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On the Solution of the Two-Sex Mixing Problem

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Differential Equations Models in Biology, Epidemiology and Ecology

Part of the book series: Lecture Notes in Biomathematics ((LNBM,volume 92))

Abstract

In this paper we describe an axiomatic framework that allows for the general incorporation of sexual structure into two-sex pair-formation models for sexuallytransmitted diseases. A representation theorem describing all solutions to this mixing framework as perturbations of particular solutions is proved. Two-sex age structured demographic and age-structured epidemiological models that make use of our framework, and are therefore capable of describing the dynamics of individuals and/or pairs of individuals, are formulated.

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References

  1. Anderson, R. M., Blythe, S. P., Gupta, S., and Konings, E. (1989): The transmission dynamics of the Human Immunodeficiency Virus Type I in the male homosexual community in the United Kindom: the influence of changes in sexual behavior. Phil. Trans. Roy. Soc. London. B 325, 145–198.

    Google Scholar 

  2. Blythe, S. P. (1991): Heterogeneous sexual mixing in populations with arbitrarily connected multiple groups. To appear in Math. Pop. Studies.

    Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Blythe, S.P. and Castillo-Chavez, C. (1991a): The one-sex mixing problems: a choice of solutions? To appear in J. Math. Biology.

    Google Scholar 

  5. Blythe, S.P. and Castillo-Chavez, C.(1990b): Like-with-like mixing and sexually transmitted disease epidemics in one-sex populations. Biometrics Unit Technical Report # BU-1078-M, Cornell University.

    Google Scholar 

  6. Blythe, S. P., Castillo-Chavez, C, and Casella, G. (1991): Empirical methods for the estimation of the mixing probabilities for socially structured populations from a single survey sample. To appear in Math. Pop. Studies.

    Google Scholar 

  7. Blythe, S.P., Castillo-Chavez, C, J.S. Palmer, Cheng, M. (1991): Towards a unified theory of mixing and pair formation. To appear in Math. Biosci.

    Google Scholar 

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

    Chapter  Google Scholar 

  9. Busenberg, S. and Castillo-Chavez C. (1991): On the role of preference in the solution of the mixing problem, and its application to risk-and age-structured epidemic models. To appear in IMA J. of Math Applic. to Med. and Biol.

    Google Scholar 

  10. Castillo-Chavez, C. (1989): Review of recent models of HIV/AIDS transmission. In (S. A. Levin, T. G. Hallam, and L. J. Gross, eds.,) Applied Mathematical Ecology, Biomathematics 18, Springer-Verlag, Berlin-Heidelberg-New York, 253–262.

    Chapter  Google Scholar 

  11. Castillo-Chavez, C. and Blythe, S. P. (1989): Mixing framework for social/sexual behavior. In (Castillo-Chavez, ed.) Mathematical and Statistical Approaches to AIDS Epidemiology. Lecture Notes in Biomathematics 83, Springer-Verlag, Berlin-Heidelberg-New York, 275–288.

    Chapter  Google Scholar 

  12. Castillo-Chavez, C. and Blythe, S. P. (1990): A “test-bed” procedure for evaluating one-sex mixing frameworks. Manuscript.

    Google Scholar 

  13. Castillo-Chavez, C, Busenberg, S., Gerow, K. (1991): Pair formation in structured populations. To appear in Differential Equations with Applications to Biology, Physics and Engineering, (J. Golstein, F. Kappel, W. Schappacher, Eds.), Marcel Dekker, New York.

    Google Scholar 

  14. Cooke, K. L. and Yorke, J. A. (1973): Some equations modelling growth processes and gonorrhea epidemics. Math. Biosci., 58, 93–109.

    MathSciNet  Google Scholar 

  15. Dietz, K. (1988): On the transmission dynamics of HIV. Math Biosci. 90, 397–414.

    Article  MATH  MathSciNet  Google Scholar 

  16. Dietz, K. and Hadeler, K. P. (1988): Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1–25.

    Article  MATH  MathSciNet  Google Scholar 

  17. Fredrickson, A. G. (1971): A mathematical theory of age structure in sexual populations: Random mating and monogamous marriage models. Math. Biosci. 20, 117–143.

    Article  Google Scholar 

  18. Gupta, S., Anderson, R.M., May, R.M. (1989): Network of sexual contacts: implications for the pattern fo spread of HIV. AIDS; 3: 1–11.

    Article  Google Scholar 

  19. Hadeler, K. P. (1989a): Pair formation in age-structured populations. Acta Applicandae Mathematicae 14, 91–102.

    Article  MATH  MathSciNet  Google Scholar 

  20. Hadeler, K. P. (1989b): Modeling AIDS in structured populations. Manuscript.

    Google Scholar 

  21. Hadeler, K. P. (1990): Homogeneous delay equations and models for pair formation. Manuscript.

    Google Scholar 

  22. Hethcote, H. W. and Yorke, J. A. (1984): Gonorrhea transmission dynamics and control. Lecture Notes in Biomathematics 56, Springer-Verlag, Berlin-Heidelberg-New York.

