Background Many networks exhibit time-dependent topologies, where an edge only exists during a certain period of time. The first measurements of such networks are very recent so that a profound theoretical understanding is still lacking. In this work, we focus on the propagation properties of infectious diseases in time-dependent networks. In particular, we analyze a dataset containing livestock trade movements. The corresponding networks are known to be a major route for the spread of animal diseases. In this context chronology is crucial. A disease can only spread if the temporal sequence of trade contacts forms a chain of causality. Therefore, the identification of relevant nodes under time-varying network topologies is of great interest for the implementation of counteractions. Methodology/Findings We find that a time-aggregated approach might fail to identify epidemiologically relevant nodes. Hence, we explore the adaptability of the concept of centrality of nodes to temporal networks using a data-driven approach on the example of animal trade. We utilize the size of the in- and out-component of nodes as centrality measures. Both measures are refined to gain full awareness of the time-dependent topology and finite infectious periods. We show that the size of the components exhibit strong temporal heterogeneities. In particular, we find that the size of the components is overestimated in time-aggregated networks. For disease control, however, a risk assessment independent of time and specific disease properties is usually favored. We therefore explore the disease parameter range, in which a time-independent identification of central nodes remains possible. Conclusions We find a ranking of nodes according to their component sizes reasonably stable for a wide range of infectious periods. Samples based on this ranking are robust enough against varying disease parameters and hence are promising tools for disease control.
References
[1]
Rushton J (2011) The Economics of Animal Health and Production. Wallingford, UK: CABI Publishing.
[2]
Green D, Kiss I, Kao R (2006) Modelling the initial spread of foot-and-mouth disease through animal movements. Proceedings of the Royal Society B: Biological Sciences 273: 2729–2735.
[3]
Bigras-Poulin M, Barfod K, Mortensen S, Greiner M (2007) Relationship of trade patterns of the danish swine industry animal movements network to potential disease spread. Preventive Veterinary Medicine 80: 143.
[4]
Christley R, Robinson S, Lysons R, French N (2005) Network analysis of cattle movement in great britain. In: Mellor D, Russell A, Wood J, editors. Proceedings of a meeting held at Nairn, Inverness, Scotland, 30th March-1st April 2005. Leicestershire, UK: Society for Veterinary Epidemiology and Preventive Medicine. 234–244.
[5]
UK Department for Environment, Food and Rural Affairs (DEFRA). Origin of the UK foot and mouth disease epidemic in 2001. Available: http://archive.defra.gov.uk/foodfarm/far？manimal/diseases/atoz/fmd/documents/fmdo？rigins1.pdf. Accessed 2013 Jan 05.
[6]
Dubé C, Ribble C, Kelton D, McNab B (2009) A review of network analysis terminology and its application to foot-and-mouth disease modelling and policy development. Transboundary and emerging diseases 56: 73–85.
[7]
Fritzemeier J, Teuffert J, Greiser-Wilke I, Staubach C, Schlüter H, et al. (2000) Epidemiology of classical swine fever in Germany in the 1990s. Veterinary Microbiology 77: 29–41.
[8]
Danon L, Ford AP, House T, Jewell CP, Keeling MJ, et al. (2011) Networks and the epidemiology of infectious disease. Interdisciplinary perspectives on infectious diseases 2011: 284909.
[9]
Martínez-López B, Perez AM, Sánchez-Vizcaíno JM (2009) Social network analysis. Review of general concepts and use in preventive veterinary medicine. Transboundary and emerging diseases 56: 109–20.
[10]
Dubé C, Ribble C, Kelton D, McNab B (2008) Comparing network analysis measures to determine potential epidemic size of highly contagious exotic diseases in fragmented monthly networks of dairy cattle movements in Ontario, Canada. Transboundary and emerging diseases 55: 382–92.
[11]
Vernon MC, Keeling MJ (2009) Representing the UK's cattle herd as static and dynamic networks. Proceedings Biological sciences/The Royal Society 276: 469–76.
[12]
Natale F, Savini L, Giovannini A, Calistri P, Candeloro L, et al. (2010) Evaluation of risk and vulnerability using a Disease Flow Centrality measure in dynamic cattle trade networks. Preventive veterinary medicine 98: 111–118.
[13]
N？remark M, H？kansson N, Lewerin SS, Lindberg A, Jonsson A (2011) Network analysis of cattle and pig movements in Sweden: Measures relevant for disease control and risk based surveillance. Preventive veterinary medicine 99: 78–90.
[14]
Bajardi P, Barrat A, Natale F, Savini L, Colizza V (2011) Dynamical Patterns of Cattle Trade Movements. PLoS ONE 6: e19869.
[15]
Bajardi P, Barrat A, Savini L, Colizza V (2012) Optimizing surveillance for livestock disease spreading through animal movements. Journal of The Royal Society Interface 9: 2814–2825.
[16]
Basu P, Bar-Noy A, Ramanathan R, Johnson MP (2010) Modeling and Analysis of Time-Varying Graphs. Arxiv preprint arXiv 10120260 11.
Lentz HHK, Selhorst T, Sokolov IM (2012) Unfolding accessibility provides a macroscopic approach to temporal networks. Arxiv preprint arXiv:12102283.
[19]
Liu S, Baronchelli A, Perra N (2012) Contagion dynamics in time-varying metapopulation networks. Arxiv preprint arXiv:12102776.
[20]
Natale F, Giovannini A, Savini L, Palma D, Possenti L, et al. (2009) Network analysis of Italian cattle trade patterns and evaluation of risks for potential disease spread. Preventive Veterinary Medicine 92: 341–50.
