Structured Population Models for HIV Infection Pair Formation and Non-constant Infectivity

  • K. P. Hadeler


The spread of a sexually transmitted disease with long incubation period such as HIV is modeled in a population which is structured by age, sex, and duration of infection. Since empirical evidence shows that in the HIV situation infectivity varies considerably from the moment of infection to the onset of AIDS, the effects of non-constant infectivity are studied in detail. A characteristic eigenvalue problem is derived which determines stability or instability of the uninfected state of the population. For the case of constant population size the basic reproduction number is calculated. The dependence of this number on the infectivity is studied by analytical and numerical methods. The results indicate that non-constant infectivity leads to a lower basic reproduction number when compared to a constant infectivity obtained by appropriate averaging.


Pair Formation Basic Reproduction Number Single Male Ordinary Differential Equation Model Constant Population Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arbogast, T., Milner F.A. (1989) A finite-difference method for a two-sex model of population dynamics. SIAM J. Numer. Anal. 26, 1474–1486Google Scholar
  2. Busenberg, S., Castillo-Chavez, C. (1989) Interaction, pair formation and force of infection in sexually transmitted diseases. In: C. Castillo-Chavez ( 1989 ), 289–300Google Scholar
  3. Busenberg, S., Hadeler, K.P. (1990) Demography and epidemics. Math. Biosc. 101, 63–74Google Scholar
  4. Castillo-Chavez, C. (Ed.) (1989) Mathematical and Statistical Approaches to AIDS Epidemiology. Lecture Notes in Biomath. 83, Springer Verlag Castillo-Chavez, C., Blythe, S.P. (1989) Mixing framework for social/sexual behavior. In: Castillo-Chavez ( 1989 ), 275–285Google Scholar
  5. Castillo-Chavez, C., Cooke, K., Huang, Levin, S.A. (1989) Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus. Applied Math. Letters 2, 327–331Google Scholar
  6. Dietz, K. (1987) Epidemiological models for sexually transmitted infections. Proc. First World Congress Bernoulli Soc., Tashkent 1986, Vol. II, 539–542, VNU Science Press, UtrechtGoogle Scholar
  7. Dietz, K. (1988) On the transmission dynamics of HIV. Math.Biosc. 90, 397–414CrossRefGoogle Scholar
  8. Dietz, K., Hadeler, K.P. (1988) Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1–25Google Scholar
  9. Feller, W. (1941) On the integral equation of renewal theory. Ann. Math. Stat. 12, 243–267Google Scholar
  10. Hadeler, K.P. (1989a) Pair formation in age structured populations. Proc. Workshop on Selected Topics in Biomathematics (eds. A.Kurzhanskij, K. Sigmund) Laxenburg, Austria 1987, Acta Appl.Math. 14, 91–102Google Scholar
  11. Hadeler, K.P. (1989b) Modeling AIDS in structured populations. Bull. Int. Stat. Inst. 53, Book 1, 83–99Google Scholar
  12. Hadeler, K.P. (1990) Homogeneous delay equations and models for pair formation. Preprint Center for Dynamical Systems, Georgia Institute of Technology, J.Math.Biol, to appear.Google Scholar
  13. Hadeler, K.P. (1992) Periodic solutions of homogeneous equations. J.Diff. Equ. 95, 183–202Google Scholar
  14. Hadeler, K.P., Ngoma, K. (1990) Homogeneous models for sexually transmitted diseases. Rocky Mtn. J. Math. 20, 967–986Google Scholar
  15. Hadeler, K.P., Waldstätter, R., Wörz-Busekros, A. (1988) Models for pair formation in bisexual populations.Google Scholar
  16. J. Math. Biol. 26, 635–649 Hoppensteadt, F. (1975) Mathematical Theories of Populations: Demographics, Genetics and Epidemics. Regional Conference Series in Applied Mathematics 20, SIAM, PhiladelphiaGoogle Scholar
  17. Jacquez, J.A., Simon, C.P., Koopman, J., Sattenspiel, L., Perry, T. (1988) Modeling and analysing HIV transmission: The effect of contact patterns. Math. Biosc. 92, 119–199Google Scholar
  18. Jacquez, J.A., Simon, C.P., Koopman, J., Structured Mixing: Heterogeneous mixing by the definition of activity groups. In: C. Castillo-Chavez (1989), 301–315Google Scholar
  19. Kendall, D.G. (1949) Stochastic processes and population growth. J. Roy. Statist. Soc. Ser.B., 11, 230–264Google Scholar
  20. Keyfitz, N. (1985) Applied Mathematical Demography, 2nd ed., Springer VerlagGoogle Scholar
  21. Kuczynski, R.R. (1932) Fertility and Reproduction. p. 36–38, New York, Falcon PressGoogle Scholar
  22. Lotka, A.J. (1922) The stability of the normal age distribution. Proc. Nat. Acad. Sci. 8, 339–345Google Scholar
  23. McKendrick, A.G. (1926) Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98–130 (1926)Google Scholar
  24. Ng, T.W., Anderson, R.M. (1989) A model for the demographic impact of AIDS in devoloping countries: Age-dependent choice of sexual partners. Bull. Int. Stat. Inst. 53, Book 4, 425–448Google Scholar
  25. Parlett, B. (1972) Can there be a marriage function? In: T.N.T.Greville ( Ed.) Population Dynamics, Academic PressGoogle Scholar
  26. Sattenspiel, L., Simon, C.P. (1988) The spread and persistence of infectious diseases in structured populations. Math. Biosc. 90, 341–366Google Scholar
  27. Sharpe, F.R., Lotka, A.J. (1911) A problem in age distribution. Phil.Mag. 21, 435–438CrossRefGoogle Scholar
  28. Thieme, H.R., Castillo-Chavez, C. (1989) On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic. In: Castillo-Chavez ( 1989 ), 157–176Google Scholar
  29. Waldstätter, R. (1989) Pair formation in sexually transmitted diseases. In: C. Castillo-Chavez ( 1989 ), 260–274Google Scholar
  30. Waldstätter, R. (1990) Models for Pair formation with Applications to Demography and Epidemiology. Dissertation Universität Tübingen 1990 Webb, G.F. ( 1985 ) Theory of Nonlinear Age-dependent Population Dynamics. M. DekkerGoogle Scholar
  31. Yellin, J., Samuelson, P.A. (1974) A dynamical model for human population. Proc.Nat. Acad.Sci. USA 71, No. 7, 2813–2817PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • K. P. Hadeler
    • 1
  1. 1.Lehrstuhl für BiomathematikUniversität TübingenTübingenDeutschland

Personalised recommendations