    Google Scholar 

  23. Hoppensteadt, F. (1974): An age dependent epidemic model. J. Franklin Instit. 297, 325–333.

    Article  MATH  Google Scholar 

  24. Hyman, J. M. and Stanley, E. A. (1988): Using mathematical models to understand the AIDS epidemic. Math Biosci. 90, 415–473.

    Article  MATH  MathSciNet  Google Scholar 

  25. Hyman, J. M. and Stanley, E. A. (1989): The effect of social mixing patterns on the spread of AIDS. In Mathematical approaches to problems in resource management and epidemiology, (C. Castillo-Chavez, S. A. Levin, and C. Shoemaker, eds.) Lecture Notes in Biomathematics 81, Springer-Verlag, Berlin-Heidelberg-New York, 190–219.

    Google Scholar 

  26. Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L., Perry, T. (1988): Modelling and analyzing HIV transmission: the effect of contact patterns. Math Biosci. 92, 119–199.

    Article  MATH  MathSciNet  Google Scholar 

  27. Jacquez, J. A., Simon, C. P., and Koopman, J. (1989): Structured mixing: heterogeneous mixing by the definition of mixing groups. Mathematical and Statistical Approaches to AIDS epidemiology (C. Castillo-Chavez, ed.) Lecture Notes in Biomathematics 83, Springer-Verlag, Berlin-Heidelberg-New York, 301–315.

    Chapter  Google Scholar 

  28. Kendall, D. G. (1949): Stochastic processes and population growth. Roy. Statist. Soc., Ser B2, 230–264.

    MathSciNet  Google Scholar 

  29. Keyfitz, N. (1949): The mathematics of sex and marriage. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. Vol. IV: Biology and Health, 89-108.

    Google Scholar 

  30. McFarland, D. D. (1972): Comparison of alternative marriage models. In Population Dynamics, (Greville, T. N. E., ed.), Academic Press, New York, London: 89–106.

    Google Scholar 

  31. Nold, A. (1980): Heterogeneity in disease-transmission modeling. Math. Biosci. 52, 227–240.

    Article  MATH  MathSciNet  Google Scholar 

  32. Parlett, B. (1972): Can there be a marriage function? In Population Dynamics (Greville, T. N. E., ed.) Academic Press, New York, London: 107–135.

    Google Scholar 

  33. Pollard, J. H. (1973): Mathematical Models for Growth of Human Populations, Chapter 7: The two sex problem. Cambridge University Press.

    Google Scholar 

  34. Pugliese, A. (1989): Contact matrices for multipopulation epidemic models: how to build a consistent matrix close to data? Technical report UTM 338, Dipartimento di Matematica, Università degli Studi di Trento.

    Google Scholar 

  35. Ross, R. (1911): The Prevention of Malaria (2nd edition, with Addendum). John Murray, London.

    Google Scholar 

  36. Ross, R., and Hudson, H. P. (1916): An application of the theory of probabilities to the study of a priori pathometry.-Part I. Proc. R. Soc. Lond. A 93, 212–225.

    Google Scholar 

  37. Waldstätter, R. (1989): Pair formation in sexually transmitted diseases. In Mathematical and Statistical Approaches to AIDS Epidemiology, (C. Castillo-Chavez, ed.). Lecture Notes in Biomathematics, 83, Springer-Verlag, Berlin-Heidelberg-New York, 260–274.

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Biometrics Unit/Center for Applied Mathematics, Cornell University, 341 Warren Hall, Ithaca, NY, 14853-7801, USA

    C. Castillo-Chavez

  2. Department of Mathematics, Harvey Mudd College, Claremont, CA, 91711, USA

    S. Busenberg

Authors
  1. C. Castillo-Chavez
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  2. S. Busenberg
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Harvey Mudd College, Claremont, CA, 91711, USA

    Stavros Busenberg

  2. Department of Mathematics, California State University, Fullerton, CA, 92634, USA

    Mario Martelli

Additional information

The work described in this paper has been motivated by work with Kenneth Cooke. Ken has used his considerable experience in the modeling and analysis of disease transmission, and most recently in the development of models that may help our fight against AIDS. Many of the ideas discussed in this article arose out of our study of Ken’s work, our discussions with him, and our collaborative efforts with Ken over the years. We dedicate this paper to him as we celebrate his 65th birthday.

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© 1991 Springer-Verlag Berlin Heidelberg

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Castillo-Chavez, C., Busenberg, S. (1991). On the Solution of the Two-Sex Mixing Problem. In: Busenberg, S., Martelli, M. (eds) Differential Equations Models in Biology, Epidemiology and Ecology. Lecture Notes in Biomathematics, vol 92. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45692-3_7

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  • DOI: https://doi.org/10.1007/978-3-642-45692-3_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54283-4

  • Online ISBN: 978-3-642-45692-3

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