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Modeling Epidemic Spreading in Complex Networks: Concurrency and Traffic

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Handbook of Optimization in Complex Networks

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 57))

Abstract

The study of complex networks sheds light on the relation between the structure and function of complex systems. One remarkable result is the absence of an epidemic threshold in infinite-size scale-free networks, which implies that any infection will perpetually propagate regardless of the spreading rate. However, real-world networks are finite and experience indicates that infections do have a finite lifetime. In this chapter, we will provide with two new approaches to cope with the problem of concurrency and traffic in the spread of epidemics. We show that the epidemic incidence is shaped by contact flow or traffic conditions. Contrary to the classical assumption that infections are transmitted as a diffusive process from nodes to all neighbors, we instead consider the scenario in which epidemic pathways are defined and driven by flows. Extensive numerical simulations and theoretical predictions show that whether a threshold exists or not depends directly on contact flow conditions. Two extreme cases are identified. In the case of low traffic, an epidemic threshold shows up, while for very intense flow, no epidemic threshold appears. In this way, the classical mean-field theory for epidemic spreading in scale free networks is recovered as a particular case of the proposed approach. Our results explain why some infections persist with low prevalence in scale-free networks, and provide a novel conceptual framework to understand dynamical processes on complex networks.

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Notes

  1. 1.

    Strictly speaking, when λ = 1, our model is not exactly the standard CP, since in that case reinfections are not considered. However, we will refer to it as a CP since only one neighbor is contacted at each time step and the critical points of both variants are the same.

References

  1. Amaral, L. A. N. , Scala, A., Barthélemy, M., Stanley, H. E. Classes of small-world networks. Proc. Nat. Acad. Sci. USA 97:11149-11152. (2000)

    Article  Google Scholar 

  2. Anderson, R. M., May, R. M. Infectious diseases of humans: Dynamics and Control. (Oxford University Press, Oxford). (1992)

    Google Scholar 

  3. Barabási, A. L., Albert, R. Emergence of scaling in random networks. Science 286:509-512. (1999)

    Article  MathSciNet  Google Scholar 

  4. Barthélemy, M., Barrat, A., Pastor-Satorras, R., Vespignani A. Velocity and hierarchical spread of epidemic outbreaks in scale-free networks. Phys. Rev. Lett. 92:178701. (2004)

    Article  Google Scholar 

  5. Barrat, A., Barthélemy, M., Pastor-Satorras, R., Vespignani, A. The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA 101:3747-3752. (2004)

    Article  Google Scholar 

  6. Boccaletti, S., Latora, V., Moreno, Y., Chávez, M., Hwang, D. U. Complex Networks: Structure and Dynamics. Phys. Rep. 424:175-308. (2006)

    Article  MathSciNet  Google Scholar 

  7. Boguña, M., Castellano, C., Pastor-Satorras, R. Langevin approach for the dynamics of the contact process on annealed scale-free networks. Phys. Rev. E 79:036110. (2009)

    Article  MathSciNet  Google Scholar 

  8. Boguñá, M., Krioukov, D. Navigating Ultrasmall Worlds in Ultrashort Time. Phys. Rev. Lett. 102:058701. (2009)

    Article  Google Scholar 

  9. Boguñá, M., Krioukov, D., Claffy, K. C. Navigability of Complex Networks. Nature Physics 5:74-80. (2009)

    Article  Google Scholar 

  10. Caldarelli, G. Scale-Free Networks. (Oxford University Press, Oxford). (2007)

    Google Scholar 

  11. Castellano, C., Pastor-Satorras, R. Non-mean-field behavior of the contact process on scale-free networks. Phys. Rev. Lett. 96:038701. (2006)

    Article  Google Scholar 

  12. Castellano, C., Pastor-Satorras, R. Reply: Non-mean-field behavior of the contact process on scale-free networks. Phys. Rev. Lett. 98:029802. (2007)

    Article  Google Scholar 

  13. Castellano, C., Pastor-Satorras, R. Routes to thermodynamic limit on scale-free networks. Phys. Rev. Lett. 100:148701. (2008)

    Article  Google Scholar 

  14. Catanzaro, M., Boguña, M., Pastor-Satorras, R. Diffusion-annihilation processes in complex networks. Phys. Rev. E 71:056104. (2005)

    Article  Google Scholar 

  15. Chakrabarti, D., Wang, Y., Wang, C., Leskovec, J., Faloutsos, C. Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur. 10(4):13. (2008)

