Advertisement

Consistent Approximation of Epidemic Dynamics on Degree-Heterogeneous Clustered Networks

  • A. Bishop
  • I. Z. Kiss
  • T. House
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 812)

Abstract

Realistic human contact networks capable of spreading infectious disease, for example studied in social contact surveys, exhibit both significant degree heterogeneity and clustering, both of which greatly affect epidemic dynamics. To understand the joint effects of these two network properties on epidemic dynamics, the effective degree model of Lindquist et al. [28] is reformulated with a new moment closure to apply to highly clustered networks. A simulation study comparing alternative ODE models and stochastic simulations is performed for SIR (Susceptible–Infected–Removed) epidemic dynamics, including a test for the conjectured error behaviour in [40], providing evidence that this novel model can be a more accurate approximation to epidemic dynamics on complex networks than existing approaches.

Keywords

Networks Epidemiology Moment Closure SIR Clustering 

References

  1. 1.
    Ball, F.: Stochastic and deterministic models for SIS epidemics among a population partitioned into households. Math. Biosci. 156(1), 41–67 (1999)Google Scholar
  2. 2.
    Ball, F., Neal, P.: Network epidemic models with two levels of mixing. Math. Biosci. 212(1), 69–87 (2008)Google Scholar
  3. 3.
    Bansal, S., Khandelwal, S., Meyers, L.A.: Exploring biological network structure with clustered random networks. BMC Bioinform. 10(1), 405 (2009)Google Scholar
  4. 4.
    Barbour, A., Reinert, G.: Approximating the epidemic curve. Electron. J. Probab. 18(54), 1–30 (2013)Google Scholar
  5. 5.
    Bohman, T., Picollelli, M.: SIR epidemics on random graphs with a fixed degree sequence. Random Struct. Algorithms 41(2), 179–214 (2012)Google Scholar
  6. 6.
    Danon, L., Ford, A.P., House, T., Jewell, C.P., Keeling, M.J., Roberts, G.O., Ross, J.V., Vernon, M.C:: Networks and the epidemiology of infectious disease. Interdiscip. Perspect. Infect. Dis. 2011 (2011)Google Scholar
  7. 7.
    Decreusefond, L., Dhersin, J.S., Moyal, P., Tran, V.C.: Large graph limit for an SIR process in random network with heterogeneous connectivity. Ann. Appl. Probab. 22(2), 541–575 (2012)Google Scholar
  8. 8.
    Del Genio, C.I., House, T.: Endemic infections are always possible on regular networks. Phys. Rev. E 88, 040,801 (2013)Google Scholar
  9. 9.
    Del Genio, C.I., Kim, H., Toroczkai, Z., Bassler, K.E.: Efficient and exact sampling of simple graphs with given arbitrary degree sequence. PloS one 5(4), e10,012 (2010)Google Scholar
  10. 10.
    Gilbert, E.N.: Random graphs. Ann. Math. Stat. 30(4), 1141–1144 (1959)Google Scholar
  11. 11.
    Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22(4), 403–434 (1976)Google Scholar
  12. 12.
    Gleeson, J.P.: Bond percolation on a class of clustered random networks. Phys. Rev. E 80(3), 036107 (2009)Google Scholar
  13. 13.
    Gleeson, J.P.: Binary-state dynamics on complex networks: pair approximation and beyond. Phys. Rev. X 3(2), 021004 (2013)Google Scholar
  14. 14.
    Green, D., Kiss, I.: Large-scale properties of clustered networks: Implications for disease dynamics. J. Biol. Dyn. 4(5), 431–445 (2010)Google Scholar
  15. 15.
    House, T.: Generalised network clustering and its dynamical implications. Adv. Complex Syst. 13(3), 281–291 (2010)Google Scholar
  16. 16.
    House, T., Davies, G., Danon, L., Keeling, M.J.: A motif-based approach to network epidemics. Bull. Math. Biol. 71(7), 1693–1706 (2009)Google Scholar
  17. 17.
    House, T., Keeling, M.J.: The impact of contact tracing in clustered populations. PLoS Comput. Biol. 6(3), e1000721 (2010)Google Scholar
  18. 18.
    House, T., Keeling, M.J.: Insights from unifying modern approximations to infections on networks. J. R. Soc. Interface 8(54), 67–73 (2011)Google Scholar
  19. 19.
    Janson, S., Luczak, M., Windridge, P.: Law of large numbers for the SIR epidemic on a random graph with given degrees. Random Struct. Algorithms 45(4), 726–763 (2014)Google Scholar
  20. 20.
    Karrer, B., Newman, M.: Random graphs containing arbitrary distributions of subgraphs. Phys. Rev. E 82, 066,118 (2010)Google Scholar
  21. 21.
    Keeling, M.J.: The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. London. Ser. B: Biol. Sci. 266(1421), 859–867 (1999)Google Scholar
  22. 22.
    Keeling, M.J., Eames, K.T.: Networks and epidemic models. J. R. Soc. Interface 2(4), 295–307 (2005)Google Scholar
  23. 23.
    Keeling, M.J., House, T., Cooper, A.J., Pellis, L.: Systematic approximations to susceptible-infectious-susceptible dynamics on networks. PLOS Comput. Biol. 12(12), e1005,296 (2016)Google Scholar
  24. 24.
    Kermack, W., McKendrick, A.: Wo kermack and ag mckendrick, proc. r. soc. london, ser. a 115, 700 (1927). Proc. R. Soc. London, Ser. A 115, 700 (1927)Google Scholar
  25. 25.
    Kirkwood, J.G., Boggs, E.M.: The radial distribution function in liquids. J. Chem. Phys. 10(6), 394–402 (1942)Google Scholar
  26. 26.
    Kiss, I.Z., Green, D.M.: Comment on ‘properties of highly clustered networks’. Phys. Rev. E 78(4), 048101 (2008)Google Scholar
  27. 27.
    Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks. Springer, Berlin (2017)Google Scholar
  28. 28.
    Lindquist, J., Ma, J., van den Driessche, P., Willeboordse, F.: Effective degree network models. J. Math. Biol. 62, 143 (2010)Google Scholar
  29. 29.
    Lindquist, J., Ma, J., van den Driessche, P., Willeboordse, F.H.: Effective degree network disease models. J. Math. Biol. 62(2), 143–164 (2011)Google Scholar
  30. 30.
    Miller, J., Slim, A., Volz, E.: Edge-based compartmental modelling for infectious disease spread. J. R. Soc. Interface 9(70), 890–906 (2012)Google Scholar
  31. 31.
    Miller, J.C.: Percolation and epidemics in random clustered networks. Phys. Rev. E 80(2), 020,901 (2009)Google Scholar
  32. 32.
    Miller, J.C.: A note on a paper by Erik Volz: SIR dynamics in random networks. J. Math. Biol. 62(3), 349–358 (2011)Google Scholar
  33. 33.
    Miller, J.C., Slim, A.C., Volz, E.M.: Edge-based compartmental modelling for infectious disease spread. J. R. Soc. Interface, rsif20110403 (2011)Google Scholar
  34. 34.
    Miller, J.C., Slim, A.C., Volz, E.M.: Edge-based compartmental modelling for infectious disease spread. J. R. Soc. Interface 9(70), 890–906 (2012)Google Scholar
  35. 35.
    Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6(2–3), 161–180 (1995)Google Scholar
  36. 36.
    Newman, M.: Networks : An Introduction. Oxford University Press, Oxford (2009)Google Scholar
  37. 37.
    Newman, M.: Random graphs with clustering. Phys. Rev. Lett. 103(5), 058701 (2009)Google Scholar
  38. 38.
    Newman, M.E.: Properties of highly clustered networks. Phys. Rev. E 68(2), 026121 (2003)Google Scholar
  39. 39.
    Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks (2014). arXiv preprint arXiv:1408.2701
  40. 40.
    Pellis, L., House, T., Keeling, M.J.: Exact and approximate moment closures for non-Markovian network epidemics. J. Theor. Biol. 382, 160–177 (2015)Google Scholar
  41. 41.
    Rand, D.: Correlation equations and pair approximations for spatial ecologies. Adv. Ecol. Theory: Princ. Appl. 100 (1999)Google Scholar
  42. 42.
    Rand, D.: Advanced ecological theory: principles and applications, chap. Correlation equations and pair approximations for spatial ecologies, pp. 100–142. Wiley, New York (2009)Google Scholar
  43. 43.
    Ritchie, M., Berthouze, L., House, T., Kiss, I.Z.: Higher-order structure and epidemic dynamics in clustered networks. J. Theor. Biol. 348, 21–32 (2014)Google Scholar
  44. 44.
    Ritchie, M., Berthouze, L., Kiss, I.Z.: Beyond clustering: Mean-field dynamics on networks with arbitrary subgraph composition (2014). arXiv preprint arXiv:1405.6234
  45. 45.
    Rogers, T.: Maximum-entropy moment-closure for stochastic systems on networks. J. Stat. Mech.: Theory Exp. 2011(05), P05,007 (2011)Google Scholar
  46. 46.
    Serrano, M.A., Boguñá, M.: Percolation and epidemic thresholds in clustered networks. Phys. Rev. Lett. 97, 088,701 (2006)Google Scholar
  47. 47.
    Simon, P., Taylor, M., Kiss, I.: Exact epidemic models on graphs using graph automorphism driven lumping. J. Math. Biol. 62, 479–508 (2010)Google Scholar
  48. 48.
    Taylor, M., Simon, P.L., Green, D.M., House, T., Kiss, I.Z.: From markovian to pairwise epidemic models and the performance of moment closure approximations. J. Math. Biol. 64(6), 1021–1042 (2012)Google Scholar
  49. 49.
    Volz, E., Miller, J., Galvani, A., Ancel-Meyers, L.: Effects of heterogeneous and clustered contact patterns on infectious disease dynamics. PLoS Comput. Biol. 7(6), e1002042 (2011)Google Scholar
  50. 50.
    Volz, E.M.: SIR dynamics in random networks with heterogeneous connectivity. J. Math. Biol. 56(3), 293–310 (2008)Google Scholar
  51. 51.
    Watts, D., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre for Complexity ScienceUniversity of WarwickCoventry, CV4 7ALUK
  2. 2.Department of Mathematics, School of Mathematical and Physical SciencesUniversity of SussexBrighton, BN1 9QHUK
  3. 3.School of MathematicsUniversity of ManchesterManchester, M13 9PLUK

Personalised recommendations