50% (442) of all trade contacts in the total three-year network took place between the multiplier holding type as supplier and the farrow-to-finishing farm as purchaser, followed by the animal transports from the farrowing farms to the finishing farms with about 25% (228) of trade contacts. In the yearly and monthly networks, the connections between these holding types showed similar percentages. Disease simulation and implemented control measures
Results Distances and number of trade contacts between different holding types Figures 1 and 2 show the geographical distribution of the holdings and the Euclidean distances of the trade contacts in the pork supply chain, respectively. The histogram (Fig. 2) illustrates the right-skewed distribution of the median Euclidean distances between the trade partners. A total of 77.6% of the trade contacts passed by a distance of
ORIGINAL ARTICLE
Epidemic Spreading in an Animal Trade Network – Comparison of Distance-Based and Network-Based Control Measures € ttner1,2, J. Krieter1, A. Traulsen2 and I. Traulsen1 K. Bu 1 2
Institute of Animal Breeding and Husbandry, Christian-Albrechts-University, Kiel, Germany €n, Germany Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, Plo
Keywords: SIR epidemic model; directed network; animal movements; network analysis; control measures Correspondence: €ttner. Institute of Animal Breeding and K. Bu Husbandry, Christian-Albrechts-University, Olshausenstr. 40, D-24098 Kiel, Germany. Tel.: +49 431 880 5457; Fax: +49 431 880 2588; E-mail: [email protected] Received for publication September 13, 2013 doi:10.1111/tbed.12245
Summary This study considered a simple SIR model for the spread of epidemics amongst holdings of a producer community in Northern Germany, based on the directed network of animal movements. The aim of this study was to evaluate the efficiency of different control measures to reduce the epidemic size substantially. The currently applied control measures based on the distance to an infected holding were compared with the control measures based on the specific network-based centrality parameters. We found that network-based measures led to a more efficient control of epidemics with a much smaller number of removed holdings. To assess the impact of different holding types, the analysed control measures were implemented by both including and excluding these holding types. The comparison revealed a crucial role of multipliers in the spread of an epidemic. The network-based control measures depending on the removal by out-degree, outgoing infection chain, betweenness centrality and outgoing closeness centrality showed the best results: In the three-year network, on average, 2.75, 4.15, 3.73 and 3.43 times more holdings had to be removed by the control measures based on the 1, 3, 5 and 10 km radius to reduce the epidemic to the same size compared with the network-based control measures. In an area with a higher holding density, the improvement of the network-based control measures may become even more obvious. The removal of holdings based on the above-mentioned centrality parameters did thus not only rapidly decompose the network into fragments, but also reduced the epidemic size most efficiently.
Introduction In addition to indirect disease transmission, for example, through the contact of persons or livestock trucks, shared equipment or the transmission due to the vicinity to another infected holding, one of the major transmission paths for highly contagious animal diseases is the direct contact with an infected animal. According to Fritzemeier et al. (2000), the majority (28%) of the secondary and follow-up outbreaks of classical swine fever in German domestic pig herds from 1993 to 1998 were caused by the trade with pigs. Also, Ribbens et al. (2004) stated that the direct virus transmission had the largest contribution to © 2014 Blackwell Verlag GmbH • Transboundary and Emerging Diseases.
the total between-herd transmission before the first infected holding was detected. During the classical swine fever outbreak in the Netherlands from 1997 to 1998, 17% of the between-herd transmission was due to direct contact of infected pigs. Additionally, in this time period, the highest transmission rate for the classical swine fever virus could be obtained for animal movements (Stegeman et al., 2002). Due to the high risk of disease spreading caused by animal movements, these transmission paths are subject to strict regulations during a disease outbreak. For example, in the EU directive 2001/89/EC (Anonymous, 2001), the control measures are written down, which are implemented in the case of a classic swine fever outbreak. Here, in an outbreak 1
€ttner et al. K. Bu
Disease Control in an Animal Trade Network
