Skip to main content
SpringerLink
Log in

Age Structured Models on Complex Networks

  • Chapter
  • First Online:
Age Structured Epidemic Modeling

Part of the book series: Interdisciplinary Applied Mathematics ((IAM,volume 52))

  • 719 Accesses

Abstract

Human behaviors affect epidemic spread. One of the main components of the disease transmission is the contact rate which in all models so far has been assumed constant or varying by age (age-since-infection) only. However, the contact rate is not constant from individual to individual; in particular some individuals have high contact rate while others may have much lower. This is particularly the case in sexually transmitted diseases but it can be observed in many others. Heterogeneity is produced due to individuals heterogeneous mixing. All the individuals and contacts generate a network. How one network structure (topology) affects the disease transmission has become a hot topic in recent years. This chapter is based on the individual contact behavior to investigate the disease transmission. In order to understand the network structure, we fist introduce some basic knowledge about networks.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 59.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. V. Andreasen, J. Lin, S. Levin, The dynamics of cocirculation influenza strains conferring partial cross-immunity. J. Math. Biol. 35, 825–842 (1997)

    Article  MathSciNet  Google Scholar 

  2. A. Barabasi, R. Albert, Emergence of scaling in random networks. Science 286, 509–512 (1999)

    Article  MathSciNet  Google Scholar 

  3. C.J. Browne, S.S. Pilyugin, Global analysis of age-structured within-host virus model. Discrete Cont. Dyn. Syst. Ser. B 18, 1999–2017 (2013)

    MathSciNet  MATH  Google Scholar 

  4. C. Castillo-Chavez, H.W. Hethcote, V. Andreasen, S.A. Levin, W.M. Liu, Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 27, 233–258 (1989)

    Article  MathSciNet  Google Scholar 

  5. C. Castillo-Chavez, W. Huang, J. Li, Competitive exclusion in gonorrhea models and other sexually transmitted diseases. SIAM J. Appl. Math. 56, 494–508 (1996)

    Article  MathSciNet  Google Scholar 

  6. Y. Chen, J. Yang, F. Zhang, Global stability of an SIRS model with infection age. Math. Biosci. Eng. 11, 449–469 (2014)

    Article  MathSciNet  Google Scholar 

  7. K.T. Eames, M.J. Keeling, Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases. Proc. Nat. Acad. Sci. 99, 13330–13335 (2002)

    Article  Google Scholar 

  8. P. Erdös, A. Rényi, On random graphs. Publ. Math. Debrecen 9, 290–297 (1959)

    MATH  Google Scholar 

  9. J. Gómez-Gardeñes, V. Latora, Y. Moreno, E. Profumo, Spreading of sexually transmitted diseases in heterosexual populations. Proc. Nat. Acad. Sci. 105, 1399–1404 (2008)

    Article  Google Scholar 

  10. M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics (Giardini, Pisa, 1995)

    Google Scholar 

  11. M. Iannelli, M. Martcheva, X.Z. Li, Strain replacement in an epidemic model with superinfection and perfect vaccination. Math. Biosci. 195, 23–46 (2005)

    Article  MathSciNet  Google Scholar 

  12. J. Zhang, Z. Jin, The analysis of an epidemic model on networks. Appl. Math. Comput. 217, 7053–7064 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Z. Jin, G. Sun, H. Zhu, Epidemic models for complex networks with demographics. Math. Biosci. Eng. 11, 1295–1317 (2014)

    Article  MathSciNet  Google Scholar 

  14. M.Y. Li, Z. Shuai, Global-stability problem for coupled systems of differential equations on networks. J. Diff. Eqs. 248, 1–20 (2010)

    Article  MathSciNet  Google Scholar 

  15. P. Magal, C. McCluskey, Two-group infection age model including and application to nosocomial infection. SIAM J. App. Math. 73, 1058–1095 (2013)

    Article  MathSciNet  Google Scholar 

  16. P. Magal, C.C. McCluskey, G.F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model. Appl. Anal. 89, 1109–1140 (2010)

    Article  MathSciNet  Google Scholar 

  17. M. Martcheva, S.S. Pilyugin, The role of coinfection in multidisease dynamics. SIAM J. Appl. Math. 66, 843–872 (2006)

    Article  MathSciNet  Google Scholar 

  18. M. Martcheva, S.S. Pilyugin, R.D. Holt, Subthreshold and superthreshold coexistence of pathogen variants: the impact of host age-structure. Math. Biosci. 207, 58–77 (2007)

