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Journal of Systems Science and Complexity

, Volume 31, Issue 5, pp 1103–1127 | Cite as

Low-Dimensional SIR Epidemic Models with Demographics on Heterogeneous Networks

  • Wenjun Jing
  • Zhen Jin
  • Juping Zhang
Article
  • 5 Downloads

Abstract

To investigate the impacts of demographics on the spread of infectious diseases, a susceptible-infectious-recovered (SIR) pairwise model on heterogeneous networks is established. This model is reduced by using the probability generating function and moment closure approximations. The basic reproduction number of the low-dimensional model is derived to rely on the recruitment and death rate, the first and second moments of newcomers’ degree distribution. Sensitivity analysis for the basic reproduction number is performed, which indicates that a larger variance of newcomers’ degrees can lead to an epidemic outbreak with a smaller transmission rate, and contribute to a slight decrease of the final density of infectious nodes with a larger transmission rate. Besides, stochastic simulations indicate that the low-dimensional model based on the log-normal moment closure assumption can well capture important properties of an epidemic. And the authors discover that a larger recruitment rate can inhibit the spread of disease.

Keywords

Complex networks demographic process moment closure approximation probability generating function. 

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References

  1. [1]
    Anderson R M, Jackson H C, May R M, et al., Population dynamics of fox rabies in Europe, Nature, 1981, 289(5800): 765–771.CrossRefGoogle Scholar
  2. [2]
    Zhang J, Jin Z, Sun G, et al., Analysis of rabies in China: Transmission dynamics and control, PLoS One, 2011, 6(7): e20891.CrossRefGoogle Scholar
  3. [3]
    Zhang J, Jin Z, Sun G, et al., Determination of original infection source of H7N9 avian influenza by dynamical model, Scientific Reports, 2014, 4(6183): 4846.Google Scholar
  4. [4]
    Ling M, Sun G, Wu Y, et al., Transmission dynamics of a multi-group brucellosis model with mixed cross infection in public farm, Applied Mathematics and Computation, 2014, 237(5): 582–594.MathSciNetMATHGoogle Scholar
  5. [5]
    Ling M, Jin Z, Sun G, et al., Modeling direct and indirect disease transmission using multi-group model, Journal of Mathematical Analysis and Applications, 2017, 446(2): 1292–1309.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    Pastor-Satorras R and Vespignani A, Epidemic spreading in scale-free networks, Physical Review Letters, 2001, 86(14): 3200–3203.CrossRefGoogle Scholar
  7. [7]
    Newman M E J, Strogatz S H, and Watts D J, Random graphs with arbitrary degree distributions and their applications, Physical Review E, 2001, 64(2): 026118.CrossRefGoogle Scholar
  8. [8]
    Pastor-Satorras R and Vespignani A, Epidemic dynamics and endemic states in complex networks, Physical Review E, 2001, 63(6): 066117.CrossRefGoogle Scholar
  9. [9]
    May R M and Lloyd A L, Infection dynamics on scale-free networks, Physical Review E, 2001, 64(6): 066112.CrossRefGoogle Scholar
  10. [10]
    House T and Keeling M J, Insights from unifying modern approximations to infections on networks, Journal of the Royal Society Interface, 2011, 8(54): 67–73.CrossRefGoogle Scholar
  11. [11]
    Miller J C, Spread of infectious disease through clustered populations, Journal of the Royal Society Interface, 2009, 6(41): 1121–1134.CrossRefGoogle Scholar
  12. [12]
    Molina C and Stone L, Modelling the spread of diseases in clustered networks, Journal of Theoretical Biology, 2012, 315: 110–118.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Volz E M, Dynamics of infectious disease in clustered networks with arbitrary degree distributions, Eprint Arxiv, 2010.Google Scholar
  14. [14]
    Miller J C, Percolation and epidemics in random clustered networks, Physical Review E, 2009, 80(2): 020901.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Lindquist J, Ma J, van den Driessche P, et al., Effective degree network disease models, Journal of Mathematical Biology, 2011, 62(2): 143–164.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Volz E M, SIR dynamics in random networks with heterogeneous connectivity, Journal of Mathematical Biology, 2008, 56(3): 293–310.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Volz E M and Meyers L A, Susceptible-infected-recovered epidemics in dynamic contact networks, Proceedings of the Royal Society, 2007, 274(1628): 2925–2933.CrossRefGoogle Scholar
  18. [18]
    Miller J C, A note on a paper by Erik Volz: SIR dynamics in random networks, Journal of Mathematical Biology, 2011, 62(3): 349–358.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Kamp C, Untangling the interplay between epidemic spread and transmission network dynamics, PLoS Computational Biology, 2010, 6(11): e1000984.MathSciNetCrossRefGoogle Scholar
  20. [20]
    Kamp C, Demographic and behavioural change during epidemics, Procedia Computer Science, 2010, 1(1): 2253–2259.CrossRefGoogle Scholar
  21. [21]
    Jin Z, Sun G, and Zhu H, Epidemic models for complex networks with demographics, Mathematical Biosciences and Engineering, 2014, 11(6): 1295–1317.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Piccardi C, Colombo A, and Casagrandi R, Connectivity interplays with age in shaping contagion over networks with vital dynamics, Physical Review E, 2015, 91(2): 022809.CrossRefGoogle Scholar
  23. [23]
    Erdös P and Rényi A, On random graphs I, Publicationes Mathematicae, 1959, 6: 290–297.MathSciNetMATHGoogle Scholar
  24. [24]
    Eames K T D and Keeling M J, Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases, Proceedings of the National Academy of Sciences of the United States of America, 2002, 99(20): 13330–13335.CrossRefGoogle Scholar
  25. [25]
    Simon P L and Kiss I Z, Super compact pairwise model for SIS epidemic on heterogeneous networks, Journal of Complex Networks, 2016, 4(2): 187–200.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Bauch C T, The spread of infectious diseases in spatially structured populations: An invasory pair approximation, Mathematical Biosciences, 2005, 198(2): 217–237.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Trapman P, Reproduction numbers for epidemics on networks using pair approximation, Mathematical Biosciences, 2007, 210(2): 464–489.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Keeling M J, Multiplicative moments and measures of persistence in ecology, Journal of Theoretical Biology, 2000, 205(2): 269–281.CrossRefGoogle Scholar
  29. [29]
    Ekanayake A J and Allen L J S, Comparison of markov chain and stochastic differential equation population models under higher-order moment closure approximations, Stochastic Analysis and Applications, 2010, 28(6): 907–927.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    Miller J C and Kiss I Z, Epidemic spread in networks: Existing methods and current challenges, Mathematical Modelling of Natural Phenomena, 2014, 9(2): 4–42.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    N˚asell I, An extension of the moment closure method, Theoretical Population Biology, 2003, 64(2): 233–239.MathSciNetCrossRefGoogle Scholar
  32. [32]
    Whittle P, On the use of the normal approximation in the treatment of stochastic processes, Journal of the Royal Statistical Society, 1957, 19(2): 268–281.MathSciNetMATHGoogle Scholar
  33. [33]
    Goutsias J and Jenkinson G, Markovian dynamics on complex reaction networks, Physics Reports, 2013, 529(2): 199–264.MathSciNetCrossRefGoogle Scholar
  34. [34]
    Hiebeler D, Moment equations and dynamics of a household SIS epidemiological model, Bulletin of Mathematical Biology, 2006, 68(6): 1315–1333.MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    Krishnarajah I, Cook A, Marion G, et al., Novel moment closure approximations in stochastic epidemics, Bulletin of Mathematical Biology, 2005, 67(4): 855–873.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    Anderson R M and May R M, Infectious Diseases of Humans, Oxford Science, Oxford, 1991.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Complex Systems Research CenterShanxi UniversityTaiyuanChina

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