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On the Estimation Problem of Mixing/Pair Formation Matrices with Applications to Models for Sexually-Transmitted Diseases

  • Carlos Castillo-Chavez
  • Shwu-Fang Shyu
  • Gail Rubin
  • David Umbach

Abstract

A problem of considerable importance lying at the interface of social dynamics, demography, and epidemiology is determining and modeling who is mixing with whom. In this article we describe a general approach, using nonlinear mixing matrices, for modeling the process of pair-formation in heterogeneous populations. Determining who is mixing with whom is complicated by a variety of factors, including the problem of denominators, which is, in our context, equivalent to the nonexistence of closely interacting social/sexual networks. We describe the use of a mark-recapture model for estimating the sizes of the missing link, that is, the size of the population having sexual contact with a specified population and hence at risk for sexually-transmitted diseases. The need to estimate the size of the sexually-active subset before estimating the size of the population at risk introduces extra variability into the problem. An estimator of the variance of the estimated size of the population at risk that accounts for this extra variability and an expression for the bias of such an estimator have been derived. We illustrate our results with data collected from a population of university undergraduates, and make use of our axiomatic modeling approach for mixing/pair formation to compute specific mixing matrices. Complete details of this work will be published elsewhere.

Keywords

Human Immunodeficiency Virus Sexual Partner Formation Matrice Sexual Contact Transmission Dynamic 
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.