    Article  Google Scholar 

  16. Chung, F., Lu, L., Vu, V. Spectra of random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 100:6313-6318. (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Colizza, V., Barrat, A., Barthélemy, M., Vespignani, A. Predictability and epidemic pathways in global outbreaks of infectious diseases: the SARS case study. BMC Medicine 5:34. (2007)

    Article  Google Scholar 

  18. Colizza, V., Pastor-Satorras, R., Vespignani A. Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nature Physics 3:276-282. (2007)

    Article  Google Scholar 

  19. Colizza, V., Vespignani, A. Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations. Journal of Theoretical Biology 251:450-467. (2008)

    Article  Google Scholar 

  20. Daley, D. J., Gani, J. Epidemic Modelling. (Cambridge University Press, Cambridge). (1999)

    Google Scholar 

  21. Dorogovtsev, S. N, Goltsev, A. V., Mendes, J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80:1275-1336. (2008)

    Google Scholar 

  22. Eubank, S., Guclu, H., Anil-Kumar, V. S., Marathe, M. V., Srinivasan, A., Toroczkai, Z., Wang N. Modelling disease outbreaks in realistic urban social networks. Nature 429:180-184. (2004)

    Article  Google Scholar 

  23. Gallos, L. K., Argyrakis, P. Absence of Kinetic Effects in Reaction-diffusion processes in scale-free Networks. Phys. Rev. Lett. 92:138301. (2004)

    Article  Google Scholar 

  24. Gardeñes, J.G., Latora, V., Moreno, Y., Profumo, E. Spreading of sexually transmitted diseases in heterosexual populations. Proc. Nat. Acad. Sci. USA 105:1399-1404. (2008)

    Article  Google Scholar 

  25. Grais, R. F., Ellis, J. H., Kress, A., Glass, G. E. Modeling the spread of annual influenza epidemics in the U.S.: The potential role of air travel. Health Care Management Science 7:127-134. (2004)

    Google Scholar 

  26. Gómez, S., Arenas, A., Borge-Holthoefer, J., Meloni, S., Moreno, Y. Discrete-time Markov chain approach to contact-based disease spreading in complex networks. Europhys. Lett. 89:38009. (2010)

    Article  Google Scholar 

  27. Guimerà, R., Díaz-Guilera, A., Vega-Redondo, F., Cabrales, A., Arenas, A. Optimal network topologies for local search with congestion. Phys. Rev. Lett. 89:248701. (2002)

    Article  Google Scholar 

  28. Guimerà, R., Mossa, S., Turtschi, A., Amaral, L. A. N. The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proc. Natl. Acad. Sci. USA 102:7794-7799. (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ha, M., Hong, H., Park, H. Comment: Non-mean-field behavior of the contact process on scale-free networks. Phys. Rev. Lett. 98:029801. (2007)

    Article  Google Scholar 

  30. Han, J.-D. J., Bertin, N., Hao, T., Goldberg, D. S. et al Evidence for dynamically organized modularity in the yeast protein-protein interaction network. Nature 430: 88-93. (2004)

    Article  Google Scholar 

  31. Hethcote, H. W. The mathematics of infectious diseases. SIAM Review 42:599-653. (2000)

    Google Scholar 

  32. Hufnagel, L., Brockmann, D., Geisel T. Forecast and control of epidemics in a globalized world. Proc. Natl. Acad. Sci. USA 101:1512415129. (2004)

    Article  Google Scholar 

  33. Liljeros F., Edling C. R., Amaral L. A. N., Stanley H. E., Aberg Y. The Web of Human Sexual Contacts. Nature 411:907-908. (2001)

    Article  Google Scholar 

  34. LLoyd, A. L., May, R. M. How viruses spread among computers and people. Science 292:1316-1317. (2001)

    Google Scholar 

  35. Marro, J., Dickman, R. Nonequilibrium phase transitions in lattice models. (Cambridge University Press, Cambridge). (1999)

    Google Scholar 

  36. Meloni, S., Gardeñes, J.G., Latora, V., Moreno, Y. Scaling Breakdown in Flow Fluctuations on Complex Networks. Phys. Rev. Lett. 100:208701. (2008)

    Article  Google Scholar 

  37. Meloni, S., Arenas, A., Moreno Y. Traffic-Driven Epidemic Spreading in Finite-Size Scale-Free Networks. Proc. Natl. Acad. Sci. USA 106:16897-16902. (2009)

    Article  Google Scholar 

  38. Moreno, Y., Pastor-Satorras, R. Vespignani, A. Epidemic outbreaks in complex heterogeneous networks. Eur. Phys. J. B 26:521-529. (2002)