situation, the disease control policy focuses on movement restrictions and contact tracing, as well as culling of infected animals, respectively, holdings and preventive culling of holdings in a specific distance to the infected farms. Therefore, based on the information of animal movements, this transmission path can be well represented. These animal movements can be illustrated as a directed network and studied with the help of network analysis. In such a network, the holdings represent the nodes of the network and the animal movements are the directed links between the nodes. Each link has one specific supplier and one specific purchaser, as animals are only transported in one direction at a time. Centrality parameters identify highly central holdings within the trade network under investigation. In the recent years, a similar kind of network analysis has been used to characterize the topology of various animal trade networks. Several studies have used static network analysis to describe the network properties of a specific observation period (Webb, 2005, Bigras-Poulin et al., 2006; 2007, Lentz et al., 2009; Volkova et al., 2010a,b; B€ uttner et al., 2013b), whereas other studies have compared the results obtained by network analysis of different time periods (Vernon and Keeling, 2009; Bajardi et al., 2011; N€ oremark et al., 2011; Rautureau et al., 2011; B€ uttner et al., 2013c). In this way, it has been possible to detect structural changes over time or stable characteristics. The studies of B€ uttner et al. (2013b, c) showed that the pork supply chain had stable values for the centrality parameters based on the outgoing trade contacts between different time periods, meaning that the pyramidal structure of this kind of network did not show temporal fluctuations. The knowledge of the network topology can be utilized to control processes, which take place in the pork supply chain, such as disease transmission or percolation. By applying percolation theory, the fragmentation of a network can be analysed (Sahimi, 1994; Stauffer and Aharony, 1994; ben-Avraham and Havlin, 2000; Newman, 2010; B€ uttner et al., 2013a). It has been shown that networks with a right-skewed distribution of the centrality parameters are highly vulnerable to the targeted removal of holdings (Albert et al., 2000; Cohen et al., 2000, 2001; May and Lloyd, 2001; Kiss et al., 2006; Dube et al., 2009; Natale et al., 2009). Thus, these kinds of networks can be decomposed into fragments by removing only a small amount of nodes. In a network that is disjointed, that is, which has many unconnected components, disease transmission is less likely than in a totally connected network. To validate this effect, a simple epidemiological SIR (susceptible–infected– recovered) model was implemented in the present study based on the trade network under investigation (Kermack and McKendrick, 1927). This SIR model made it possible to illustrate not only the efficiency of the network-based con2
trol measures but also the differences to the control measures based on the radius around an infected holding. Therefore, the objective of this study was to use simulation modelling to compare the application of distance-based control measures to control measures based on network characteristics. Thus, the most appropriate control measure to reduce the epidemic size efficiently could be identified. Materials and Methods Data and network construction Pig movement data from a producer community in Northern Germany were analysed in an observation period between 1 June 2006 and 31 May 2009. The date of the movements, the supplier, the purchaser as well as the batch size and the type and age group of the delivered livestock were recorded. For data protection reasons, the exact coordinates of the holdings were not available. To approximate the geographical location (longitude and latitude) of the holdings, information of the town and the zip code was used. The geographic distances were calculated as the Euclidean distance between the holdings. In total, the data contained 15 372 directed animal movements between 658 holdings, meaning that each of the animal movements had one specific supplier and one specific purchaser. For the analysis of different time periods, the data were separated into the three-year network, three yearly networks and 36 monthly networks. Multiple trade contacts throughout the analysed time periods were aggregated into a single one. The production type of the different holdings was classified into five categories by means of number, type and age group of the delivered livestock: multipliers, farrowing farms, finishing farms, farrow-to-finishing farms and abattoirs. Due to the dead-end characteristic of abattoirs for the transport of live animals, this holding type was excluded from the analysis. Thus, in the total three-year network, the number of holdings reduced to 483 and 4635 animal movements, which were aggregated to 926 group movements (B€ uttner et al., 2013b). In the yearly networks, the number of holdings was on average 322 (319–323), which were connected by 1545 (1522–1571) animal movements, that is, 449 (431–468) aggregated group movements. The monthly networks had on average 129 (107 to 148) holdings and 427 (359 to 479) animal movements, which were aggregated to 114 (93–134) group movements (B€ uttner et al., 2013c). The proportions of the four holding types in the specific time intervals are reported in B€ uttner et al. (2013b,c). Disease simulation and implemented control measures In this study, the disease spread in the three observed time periods was modelled using a simple SIR model. Each © 2014 Blackwell Verlag GmbH • Transboundary and Emerging Diseases.