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References

  1. Anderson, R. M., May, R. M. and Medley, G. F. (1986). A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA J. Math. Med. Biol. 3, 229–263.CrossRefGoogle Scholar
  2. Anderson, R. M., Blythe, S. P., Gupta, S. and Konings, E. (1989). The transmission dynamics of the Human Immunodeficiency Virus Type 1 in the male homosexual community in the United Kingdom: the influence of changes in sexual behavior. Phil. Trans. R. Soc. Lond. B 325, 145–198.Google Scholar
  3. Bailey, N. T. J. (1951). On estimating the size of mobile populations from recapture data. Biometrika 38, 293–306.Google Scholar
  4. Blythe, S. P. and Castillo-Chavez, C. (1989). Like-with-like preference and sexual mixing models. Math. Biosci. 96, 221–238.PubMedCrossRefGoogle Scholar
  5. Blythe, S. P., Castillo-Chavez, C. and Casella, G. (1992). Empirical methods for the estimation of the mixing probabilities for socially-structured populations from a single survey sample. Mathematical Population Studies (in press).Google Scholar
  6. Blythe, S. P., Castillo-Chavez, C., Palmer, J. and Cheng, M. (1991). Towards unified theory of mixing and pair formation. Math. Biosci. 107: 379–405.PubMedCrossRefGoogle Scholar
  7. Blythe, S. P., Cooke, K. and Castillo-Chavez, C. (1991). Autonomous risk-behavior change, and non-linear incidence rate, in models of sexually transmitted diseases. Biometrics Unit Technical Report BU-1048-M, Cornell University, Ithaca, NY.Google Scholar
  8. Busenberg, S. and Castillo-Chavez, C. (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 83, 289–300. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  9. Busenberg, S. and Castillo-Chavez, C. (1991). A general solution of the problem of mixing subpopulations, and its application to risk-and age-structured epidemic models for the spread of AIDS. IMA J. of Mathematics Applied in Med. and Biol. 8, 1–29.CrossRefGoogle Scholar
  10. Castillo-Chavez, C. (1989). Review of recent models of HIV/AIDS transmission. In Applied Mathematical Ecology, S. A. Levin, T. G. Hallam and L. J. Gross (eds.), Biomathematics 18, 253–262. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  11. Castillo-Chavez, C. (ed.). (1989). Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics 83. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  12. Castillo-Chavez, C. and Blythe, S. P. (1989). Mixing framework for social/sexual behavior. In Mathematical and statistical approaches to AIDS epidemiology, C. Castillo-Chavez (ed.), Lecture Notes in Biomathematics 83, 275–288. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  13. Castillo-Chavez, C. and Busenberg, S. (1991). On the solution of the two-sex moxong problem. In Proceedings of the International Conference on Differential Equations and Applications to Biology and Population Dynamics, S. Busenberg and M. Martelli (eds.), Lecture Notes in Biomathematics 92, 80–98. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, Budapest: Springer-Verlag.Google Scholar
  14. Castillo-Chavez, C., Busenberg, S. and Gerow, K. (1991). Pair formation in structured populations. In Differential Equations with Applications in Biology, Physics and Engineering, J. Goldstein, F. Kappel and W. Schappacher (eds.), 47–65. New York: Marcel Dekker.Google Scholar
  15. Castillo-Chavez, C., Cooke, K. L., Huang, W. and Levin, S. A. (1989a). Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus. Applied Math. Letters 2, 327–331.CrossRefGoogle Scholar
  16. Castillo-Chavez, C., Cooke, K. L., Huang, W. and Levin, S. A. (1989b). On the role of long incubation periods in the dynamics of HIV/AIDS, Part 2: Multiple group models. In Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez (ed.), Lecture Notes in Biomathematics 83, 200–217. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  17. Centers for Disease Control. (1985). Self-reported behavioral change among gay and bisexual men, San Francisco. MMWR 34, 613–615.Google Scholar
  18. Crawford, C. M., Schwager, S. J. and Castillo-Chavez, C. (1990). A methodology for asking sensitive questions among college undergraduates. Biometrics Unit Tech. Report BU-1105-M Cornell University, Ithaca, New York.Google Scholar
  19. Dietz, K. (1988). On the transmission dynamics of HIV. Math. Biosci. 90, 397–414.CrossRefGoogle Scholar
  20. Dietz, K. and Hadeler, K. P. (1988). Epidemiological models for sexually transmitted diseases. J. Math. Biol. 26, 1–25.PubMedCrossRefGoogle Scholar
  21. Fredrickson, A. G. (1971). A mathematical theory of age structure in sexual populations: Random mating and monogamous marriage models. Math. Biosci. 20, 117–143.CrossRefGoogle Scholar
  22. Gupta, S., Anderson, R. M. and May, R. M. (1989). Networks of sexual contacts: implications for the pattern of spread of HIV. AIDS 3, 1–11.CrossRefGoogle Scholar
  23. Hadeler, K. P. (1989a). Pair formation in age-structured populations. Acta Applicandae Mathematicae 14, 91–102.PubMedCrossRefGoogle Scholar
  24. Hadeler, K. P. (1989b). Modeling AIDS in structured populations. 47th Session of the International Statistical Institute, Paris, August /September. Conference Proc. C1–2. 1, 83–99.Google Scholar
  25. Hadeler, K. P. and Ngoma, K. (1990). Homogeneous models for sexually transmitted diseases. Rocky Mountain Journal of Mathematics 20, 967–986.Google Scholar
  26. Hethcote, H. W. and Yorke, J. A. (1984). Gonorrhea transmission dynamics and control, Lect. Notes Biomath. 56. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  27. Hethcote, H. W. and Van Ark, J. W. (1987). Epidemiological models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs. Math. Biosci. 84, 85–111.CrossRefGoogle Scholar