    Google Scholar 

  39. Murray J. D. Mathematical Biology. (Springer-Verlag, Germany, Berlin). (2002)

    Google Scholar 

  40. Murray, J. D. Mathematical Biology. (Springer-Verlag, 3rd Edition). (2007)

    Google Scholar 

  41. Newman, M. E. J. The spread of epidemic disease on networks. Phys. Rev. E 66:016128. (2002)

    Google Scholar 

  42. Newman, M. E. J. The structure and function of complex networks. SIAM Review 45:167-256. (2003)

    Google Scholar 

  43. Newman, M. E. J., Forrest, S., Balthrop, J. Email networks and the spread of computer viruses. Phys. Rev. E 66:035101. (2002)

    Article  Google Scholar 

  44. Noh, J. D., Rieger, H. Random Walks on Complex Networks. Phys. Rev. Lett. 92:118701. (2004)

    Article  Google Scholar 

  45. Pastor-Satorras, R., Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86:3200-3203. (2001)

    Article  Google Scholar 

  46. Pastor-Satorras, R., Vespignani A. Epidemic dynamics and endemic states in complex networks. Phys. Rev. E 63:066117. (2001)

    Article  Google Scholar 

  47. Pastor-Satorras, R., Vespignani A. Evolution and Structure of the Internet: a statistical physics approach. (Cambridge University Press, Cambridge). (2004)

    Google Scholar 

  48. Restrepo, J. G., Ott, E., Hunt, B. R. Approximating the largest eigenvalue of network adjacency matrices. Phys. Rev. E 76:056119. (2007)

    Article  MathSciNet  Google Scholar 

  49. Rosato, V., Meloni, S., Issacharoff, L., Tiriticco, F. Is the topology of the internet network really fit to its function? Physica A 387:1689-1704. (2008)

    Article  Google Scholar 

  50. Serrano, M. A. , Krioukov, D., Boguñá, M. Self-Similarity of Complex Networks and Hidden Metric Spaces. Phys. Rev. Lett. 100:078701. (2008)

    Article  Google Scholar 

  51. Shao, J., Buldyrev, S. V., Braunstein, L. A., Havlin, S., Stanley, E. Structure of shells in complex networks. Phys. Rev. E 80:036105. (2009)

    Article  Google Scholar 

  52. Sreenivasan, S., Cohen, R., Lopez, E., Toroczkai, Z., Stanley, H. E. Structural Bottlenecks for Communication in Networks. Phys. Rev. E 75:036105. (2007)

    Article  Google Scholar 

  53. Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena. (Oxford University Press, Oxford). (1987)

    Google Scholar 

  54. Zhao, L., Lai, Y.-C., Park, K., Ye, N. Onset of traffic congestion in complex networks. Phys. Rev. E 71:026125. (2005)

    Article  Google Scholar 

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Acknowledgements

We acknowledge the group of Prof. L. A. N. Amaral for sharing the airports data set. This work was supported by Spanish MICINN FIS2009-13730-C02-02, FIS2008-01240 and FIS2009-13364-C02-01, and the Generalitat de Catalunya 2009-SGR-838. A. A. acknowledges partial support by the Director, Office of Science, Computational and Technology Research, U.S. Department of Energy under Contract DE-AC02-05CH11231. Y. M. acknowledges support from the DGA through Project PI038/08 and a grant to FENOL.

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Authors and Affiliations

  1. Department of Informatics and Automation, University of Rome “Roma Tre”, Rome, 00146, Italy

    Sandro Meloni

  2. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007, Tarragona, Spain

    Alex Arenas, Sergio Gómez & Javier Borge-Holthoefer

  3. Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, 50009, Zaragoza, Spain

    Alex Arenas, Javier Borge-Holthoefer & Yamir Moreno

  4. Departamento de Física Teórica, Universidad de Zaragoza, 50009, Zaragoza, Spain

    Yamir Moreno

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  1. Sandro Meloni
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Editors and Affiliations

  1. Science and Engineering, Department of Computer and Information, University of Florida, Gainesville, 3260, Florida, USA

    My T. Thai

  2. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

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Meloni, S., Arenas, A., Gómez, S., Borge-Holthoefer, J., Moreno, Y. (2012). Modeling Epidemic Spreading in Complex Networks: Concurrency and Traffic. In: Thai, M., Pardalos, P. (eds) Handbook of Optimization in Complex Networks. Springer Optimization and Its Applications(), vol 57. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-0754-6_15

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