€ttner et al. K. Bu
Disease Control in an Animal Trade Network
holding represented a single epidemiological unit, which was connected to its trade partners by animal movements. Each holding could be in three basic states: susceptible, infected or recovered, respectively, removed. All holdings had the state susceptible at the beginning of the simulation. The epidemic started with a single infected holding chosen uniformly at random from the whole number of holdings in the network. The successors, that is, the holdings that received animals from the infected holding, became themselves infected by different transmission probabilities T = 0.1, 0.2 . . . 0.9, 0.95. For each of the new infected holdings, their successors were determined and became again infected with the transmission probability T. A holding immediately becomes recovered, respectively, removed after its successors had been determined and it played no further part in the course of the epidemic. Removal in this context does not necessarily mean depopulation, but can also stand for trade restrictions. No uncertainty or variability in the detection or the control was implemented, and the detection of infected holdings was assumed to be instantaneous. The epidemic terminated when there were no more successors remaining in the network which could be infected. This model was implemented in Python (van Rossum and Drake, 2001). Temporal aspects were not included in the model as the network-based control measures were based on the centrality parameters calculated for the aggregated networks. Control measures based on the removal of holdings in a specific radius around the infected holdings The first control measure implemented in the SIR model was the removal of holdings in a specific radius around the infected holding, that is, 1, 3, 5 and 10 km. When the simulation started and the first holding was infected, the successors of this holding were detected. The next step was to identify the holdings, which were located in one of the specified radii (e.g. 5 km) around the infected holding. These holdings were removed from the network and played therefore no further role in the course of the epidemic. All
other successors became infected with a specific transmission probability (e.g. 0.95). The epidemic terminated when there were no more successors remaining in the network, which could be infected. For each transmission probability and radius, 10 000 independent simulations were performed. To evaluate the impact of the four holding types on the implementation of the control strategies, each control measure was performed additionally excluding one specific holding type. Control measures based on the removal of holdings using the rank of specific centrality parameters The second control measure implemented in the SIR model was the removal of holdings using the rank of the centrality parameters in-degree and out-degree, ingoing and outgoing infection chain, betweenness centrality and ingoing and outgoing closeness centrality (Table 1). For each time period, the analysis of the centrality parameters was performed separately, see B€ uttner et al. (2013b,c). The centrality parameters as well as the general network properties were calculated using the Python module NetworkX (Hagberg et al., 2008). Holdings were removed from the network in descending order of the centrality parameters. More precisely, for each holding in the network, the centrality parameters were calculated and then sorted in descending order. According to this order, the most central holdings were removed from the networks, meaning the holdings with the highest centrality parameter. Then, the simulation started as described above. For each value of the centrality parameters obtained, the simulation was performed 10 000 times. The control measures based on the removal of holdings using the rank of specific centrality parameters were also carried out excluding the different holding types, as mentioned above. Comparison of control methods The control measures based on the radius around an infected holding were compared with the control measures based on the targeted removal of holdings according to
Table 1. Definition of parameters used in network analysis for the characterization of animal movements Parameter
Definition
References
In-degree Out-degree Ingoing infection chain
Number of trade partners which deliver animals to a specific holding Number of trade partners which receive animals from a specific holding Number of direct and indirect trade contacts which lead to a specific holding taking the chronological order of the contacts into account Number of direct and indirect trade contacts which originate at a specific holding taking the chronological order of the contacts into account The betweenness centrality measures the extent to which a holding lies on paths between other holdings Mean distance from all other reachable holdings to one specific holding Mean distance from one holding to all other reachable holdings
Newman (2010) Newman (2010) €remark et al. (2011) No
Outgoing infection chain Betweenness Ingoing closeness Outgoing closeness
© 2014 Blackwell Verlag GmbH • Transboundary and Emerging Diseases.
Webb (2006); Dube et al. (2008) Wasserman and Faust (1994) Wasserman and Faust (1994) Wasserman and Faust (1994)
3
Disease Control in an Animal Trade Network
specific centrality parameters. The efficiencies of both approaches based on the resulting epidemic size were analysed. In contrast to the distance-based control measures, the network-based control measures are independent from the course of the epidemic, meaning the first infected holding. Furthermore, for each rank of the centrality parameters, a mean epidemic size was obtained, whereas the distance-based control measures had only one mean epidemic size for each analysed transmission probability and radius. To facilitate a comparison of these different results, the average values for the distance-based control measures were approximated with the help of the slope obtained for the network-based control measures based on the same epidemic size. By means of this approximation, the difference between the network-based and the distance-based control measures could be obtained represented as a factor. As reference value the fraction of removed holdings in the distance-based control measures was set to 1. The approximation was performed using SAS statistical software package (SASâ Institute Inc., 2008).
€ttner et al. K. Bu
61.8% (572) of the contacts originated from a multiplier and 25.6% (237) from a farrowing farm. The yearly and monthly networks showed similar results with 65.3% (293) and 68.4% (78) for the contacts starting at a multiplier and 24.1% (108) and 23.7% (27) for the farrowing farms as supplier. Otherwise, the majority of the direct trade contacts in the three-year network ended at finishing farms with 40.1% (371) and farrow-to-finishing farms with 49.8% (461). In the yearly and monthly networks, 37.4% (168) and 36.0% (41) of the trade contacts ended at a finishing farm and 51.9% (233) and 52.6% (60) ended at a farrow-to-finishing farm. Furthermore, almost 50% (442) of all trade contacts in the total three-year network took place between the multiplier holding type as supplier and the farrow-to-finishing farm as purchaser, followed by the animal transports from the farrowing farms to the finishing farms with about 25% (228) of trade contacts. In the yearly and monthly networks, the connections between these holding types showed similar percentages. Disease simulation and implemented control measures
Results Distances and number of trade contacts between different holding types Figures 1 and 2 show the geographical distribution of the holdings and the Euclidean distances of the trade contacts in the pork supply chain, respectively. The histogram (Fig. 2) illustrates the right-skewed distribution of the median Euclidean distances between the trade partners. A total of 77.6% of the trade contacts passed by a distance of