[21]
Vazquez A, Rácz B, Lukács A, Barabási AL (2007) Impact of Non-Poissonian Activity Patterns on Spreading Processes. Physical Review Letters 98: 1–4.
[22]
Karsai M, Kivel？ M, Pan R, Kaski K, Kertész J, et al. (2011) Small but slow world: How network topology and burstiness slow down spreading. Physical Review E 83: 1–4.
[23]
Rocha LEC, Liljeros F, Holme P (2011) Simulated Epidemics in an Empirical Spatiotemporal Network of 50,185 Sexual Contacts. PLoS Computational Biology 7: e1001109.
[24]
Miritello G, Moro E, Lara R (2011) Dynamical strength of social ties in information spreading. Physical Review E 83: 3–6.
[25]
Stehlé J, Voirin N, Barrat A, Cattuto C, Colizza V, et al. (2011) Simulation of an SEIR infectious disease model on the dynamic contact network of conference attendees. BMC Medicine 9: 87.
[26]
Rocha LEC, Decuyper A, Blondel VD (2012) Epidemics on a stochastic model of temporal network. Arxiv preprint arXiv:12045421.
[27]
Kitsak M, Gallos LK, Havlin S, Liljeros F, Muchnik L, et al. (2010) Identification of inuential spreaders in complex networks. Nature Physics 6: 888–893.
[28]
Lee S, Rocha LEC, Liljeros F, Holme P (2012) Exploiting temporal network structures of human interaction to effectively immunize populations. PLoS ONE 7: e36439.
[29]
Cohen R, Havlin S, Ben-Avraham D (2003) Efficient Immunization Strategies for Computer Networks and Populations. Physical Review Letters 91: 2–5.
[30]
Directive 2000/15/EC of the European Parliament and the Council of 10 April 2000 amending Council Directive 64/432/EEC on health problems affecting intra-community trade in bovine animals and swine. Available: http://eu.vlex.com/vid/problems-affectin？g-intra-animals-swine-24527186. Accessed 2013 Jan 14.
[31]
Tang J, Scellato S, Musolesi M, Mascolo C, Latora V (2010) Small-world behavior in time-varying graphs. Physical Review E 81: 81–84.
[32]
Tang J, Musolesi M, Mascolo C, Latora V, Nicosia V (2010) Analysing information ows and key mediators through temporal centrality metrics. In: Proceedings of the 3rd Workshop on Social Network Systems - SNS '10. New York: ACM Press. 1–6. doi:10.1145/1852658. 1852661.
[33]
Casteigts A, Flocchini P, Quattrociocchi W, Santoro N (2012) Time-varying graphs and dynamic networks. International Journal of Parallel, Emergent and Distributed Systems 27: 387–408.
[34]
Pan R, Saram？ki J (2011) Path lengths, correlations, and centrality in temporal networks. Physical Review E 84: 1–10.
[35]
Grindrod P, Parsons M, Higham D, Estrada E (2011) Communicability across evolving networks. Physical Review E 83: 1–10.
[36]
Nicosia V, Tang J, Musolesi M, Russo G, Mascolo C, et al. (2012) Components in time-varying graphs. Chaos: An Interdisciplinary Journal of Nonlinear Science 22: 023101.
[37]
Kim H, Anderson R (2012) Temporal node centrality in complex networks. Physical Review E 85: 026107.
[38]
Lentz HHK, Selhorst T, Sokolov IM (2012) Spread of infectious diseases in directed and modular metapopulation networks. Physical Review E 85: 066111.
[39]
Holme P (2005) Network reachability of real-world contact sequences. Physical Review E 71: 046119.
[40]
Riolo CS, Koopman JS, Chick SE (2001) Methods and measures for the description of epidemiologic contact networks. Journal of urban health: bulletin of the New York Academy of Medicine 78: 446–57.
[41]
Kivel？ M, Pan RK, Kaski K, Kertész J, Saram？ki J, et al. (2012) Multiscale analysis of spreading in a large communication network. J Stat Mech 2012: P03005.
[42]
Kao RR, Danon L, Green DM, Kiss IZ (2006) Demographic structure and pathogen dynamics on the network of livestock movements in Great Britain. Proceedings Biological sciences/The Royal Society 273: 1999–2007.
[43]
Kiss IZ, Green DM, Kao RR (2006) The network of sheep movements within Great Britain: Network properties and their implications for infectious disease spread. Journal of the Royal Society, Interface/the Royal Society 3: 669–77.
[44]
Robinson SE, Everett MG, Christley RM (2007) Recent network evolution increases the potential for large epidemics in the British cattle population. Journal of the Royal Society, Interface/the Royal Society 4: 669–74.
[45]
The HI-Tier database is administered by the Bavarian State Ministry for Agriculture and Forestry on behalf of the German federal states.
[46]
Dorogovtsev S, Mendes J, Samukhin A (2001) Giant strongly connected component of directed networks. Physical Review E 64: 1–4.
[47]
Horst H (1998) Risk and economic consequences of contagious animal disease introduction. Wageningen University and Researchcenter Publications (Netherlands). Available: http://library.wur.nl/WebQuery/wda/abstr？act/945483. Accessed 2013 Jan 14.
[48]
Council Directive 2001/89/EC of 23 October 2001 on Community measures for the control of classical swine fever. Available: http://eur-lex.europa.eu/LexUriServ/LexU？riServ.do?uri=OJ:L:2001:316:0005:0035:EN？:PDF. Accessed 2013 Jan 14.
