Skip to main content
SpringerLink
Log in

Homogeneous Dynamical Systems and the Age-Structured SIR Model with Proportionate Mixing Incidence

  • Conference paper
Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 55))

Abstract

The simplest model of a population is Malthus model. It states that the rate of change of the number of individuals in a population is proportional to the number of individuals so that populations modeled by this law grow or decay exponentially. Malthus model describes an unstructured population, that is a population in which individuals are not distinguishable. Models for structured populations may also display an overall exponential behavior. This is the case with the LotkaMcKendrick model for age-structured populations, where the so called persistent solutions occur, namely solutions that can be expressed as a product of a function of age (the stable distribution) and an exponentially growing function of time. The Lotka-McKendrick model has played an important role in demography, due to its simplicity and relatively good fit to data on a short time scale. Consequently, it remains the preferred theoretical paradigm in the study of human demography.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Busenberg and M. IannelliSeparable models in age-dependent population dynamics.J. Math. Biol. 22 (1985), 145–173.

    Article  MathSciNet  MATH  Google Scholar 

  2. M.E. Curtin and R.C. MacCamyNonlinear age-dependent population dynamics. Archive for Rational Mechanics and Analysis 54 (1974), 281–300.

    MathSciNet  Google Scholar 

  3. M.E. Curtin and R.C. MacCamySome simple models for nonlinear age-dependent population dynamics. Math. Biosci. 43 (1979), 199–211.

    Article  MathSciNet  Google Scholar 

  4. S. Busenberg, C. Castillo-Chavez, AGeneral solution of the problem of mixing of subpopulations and its application to Risk and age-structured epidemic models for the spread of AIDS.IMA J. Math. Appl. Med. Biol. 8 (1991), 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  5. Ph. Clément, H. Heijmans, S. Angenent, C. van Duijn, B. de PagterOne parameter semigroups.North-Holland, Amsterdam, New York, Oxford, Tokio, 1987.

    Google Scholar 

  6. W. FellerOn the integral equation of renewal theory.Ann. Math. Stat. 12 (1941), 243–267.

    Article  MathSciNet  Google Scholar 

  7. A. FredricksonA Mathematical theory of age structure in sexual populations: random mating and monogamous marriage models.Math. Biosci. 10 (1971), 117–143.

    Article  Google Scholar 

  8. K. Hadeler, R. Waldstätter.A. Wörz-Busekros, Models for pair formation in bisexual populations.J. Math. Biol. 26 (1988), 635–649.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. HadelerPair formation with maturation period.J. Math. Biol. 32 (1993), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. HadelerPeriodic solutions of homogeneous equation.J. Diff. Eq. 95 (1992), 183–202.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Hadeler, J. MüllerVaccination in age structured population I: The reproduction number. Epidemic models: Their structure and relation to data (V. Isham and G. Medley, eds.), Cambridge University Press, Cambridge, UK, 1996.

    Google Scholar 

  12. K. Hadeler, J. MüllerVaccination in age structured population II:Optimal vaccination strategiesEpidemic models: Their structure and relation to data (V. IshamandG. Medley, eds.),Cambridge University Press, Cambridge, UK, 1996.

    Google Scholar 

  13. H. HethcoteThe Mathematics of infectious diseases. SIAM Rev. 42 no. 4 (2000), 599–653.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. HoppensteadtMathematical Theories of Populations: Demographics Genetics and Epidemics. Society for Industrial and Applied Mathematics, Philadelphia, 1975.

    Book  Google Scholar 

  15. M. IannelliMathematical Theory of Age-Structured Population Dynamics.Applied Mathematics Monographs, Comitato Nazionale per le Scienze Matematiche, Consiglio Nazionale delle Ricerche (C.N.R.), Vol. 7, Giardini, Pisa, 1995.

    Google Scholar 

  16. M. Iannelli, M.-Y. Kim, E.-J. ParkAsymptotic behavior for an S-I-S epidemic model and its approximation.Nonlin.Anal. 35 (1999), 797–814.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Iannelli, M. Martcheva, ASemigroup approach to the wellposedness of the two-sex population model.Dynam. Sys. Appl. 5 (1997), 353–370.

    MathSciNet  Google Scholar 

  18. H. InabaThreshold and stability results for an age-structured epidemic model.J. Math. Biol. 28 (1990), 411–434.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. MartchevaExponential growth in age-structured two-sex populations.Math. Biosci. 157 (1999), 1–22.

    Article  MathSciNet  Google Scholar 

  20. M. Martcheva, F. Milner, Atwo-sex age-structured populations model: wellposedness.Math. Popul. Studies 7(2) (1999), 111–129.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. MüllerOptimal vaccination patterns in age-structured populations. SIAM J. Appl. Math. 59(1) (1998), 222–241.

    Google Scholar 

  22. J. MüllerOptimal vaccination patterns in age-structured populations: endemic case. Proceedings of the Conference on Dynamical Systems in Biology and Medicine (Veszprém, 1997). Math. Comput. Modelling 31 no. 4–5 (2000), 149–160.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. PrüssEquilibrium solutions of age specific population dynamics of several species. J. Math. Biol. 11 (1981), 65–84.

    Google Scholar 

  24. J. PrüssOn the qualitative behaviour of populations with age-specific interactions.Comput. Math. Appl. 9 (1983), 327–339.

    Google Scholar 

  25. J. Prüss, W. SchappacherPersistent age-distributions for a pair-formation model. J. Math. Biol. 33 (1994), 17–33.

    Google Scholar 

  26. G. WebbAsynchronous exponential growth in differential equations with homogeneous nonlinearities. In: Differential equations models in Banach spaces, Bologna, 1991 (Lecture Notes inPure and Applied Mathematics, 148)Dekker, New York, 1993.

    Google Scholar 

  27. G. WebbTheory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York, 1985.

    Google Scholar 

  28. A. TaylorIntroduction to functional analysis.John Wiley & Sons, New York, 1958.

    MATH  Google Scholar 

  29. H. ThiemeStability change of the endemic equilibrium in age-structured models for the spread of S - I - R type infectious diseases.In: Differential equations models in Biology, Epidemiology and Ecology (Lecture notes in Biomathematics 92), Springer-Verlag, New York, 1991.

    Google Scholar 

  30. K. YosidaFunctional Analysis.(second edition) Springer-Verlag, Berlin, Heidelberg, New York, 1968.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, I-38050, Povo (Trento), Italy

    Mimmo Iannelli

  2. Department of Mathematics, Polytechnic University, Brooklyn, 11201, NY, USA

    Maia Martcheva

Authors
  1. Mimmo Iannelli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maia Martcheva
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Trento, via Sommarive 14, 38050, Povo, Trento, Italy

    Mimmo Iannelli

  2. Institute of Mathematics and Informatics, University of Mons-Hainaut, 20, place du Parc, 7000, Mons, Belgium

    Günter Lumer

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Basel AG

About this paper

Cite this paper

Iannelli, M., Martcheva, M. (2003). Homogeneous Dynamical Systems and the Age-Structured SIR Model with Proportionate Mixing Incidence. In: Iannelli, M., Lumer, G. (eds) Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics. Progress in Nonlinear Differential Equations and Their Applications, vol 55. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8085-5_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8085-5_17

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9433-3

  • Online ISBN: 978-3-0348-8085-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation