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Temporal Network Epidemiology pp 267–289Cite as

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

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.

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References

  1. Ball, F., Neal, P.: Network epidemic models with two levels of mixing. Math. Biol. 212(1), 69–87 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Chung, F.R., Lu, L.: Complex Graphs and Networks, vol. 107. American Mathematical Society Providence, Providence (2006)

    MATH  Google Scholar 

  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. Durrett, R.: Random Graph Dynamics. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  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)

    CrossRef  Google Scholar 

  6. Gross, T., Blasius, B.: Adaptive coevolutionary networks: a review. J. R. Soc. Interface 5(20), 259–271 (2008)

    CrossRef  Google Scholar 

  7. Gross, T., D’Lima, C.J.D., Blasius, B.: Epidemic dynamics on an adaptive network. Phys. Rev. Lett. 96(20), 208701 (2006)

    CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  9. House, T., Keeling, M.: Insights from unifying modern approximations to infections on networks. J. R. Soc. Interface 8(54), 67–73 (2011)

    CrossRef  Google Scholar 

  10. Keeling, M.J.: The effects of local spatial structure on epidemiological invasions. Proc. R. Soc. B Biol. Sci. 266(1421), 859–867 (1999)

    CrossRef  Google Scholar 

  11. Kiss, I.Z., Miller, J.C., Simon, P.L.: Mathematics of Epidemics on Networks: From Exact to Approximate Models. IAM. Springer (2017)

    CrossRef  MATH  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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

    MathSciNet  CrossRef  MATH  Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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

    CrossRef  Google Scholar 

  17. Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87(3), 925–979 (2015)

    MathSciNet  CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  23. Szabó-Solticzky, A.: Dynamics of a link-type independent adaptive epidemic model. Differ. Equ. Appl. 9, 105–122 (2017)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  25. Taylor, M., Taylor, T.J., Kiss, I.Z.: Epidemic threshold and control in a dynamic network. Phys. Rev. E 85(1), 016103 (2012)

    CrossRef  Google Scholar 

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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).

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, BN1 9QH, Falmer, Brighton, UK

    István Z. Kiss

  2. Centre for Computational Neuroscience and Robotics, University of Sussex, BN1 9QH, Falmer, Brighton, UK

    Luc Berthouze

  3. School of Mathematical Sciences, School of Biological Sciences and Monash Academy for Cross and Interdisciplinary Mathematics, Monash University, Clayton, VIC, Australia

    Joel C. Miller

  4. Institute for Disease Modeling, Bellevue, WA, USA

    Joel C. Miller

  5. Institute of Mathematics, Eötvös Loránd University Budapest, and Numerical Analysis, and Large Networks Research Group, Hungarian Academy of Sciences, Budapest, Hungary

    Péter L. Simon

Authors
  1. István Z. Kiss
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  2. Luc Berthouze
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  3. Joel C. Miller
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  4. Péter L. Simon
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Corresponding author

Correspondence to István Z. Kiss .

Editor information

Editors and Affiliations

  1. Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom

    Dr. Naoki Masuda

  2. Institute of Innovative Research, Tokyo Institute of Technology, Yokohama, Japan

    Prof. Dr. Petter Holme

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Kiss, I.Z., Berthouze, L., Miller, J.C., Simon, P.L. (2017). Mapping Out Emerging Network Structures in Dynamic Network Models Coupled with Epidemics. In: Masuda, N., Holme, P. (eds) Temporal Network Epidemiology. Theoretical Biology. Springer, Singapore. https://doi.org/10.1007/978-981-10-5287-3_12

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