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Disease Transmission Models with Age Structure

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Part of the book series: Texts in Applied Mathematics ((TAM,volume 69))

Abstract

Age is one of the most important characteristics in the modeling of populations and infectious diseases. Because age groups frequently mix heterogeneously it may be appropriate to include age structure in epidemiological models. While there are other aspects of heterogeneity in disease transmission models, such as behavioral and spatial heterogeneity, age structure is one of the most important aspects of heterogeneity in disease modeling.

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Change history

  • 15 December 2019

    The book was inadvertently published with the following errors in Chaps. 5 and 13. The same has now been corrected in the book.

References

  1. Anderson, R.M. & R.M. May (1991) Infectious Diseases of Humans. Oxford University Press (1991)

    Google Scholar 

  2. Anderson, R.M. & R.M. May (1979) Population biology of infectious diseases I, Nature 280: 361–367.

    Article  Google Scholar 

  3. Anderson, R.M., and R.M. May (1984) Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes, Math. Med. Biol. 1(3), 233–266.

    Article  MathSciNet  Google Scholar 

  4. Andreasen, V. (1995) Instability in an SIR-model with age dependent susceptibility, Mathematical Population Dynamics: Analysis of Heterogeneity, Vol. 1, Theory of Epidemics (O. Arino, D. Axelrod, M. Kimmel, M. Langlais, eds.), Wuerz, Winnipeg: 3–14.

    Google Scholar 

  5. Blythe, S.P. and C. Castillo-Chavez (1989) Like-with-like preference and sexual mixing models, Math. Biosci. 96: 221–238.

    Article  Google Scholar 

  6. Blythe, S.P., C. Castillo-Chavez, J. Palmer & M. Cheng (1991) Towards a unified theory of mixing and pair formation, Math. Biosc. 107: 379–405.

    Article  Google Scholar 

  7. Busenberg, S. and C. Castillo-Chavez (1989) Interaction, pair formation and force of infection terms in sexually transmitted diseases, in: Mathematical and Statistical Approaches to AIDS Epidemiology, (C. Castillo-Chavez, ed.), Lecture Notes in Biomathematics, Vol. 83, Springer-Verlag, New York, (1989), pp. 289–300.

    Google Scholar 

  8. Busenberg, S. and C. Castillo-Chavez (1991) A general 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: 1–29.

    Article  MathSciNet  Google Scholar 

  9. Capasso, V. (1993) Mathematical Structures of Epidemic Systems, Lect. Notes in Biomath. 83, Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  10. Castillo-Chavez, C. and Z. Feng (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies, Math. Biosci. 151(2), 135–154.

    MATH  Google Scholar 

  11. Castillo-Chavez, C., H.W. Hethcote, V. Andreasen, S.A. Levin, and W.M. Liu (1989) Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. Math. Biol. 27(3), 233–258.

    Article  MathSciNet  Google Scholar 

  12. Castillo-Chavez, C. and S. P. Blythe (1989) Mixing Framework for Social/Sexual Behavior, in: Mathematical and Statistical Approaches to AIDS Epidemiology, (C. Castillo-Chavez, ed.), Lecture Notes in Biomathematics 83, Springer-Verlag, New York, pp. 275–288.

    Google Scholar 

  13. Cha, Y., M. Ianelli, & F. Milner (1998) Existence and uniqueness of endemic states for the age-structured S-I-R epidemic model, Math. Biosc. 150: 177–190.

    Article  MathSciNet  Google Scholar 

  14. Chitnis, N., J.M. Hyman, and J.M. Cushing (2008) Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol 70:1272–1296.

    Article  MathSciNet  Google Scholar 

  15. Dietz, K. (1975) Transmission and control of arbovirus diseases, Epidemiology (D. Ludwig, K. Cooke, eds.), SIAM, Philadelphia: 104–121.

    Google Scholar 

  16. Dietz, K., and D. Schenzle (1985) Proportionate mixing models for age-dependent infection transmission, J. Math. Biol. 22(1), 117–120.

    Article  MathSciNet  Google Scholar 

  17. Garret-Jones, C. and G.R. Shidrawi (1969) Malaria vectorial capacity of a population of Anopheles gambiae. Bull. Wld Hlth Org. 40:531–545.

    Google Scholar 

  18. Gurtin, M.L. and R.C. MacCamy (1974) Nonlinear age dependent population dynamics, Arch. Rat. Mech. Analysis, 54: 281–300.

    Article  Google Scholar 

  19. Hadeler, K.P., R. Waldstatter, and A. Worz-Busekros (1988) Models for pair-formation in bisexual populations, J. Math. Biol. 26: 635–649.

    Article  MathSciNet  Google Scholar 

  20. Hethcote, H.W. (2000) The mathematics of infectious diseases, SIAM Review 42: 599–653.

    Article  MathSciNet  Google Scholar 

  21. Hoppensteadt, F.C. (1975) Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM, Philadelphia.

    Google Scholar 

  22. Iannelli, M. (1995) Mathematical Theory of Age-Structured Population Dynamics, Appl. Mathematical Monographs, C.N.R..

    Google Scholar 

  23. Kot, M. (2001) Elements of Mathematical Ecology. Cambridge University Press.

    Book  Google Scholar 

  24. May, R.M. (1986) Population biology of macroparasitic infections, in Mathematical Ecology; An Introduction, (Hallam, T.G., Levin, S.A. eds.), Biomathematics 18, Springer-Verlag, Berlin-Heidelberg-New York: 405–442.

    Google Scholar 

  25. May, R.M. & Anderson, R.M. (1979) Population biology of infectious diseases II, Nature 280: 455–461.

    Article  Google Scholar 

  26. McKendrick, A.G. (1926) Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc. 44: 98–130.

    Article  Google Scholar 

  27. Müller, J. (1998) Optimal vaccination patterns in age structured populations, SIAM J. Appl. Math. 59: 222–241.

    MATH  Google Scholar 

  28. Murray, J.D. (2002) Mathematical Biology, Vol. I, Springer-Verlag, Berlin-Heidelberg-New York.

    Book  Google Scholar 

  29. Paaijmans, K. Weather, water and malaria mosquito larvae, 2008.

    Google Scholar 

  30. Park, T. (1984) Age-Dependence in Epidemic Models of Vector-Borne Infections, Ph.D. Dissertation, University of Alabama in Huntsville, 2004.

    Google Scholar 

  31. Sharpe, F.R., and A.J. Lotka (1911) A problem in age-distribution. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(124): 435–438.

    Article  Google Scholar 

  32. Thieme, H.R. (2003) Mathematics in Population Biology, Princeton University Press, Princeton, N.J.

    Book  Google Scholar 

  33. United nations, department of economic and social affairs, population division, population estimates and projections sections. Retrieved January, 2013 from http://esa.un.org/wpp/index.htm.

  34. Waltman, P. (1974) Deterministic Threshold Models in the Theory of Epidemics, Lect. Notes in Biomath. 1, Springer-Verlag, Berlin-Heidelberg-New York.

    Book  Google Scholar 

  35. Webb, G.F. (1985) Theory of Nonlinear Age-dependent Population Dynamics, Marcel Dekker, New York.

    MATH  Google Scholar 

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Brauer, F., Castillo-Chavez, C., Feng, Z. (2019). Disease Transmission Models with Age Structure. In: Mathematical Models in Epidemiology. Texts in Applied Mathematics, vol 69. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9828-9_13

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