Journal of Mathematical Biology

, Volume 69, Issue 6–7, pp 1627–1660 | Cite as

Spreading dynamics on complex networks: a general stochastic approach

  • Pierre-André Noël
  • Antoine Allard
  • Laurent Hébert-Dufresne
  • Vincent Marceau
  • Louis J. Dubé
Article

Abstract

Dynamics on networks is considered from the perspective of Markov stochastic processes. We partially describe the state of the system through network motifs and infer any missing data using the available information. This versatile approach is especially well adapted for modelling spreading processes and/or population dynamics. In particular, the generality of our framework and the fact that its assumptions are explicitly stated suggests that it could be used as a common ground for comparing existing epidemics models too complex for direct comparison, such as agent-based computer simulations. We provide many examples for the special cases of susceptible-infectious-susceptible and susceptible-infectious-removed dynamics (e.g., epidemics propagation) and we observe multiple situations where accurate results may be obtained at low computational cost. Our perspective reveals a subtle balance between the complex requirements of a realistic model and its basic assumptions.

Keywords

Spreading dynamics Complex networks Stochastic processes Contact networks Epidemics Markov processes 

Mathematics Subject Classification (2000)

93A30 05C82 60J28 92D25 92D30 

Notes

Acknowledgments

The research team acknowledges to the Canadian Institutes of Health Research (CIHR), the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec—Nature et technologies (FRQ–NT) for financial support. We are grateful to the anonymous referees that have helped improve our presentation.