    Article  MathSciNet  Google Scholar 

  19. R. May, M. Nowak, Coinfection and the evolution of parasite virulence. Proc. R. Soc. Lond. B 261, 209–215 (1995)

    Article  Google Scholar 

  20. CC. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Math. Biosci. Eng. 9, 819–841 (2012)

    Google Scholar 

  21. J.C. Miller, Mathematical models of sir disease spread with combined non-sexual and sexual transmission routes. Infect. Dis. Model. 2, 35–55 (2017)

    Google Scholar 

  22. M.E. Newman, Spread of epidemic disease on networks. Phys. Rev. E 66, 016128 (2002)

    Article  MathSciNet  Google Scholar 

  23. O. Ore, R.Ore, Theory of Graphes, vol. 38 (AMS, Providence, 1962)

    Google Scholar 

  24. R. Pastor-Satorras, A. Vespignani, Epidemic dynamics and endemic states in complex networks. Phys. Rev. E. 63, 056109 (2001)

    Article  Google Scholar 

  25. R. Pastor-Satorras, A. Vespignani, Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86, 3200–3203 (2001)

    Article  Google Scholar 

  26. D.L. Qian, X.Z. Li, M. Ghosh, Coexistence of the strains induced by mutation. Int. J. Biomath. 5, 1–25 (2012)

    Article  MathSciNet  Google Scholar 

  27. Q.C. Wu, X.C. Fu, M. Yang, Epidemic thresholds in a heterogenous population with competing strains. Chin. Phys. B 20, 046401 (2011)

    Article  Google Scholar 

  28. H.L. Smith, H.R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, (American Mathematical Society, Providence, 2011)

    MATH  Google Scholar 

  29. H.R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166, 173–201 (2000)

    Article  MathSciNet  Google Scholar 

  30. L. Wang, G.Z. Dai, Global stability of virus spreading in complex heterogeneous networks. SIAM J. Appl. Math. 68, 1495–1502 (2008)

    Article  MathSciNet  Google Scholar 

  31. Y. Wang, J. Jin, Z. Yang, Z. Zhang, T. Zhou, G. Sun, Global analysis of an SIS model with an infective vector on complex networks. Nonlinear Anal. Real World Appl. 13, 543–557 (2012)

    Article  MathSciNet  Google Scholar 

  32. X. Wang, Y. Bai, J. Yang, F. Zhang, Global stability of an epidemic model for HIV-TB co-infection with infection-age. Int. J. Biomath. 7, 1450043 (2014)

    Article  MathSciNet  Google Scholar 

  33. D. Watts, S. Strogatz, Collective dynamics of small-world networks. Nature 393, 440–442 (1998)

    Article  Google Scholar 

  34. G.F. Webb, A semigroup proof of the Sharpe-Lotka theorem, in Infinite-Dimensional Systems (Retzhof, 1983). Lecture Notes in Mathematics, vol. 1076 (Springer, Berlin, 1984), pp. 254–268

    Google Scholar 

  35. G. Webb, Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and Textbooks in Pure and Applied Mathematics, vol. 89 (Marcel Dekker, New York, 1985)

    Google Scholar 

  36. Q. Wu, M. Small, H. Liu, Superinfection behaviors on scale-free networks with competing strains. J. Nonlinear Sci. 23, 113–127 (2013)

    Article  MathSciNet  Google Scholar 

  37. J. Yang, Y. Chen, F. Xu, Effect of infection age on an SIS epidemic model on complex networks. J. Math. Biol. 73, 1227–1249 (2016)

    Article  MathSciNet  Google Scholar 

  38. T. Zhou, J. Liu, W. Bai, G. Chen, B. Wang, Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity. Phys. Rev. E 74, 056109 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Information Sciences, Henan Normal University Xinxiang, Xinxiang, China

    Xue-Zhi Li

  2. Complex Systems Research Center, Shanxi University Taiyuan, Shanxi, China

    Junyuan Yang

  3. Dept of Math, Little Hall 358, University of Florida, Gainesville, FL, USA

    Maia Martcheva

Authors
  1. Xue-Zhi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junyuan Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Maia Martcheva
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Li, XZ., Yang, J., Martcheva, M. (2020). Age Structured Models on Complex Networks. In: Age Structured Epidemic Modeling. Interdisciplinary Applied Mathematics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-030-42496-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation