Mapping Out Emerging Network Structures in Dynamic Network Models Coupled with Epidemics

  • István Z. Kiss
  • Luc Berthouze
  • Joel C. Miller
  • Péter L. Simon
Chapter
Part of the Theoretical Biology book series (THBIO)

Abstract

We consider the susceptible – infected – susceptible (SIS) epidemic on a dynamic network model with addition and deletion of links depending on node status. We analyse the resulting pairwise model using classical bifurcation theory to map out the spectrum of all possible epidemic behaviours. However, the major focus of the chapter is on the evolution and possible equilibria of the resulting networks. Whereas most studies are driven by determining system-level outcomes, e.g., whether the epidemic dies out or becomes endemic, with little regard for the emerging network structure, here, we want to buck this trend by augmenting the system-level results with mapping out of the structure and properties of the resulting networks. We find that depending on parameter values the network can become disconnected and show bistable-like behaviour whereas the endemic steady state sees the emergence of networks with qualitatively different degree distributions. In particular, we observe de-phased oscillations of both prevalence and network degree during which there is role reversal between the level and nature of the connectivity of susceptible and infected nodes. We conclude with an attempt at describing what a potential bifurcation theory for networks would look like.

Notes

Acknowledgements

Joel C. Miller was funded by the Global Good Fund through the Institute for Disease Modeling and by a Larkins Fellowship from Monash University. Péter L. Simon acknowledges support from Hungarian Scientific Research Fund, OTKA, (grant no. 115926).

References

  1. 1.
    Ball, F., Neal, P.: Network epidemic models with two levels of mixing. Math. Biol. 212(1), 69–87 (2008)MathSciNetMATHGoogle Scholar
  2. 2.
    Chung, F.R., Lu, L.: Complex Graphs and Networks, vol. 107. American Mathematical Society Providence, Providence (2006)MATHGoogle Scholar
  3. 3.
    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. Interdisciplinary Perspectives on Infectious Diseases 2011 (2011)Google Scholar
  4. 4.
    Durrett, R.: Random Graph Dynamics. Cambridge University Press, Cambridge (2007)MATHGoogle Scholar
  5. 5.
    Eames, K., Keeling, M.: Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc. Natl. Acad. Sci. USA 99(20), 13330–13335 (2002)CrossRefGoogle Scholar
  6. 6.
    Gross, T., Blasius, B.: Adaptive coevolutionary networks: a review. J. R. Soc. Interface 5(20), 259–271 (2008)CrossRefGoogle Scholar
  7. 7.
    Gross, T., D’Lima, C.J.D., Blasius, B.: Epidemic dynamics on an adaptive network. Phys. Rev. Lett. 96(20), 208701 (2006)CrossRefGoogle Scholar
  8. 8.
    Gross, T., Kevrekidis, I.G.: Robust oscillations in sis epidemics on adaptive networks: coarse graining by automated moment closure. Europhys. Lett. 82(3), 38004 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    House, T., Keeling, M.: Insights from unifying modern approximations to infections on networks. J. R. Soc. Interface 8(54), 67–73 (2011)CrossRefGoogle Scholar
  10. 10.
    Keeling, M.J.: The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. B Biol. Sci. 266(1421), 859–867 (1999)CrossRefGoogle Scholar
  11. 11.
    Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks: From Exact to Approximate Models. IAM. Springer (2017)CrossRefMATHGoogle Scholar
  12. 12.
    Kiss, I.Z., Berthouze, L., Taylor, T.J., Simon, P.L.: Modelling approaches for simple dynamic networks and applications to disease transmission models. Proc. R. Soc. A 468(2141), 1332–1355 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lindquist, J., Ma, J., van den Driessche, P., Willeboordse, F.: Effective degree network disease models. J. Math. Biol. 62(2), 143–164 (2011). doi:  10.1007/s00285-010-0331-2 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Marceau, V., Noël, P.A., Hébert-Dufresne, L., Allard, A., Dubé, L.J.: Adaptive networks: coevolution of disease and topology. Phys. Rev. E 82(3), 036116 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Miller, J.C., Kiss, I.Z.: Epidemic spread in networks: existing methods and current challenges. Math. Model. Nat. Phenom. 9(02), 4–42 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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). doi:  10.1098/rsif.2011.0403 CrossRefGoogle Scholar
  17. 17.
    Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87(3), 925–979 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rogers, T., Clifford-Brown, W., Mills, C., Galla, T.: Stochastic oscillations of adaptive networks: application to epidemic modelling. J. Stat. Mech: Theory Exp. 2012(08), P08018 (2012)CrossRefGoogle Scholar
  20. 20.
    Sayama, H., Pestov, I., Schmidt, J., Bush, B.J., Wong, C., Yamanoi, J., Gross, T.: Modeling complex systems with adaptive networks. Comput. Math. Appl. 65(10), 1645–1664 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Silk, H., Demirel, G., Homer, M., Gross, T.: Exploring the adaptive voter model dynamics with a mathematical triple jump. New J. Phys. 16(9), 093051 (2014)CrossRefGoogle Scholar
  22. 22.
    Szabó, A., Simon, P.L., Kiss, I.Z.: Detailed study of bifurcations in an epidemic model on a dynamic network. Differ. Equ. Appl. 4, 277–296 (2012)MathSciNetMATHGoogle Scholar
  23. 23.
    Szabó-Solticzky, A.: Dynamics of a link-type independent adaptive epidemic model. Differ. Equ. Appl. 9, 105–122 (2017)MathSciNetMATHGoogle Scholar
  24. 24.
    Szabó-Solticzky, A., Berthouze, L., Kiss, I.Z., Simon, P.L.: Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis. J. Math. Biol. 72(5), 1153–1176 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Taylor, M., Taylor, T.J., Kiss, I.Z.: Epidemic threshold and control in a dynamic network. Phys. Rev. E 85(1), 016103 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • István Z. Kiss
    • 1
  • Luc Berthouze
    • 2
  • Joel C. Miller
    • 3
    • 4
  • Péter L. Simon
    • 5
  1. 1.Department of Mathematics, School of Mathematical and Physical SciencesUniversity of SussexFalmerUK
  2. 2.Centre for Computational Neuroscience and RoboticsUniversity of SussexFalmerUK
  3. 3.School of Mathematical Sciences, School of Biological Sciences and Monash Academy for Cross and Interdisciplinary MathematicsMonash UniversityClaytonAustralia
  4. 4.Institute for Disease ModelingBellevueUSA
  5. 5.Institute of Mathematics, Eötvös Loránd University Budapest, and Numerical Analysis, and Large Networks Research GroupHungarian Academy of SciencesBudapestHungary

Personalised recommendations