  28. Hethcote, H. W., Van Ark, J. W. and Karon, J. M. (1991). A simulation model of AIDS in San Francisco, II. Simulations, therapy, and sensitivity analysis. Math, Biosci. 106, 223–247.CrossRefGoogle Scholar
  29. Iethcote, H. W. and Van Ark, J. W. (1992). Weak linkage between HIV epidemics in homosexual men and intravenous drug users (in this volume).Google Scholar
  30. Ilyman, J. M. and Stanley, E. A. (1988). Using mathematical models to understand the AIDS epidemic. Math. Biosci. 90, 415–473.CrossRefGoogle Scholar
  31. 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. A. Shoemaker (eds.), Lect. Notes Biomath. 81, 190–219. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  32. Huang, W., Cooke, K. and Castillo-Chavez, C. (1992). Stability and bifurcation for a multiple group model for the dynamics of HIV/AIDS transmission. SIAM J. of Applied Math. (in press).Google Scholar
  33. Jacquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L. and Perry, T. (1988). Modeling and analyzing HIV transmission: the effect of contact patterns. Math. Biosci. 92, 119–199.CrossRefGoogle Scholar
  34. Jacqez, J. A., Simon, C. P. and Koopman, J. (1989). Structured mixing: heterogeneous mixing by the definition of mixing groups. In Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez (ed.), Lecture Notes in Biomathematics 83, 301–315. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  35. Kendall, D. G. (1949). Stochastic processes and population growth. Roy. Statist. Soc., Ser. B2, 230–264.Google Scholar
  36. Keyfitz, N. (1949). The mathematics of sex and marriage. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. IV: Biology and Health, 89–108.Google Scholar
  37. McFarland, D. D. (1972). Comparison of alternative marriage models. In Population Dynamics, T. N. E. Greville (ed.), 89–106. New York, London: Academic Press.Google Scholar
  38. Martin, J. L. (1986a). AIDS risk reduction recommendations and sexual behavior patterns among gay men: a multifactorial categorical approach to assessing change. Health Educ. Qtly. 13, 347–358.CrossRefGoogle Scholar
  39. Martin, J. L. (1986b). The impact of AIDS in gay male sexual behavior patterns in New York City. Am. J. of Pub. Health 77, 578–581.CrossRefGoogle Scholar
  40. McKusick, L., Horstman, W. and Coates, T. J. (1985a). AIDS and sexual behavior reported by gay men in San Francisco. Public Health Reports 75, 493–496.Google Scholar
  41. McKusick, L., Wiley, J. A., Coates, T. J., Stall, R., Saika, B., Morin, S., Horstman, C. K. and Conant, M. A. (1985b). Reported changes in the sexual behavior of men at risk for AIDS, San Francisco, 1983–1984: the AIDS behavioral research project. Public Health Reports 100, 622–629.PubMedGoogle Scholar
  42. Nold, A. (1980). Heterogeneity in disease-transmission modeling. Math. Biosci. 52, 227–240.CrossRefGoogle Scholar
  43. Palmer, J. S., Castillo-Chavez, C. and Blythe, S. P. (1991). State-dependent mixing and state-dependent contact rates in epidemiological models. Biometrics Unit Technical Report BU-1122-M, Cornell University, Ithaca, NY.Google Scholar
  44. Parlett, B. (1972). Can there be a marriage function?. In Population Dynamics, T. N. E. Greville (ed.), 107–135. New York, London: Academic Press.Google Scholar
  45. Pollard, J. H. (1973). The two-sex problem. In Mathematical Models for the Growth of Human Populations,Chapter 7. Cambridge University Press.Google Scholar
  46. Rubin, G., Umbach, D., Shyu, S-F. and Castillo-Chavez, C. (1991). Application of capture-recapture methodology to estimation of size of population at risk of AIDS and/or other sexually-transmitted diseases. Biometrics Unit Technical Report BU-1112-M, Cornell University, Ithaca, NY.Google Scholar
  47. Saltzman, S. P., Stoddard, A. M., McCusker, J., Moon, M. W. and Mayer, K. H. (1987). Reliability of self-reported sexual behavior risk factors for HIV infection in homosexual men. Public Health Reports 102, 692–697.PubMedGoogle Scholar
  48. Sattenspiel, L. and Castillo-Chavez, C. (1990). Environmental context, social interactions, and the spread of HIV. American Journal of Human Biology 2, 397–417.CrossRefGoogle Scholar
  49. Seber, G. A. F. (1082). The estimation of animal abundance and related parameters. New York: MacMillan.Google Scholar
  50. Shilts, R. (1987). And the band played on. New York: St. Martin’s Press.Google Scholar
  51. Thieme, H. R. and Castillo-Chavez, C. (1989). On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic. In Mathematical and Statistical Approaches to AIDS Epidemiology, C. Castillo-Chavez (ed.), Lecture Notes in Biomathematics 83, 157–176. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  52. Thieme, H. R. and Castillo-Chavez, C. (1990). On the possible effects of infection-age-dependent infectivity in the dynamics of HIV/AIDS. Biometrics Unit Technical Report BU-1102-M, Cornell University, Ithaca, NY.Google Scholar
  53. Waldstatter, 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, 260–274. Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong: Springer-Verlag.Google Scholar
  54. Winkelstein, W. Jr., Wiley, J. A., Padian, N. S., et al. (1988). The San Francisco Men’s Health Study, continued decline in HIV seroconversion rates among homosexual/bisexual men. Am. J. Pub. Health 78, 1472–1474.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Carlos Castillo-Chavez
    • 1
  • Shwu-Fang Shyu
    • 1
  • Gail Rubin
    • 1
  • David Umbach
    • 2
  1. 1.Biometrics UnitCornell UniversityIthacaUSA
  2. 2.National Institute of Environmental Health Sci.Research Triangle ParkUSA

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