Abstract
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.
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References
Arbogast, T., Milner F.A. (1989) A finite-difference method for a two-sex model of population dynamics. SIAM J. Numer. Anal. 26, 1474–1486
Busenberg, S., Castillo-Chavez, C. (1989) Interaction, pair formation and force of infection in sexually transmitted diseases. In: C. Castillo-Chavez ( 1989 ), 289–300
Busenberg, S., Hadeler, K.P. (1990) Demography and epidemics. Math. Biosc. 101, 63–74
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 ), 275–285
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, 327–331
Dietz, K. (1987) Epidemiological models for sexually transmitted infections. Proc. First World Congress Bernoulli Soc., Tashkent 1986, Vol. II, 539–542, VNU Science Press, Utrecht
Dietz, K. (1988) On the transmission dynamics of HIV. Math.Biosc. 90, 397–414
Dietz, K., Hadeler, K.P. (1988) Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1–25
Feller, W. (1941) On the integral equation of renewal theory. Ann. Math. Stat. 12, 243–267
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, 91–102
Hadeler, K.P. (1989b) Modeling AIDS in structured populations. Bull. Int. Stat. Inst. 53, Book 1, 83–99
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.
Hadeler, K.P. (1992) Periodic solutions of homogeneous equations. J.Diff. Equ. 95, 183–202
Hadeler, K.P., Ngoma, K. (1990) Homogeneous models for sexually transmitted diseases. Rocky Mtn. J. Math. 20, 967–986
Hadeler, K.P., Waldstätter, R., Wörz-Busekros, A. (1988) Models for pair formation in bisexual populations.
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, Philadelphia
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, 119–199
Jacquez, J.A., Simon, C.P., Koopman, J., Structured Mixing: Heterogeneous mixing by the definition of activity groups. In: C. Castillo-Chavez (1989), 301–315
Kendall, D.G. (1949) Stochastic processes and population growth. J. Roy. Statist. Soc. Ser.B., 11, 230–264
Keyfitz, N. (1985) Applied Mathematical Demography, 2nd ed., Springer Verlag
Kuczynski, R.R. (1932) Fertility and Reproduction. p. 36–38, New York, Falcon Press
Lotka, A.J. (1922) The stability of the normal age distribution. Proc. Nat. Acad. Sci. 8, 339–345
McKendrick, A.G. (1926) Applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98–130 (1926)
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, 425–448
Parlett, B. (1972) Can there be a marriage function? In: T.N.T.Greville ( Ed.) Population Dynamics, Academic Press
Sattenspiel, L., Simon, C.P. (1988) The spread and persistence of infectious diseases in structured populations. Math. Biosc. 90, 341–366
Sharpe, F.R., Lotka, A.J. (1911) A problem in age distribution. Phil.Mag. 21, 435–438
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 ), 157–176
Waldstätter, R. (1989) Pair formation in sexually transmitted diseases. In: C. Castillo-Chavez ( 1989 ), 260–274
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. Dekker
Yellin, J., Samuelson, P.A. (1974) A dynamical model for human population. Proc.Nat. Acad.Sci. USA 71, No. 7, 2813–2817
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Hadeler, K.P. (1992). Structured Population Models for HIV Infection Pair Formation and Non-constant Infectivity. In: Jewell, N.P., Dietz, K., Farewell, V.T. (eds) AIDS Epidemiology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-1229-2_8
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DOI: https://doi.org/10.1007/978-1-4757-1229-2_8
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