References

  1. Allard A, Noël PA, Dubé LJ, Pourbohloul B (2009) Heterogeneous bond percolation on multitype networks with an application to epidemic dynamics. Phys Rev E 79(036):113. doi: 10.1103/PhysRevE.79.036113 Google Scholar
  2. Allard A, Hébert-Dufresne L, Noël PA, Marceau V, Dubé LJ (2012) Bond percolation on a class of correlated and clustered random graphs. J Phys A 45(405):005. doi: 10.1088/1751-8113/45/40/405005 Google Scholar
  3. Auchincloss AH, Diez Roux AV (2008) A new tool for epidemiology: the usefulness of dynamic-agent models in understanding place effects on health. Am J Epidemiol 168:1–8. doi: 10.1093/aje/kwn118 CrossRefGoogle Scholar
  4. Ball F, Neal P (2008) Network epidemic models with two levels of mixing. Math Biosci 212:69–87CrossRefMATHMathSciNetGoogle Scholar
  5. Bansal S, Grenfell BT, Meyers LA (2007) When individual behaviour matters: homogeneous and network models in epidemiology. J R Soc Interface 4:879–891. doi: 10.1098/rsif.2007.1100 CrossRefGoogle Scholar
  6. Barrat A, Barthélemy M, Vespignani A (2008) Dynamical processes on complex networks. Cambridge University Press, New YorkCrossRefMATHGoogle Scholar
  7. Bascompte J, Stouffer DB (2009) The assembly and disassembly of ecological networks. Philos Trans R Soc Lond B 364:1781–1787. doi: 10.1098/rstb.2008.0226 CrossRefGoogle Scholar
  8. Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU (2006) Complex networks: structure and dynamics. Phys Rep 424:175–308. doi: 10.1016/j.physrep.2005.10.009 CrossRefMathSciNetGoogle Scholar
  9. Broeck W, Gioannini C, Goncalves B, Quaggiotto M, Colizza V, Vespignani A (2011) The GLEaMviz computational tool, a publicly available software to explore realistic epidemic spreading scenarios at the global scale. BMC Infect Dis 11(1):37. doi: 10.1186/1471-2334-11-37 CrossRefGoogle Scholar
  10. Dangerfield CE, Ross JV, Keeling MJ (2009) Integrating stochasticity and network structure in an epidemic model. J R Soc Interface 6:761–774. doi: 10.1098/rsif.2008.0410 Google Scholar
  11. Danon L, Ford A, House T, Jewell CP, Keeling MJ, Roberts GO, Ross JV, Vernon MC (2011) Networks and the epidemiology of infectious disease. Interdiscip Perspect Infect Dis 284:909:1-28. doi: 10.1155/2011/284909 Google Scholar
  12. Decreusefond L, Dhersin JS, Moyal P, Tran VC (2012) Large graph limit for an sir process in random network with heterogeneous connectivity. Ann Appl Probab 22:541–575CrossRefMATHMathSciNetGoogle Scholar
  13. Dunne JA, Williams RJ (2009) Cascading extinctions and community collapse in model food webs. Philos Trans R Soc Lond B 364:1711–1723. doi: 10.1098/rstb.2008.0219 CrossRefGoogle Scholar
  14. Durrett R (2007) Random graph dynamicsGoogle Scholar
  15. Eames KTD, Keeling MJ (2002) Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. PNAS 99:13,330–13,335. doi: 10.1073/pnas.202244299 CrossRefGoogle Scholar
  16. Gardiner CW (2004) Handbook of stochastic methods for physics. Chemistry and the natural sciences. Springer, BerlinCrossRefMATHGoogle Scholar
  17. Gleeson JP (2011) High-accuracy approximation of binary-state dynamics on networks. Phys Rev Lett 107(068):701. doi: 10.1103/PhysRevLett.107.068701 Google Scholar
  18. Hébert-Dufresne L, Noël PA, Marceau V, Allard A, Dubé LJ (2010) Propagation dynamics on networks featuring complex topologies. Phys Rev E 82(3):036,115. doi: 10.1103/PhysRevE.82.036115 CrossRefGoogle Scholar
  19. House T, Keeling MJ (2011) Insights from unifying modern approximations to infections on networks. J R Soc Int 8(54):67–73. doi: 10.1098/rsif2010.0179 Google Scholar
  20. House T, Davies G, Danon L, Keeling MJ (2009) A motif-based approach to network epidemics. Bull Math Biol 71:1693–1706. doi: 10.1007/s11538-009-9420-z CrossRefMATHMathSciNetGoogle Scholar
  21. Karrer B, Newman MEJ (2010) Random graphs containing arbitrary distributions of subgraphs. Phys Rev E 82(6):066118. doi: 10.1103/PhysRevE.82.066118 CrossRefMathSciNetGoogle Scholar
  22. Keeling MJ, Eames KTD (2005) Networks and epidemic models. J R Soc Interface 2(4):295–307. doi: 10.1098/rsif2005.0051 Google Scholar
  23. Keeling MJ, Rand DA, Morris AJ (1997) Correlation models for childhood epidemics. Proc R Soc B 264(1385):1149–1156. doi: 10.1098/rspb.1997.0159 CrossRefGoogle Scholar
  24. Marceau V, Noël PA, Hébert-Dufresne L, Allard A, Dubé LJ (2010) Adaptive networks: coevolution of disease and topology. Phys Rev E 82(3):036116. doi: 10.1103/PhysRevE.82.036116 CrossRefMathSciNetGoogle Scholar
  25. Marceau V, Noël PA, Hébert-Dufresne L, Allard A, Dubé LJ (2011) Modeling the dynamical interaction between epidemics on overlay networks. Phys Rev E 84(2):026105. doi: 10.1103/PhysRevE.84.026105 CrossRefGoogle Scholar
  26. McLane AJ, Semeniuk C, McDermid GJ, Marceau DJ (2011) The role of agent-based models in wildlife ecology and management. Ecol Model 222:1544–1556. doi: 10.1016/j.ecolmodel.2011.01.020 CrossRefGoogle Scholar
  27. Miller JC (2010) A note on a paper by Erik Volz: SIR dynamics in random networks. J Math Biol 62(3):349–358. doi: 10.1007/s00285-010-0337-9 CrossRefGoogle Scholar
  28. Miller JC, Slim AC, Volz EM (2011) Edge-based compartmental modeling for infectious disease spread. J R Soc Interface. doi: 10.1098/rsif.2011.0403 Google Scholar
  29. Newman MEJ (2010) Networks: an introduction. Oxford University Press, OxfordCrossRefGoogle Scholar
  30. Newman MEJ, Strogatz SH, Watts DJ (2001) Random graphs with arbitrary degree distributions and their applications. Phys Rev E 64(026):118. doi: 10.1103/PhysRevE.64.026118 Google Scholar
  31. Noël PA, Allard A, Hébert-Dufresne L, Marceau V, Dubé LJ (2012) Propagation on networks: an exact alternative perspective. Phys Rev E 85(031):118. doi: 10.1103/PhysRevE.85.031118 Google Scholar
  32. Park J, Newman MEJ (2004) Statistical mechanics of networks. Phys Rev E 70(1—-13):066117CrossRefMathSciNetGoogle Scholar
  33. Rezende EL, Lavabre JE, Guimarães PR, Jordano P, Bascompte JB (2007) Non-random coextinctions in phylogenetically structured mutualistic networks. Nature 448:925–928. doi: 10.1038/nature05956 CrossRefGoogle Scholar
  34. Rogers T (2011) Maximum-entropy moment-closure for stochastic systems on networks. J Stat Mech (05):P05007Google Scholar
  35. Sharkey KJ (2011) Deterministic epidemic models on contact networks: correlations and unbiological terms. Theor Popul Biol 79(4):115–129. doi: 10.1016/j.tpb.2011.01.004 CrossRefMathSciNetGoogle Scholar
  36. Taylor M, Simon PL, Green DM, House T, Kiss IZ (2012) From Markovian to pairwise epidemic models and the performance of moment closure approximations. J Math Biol 64(6):1021–1042. doi: 10.1007/s00285-011-0443-3 CrossRefMATHMathSciNetGoogle Scholar
  37. Van Kampen NG (2007) Stochastic processes in physics and chemistry, 3rd ednGoogle Scholar
  38. Volz E (2008) SIR dynamics in random networks with heterogeneous connectivity. J Math Biol 56(3):293–310. doi: 10.1007/s00285-007-0116-4 CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Pierre-André Noël
    • 1
  • Antoine Allard
    • 2
  • Laurent Hébert-Dufresne
    • 2
  • Vincent Marceau
    • 2
  • Louis J. Dubé
    • 2
  1. 1.University of CaliforniaDavisUSA
  2. 2.Département de Physique, de Génie Physique et d’OptiqueUniversité Laval, QuébecQuébecCanada

Personalised recommendations