Notes from the heterogeneous: a few observations on the implications and necessity of affinity

References

• Anderson , R. M. 1982 . Population Dynamics of Infectious Diseases: Theory and Applications , Edited by: Anderson , R. M. London, New York : Chapman and Hall .
   Google Scholar
• Anderson , R. M. and May , R. M. 1991 . Infectious Diseases of Humans , Oxford, New York : Oxford Science Publications .
   Google Scholar
• Bailey , N. T.J. 1975 . The Mathematical Theory of Infectious Diseases and Its Applications , London : Griffin .
   Google Scholar
• Blythe , S. P. and Castillo-Chavez , C. 1989 . Like-with-like preference and sexual mixing models . Math. Biosci. , 96 : 221 – 238 .
   PubMedGoogle Scholar
• Blythe , S. P. , Castillo-Chavez , C. , Palmer , J. and Cheng , M. 1991 . Towards a unified theory of mixing and pair formation . Math. Biosci. , 107 : 379 – 405 .
   Google Scholar
• 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 . Math. Popul. Stud. , 3 ( 3 ) : 199 – 225 .
   PubMedGoogle Scholar
• Blythe , S. P. , Busenberg , S. and Castillo-Chavez , C. 1995 . Affinity and paired-event probability . Math. Biosci. , 128 : 265 – 284 .
   Google Scholar
• Busenberg , S. and Castillo-Chavez , C. 1989 . Interaction, Pair Formation and Force of Infection Terms in Sexually Transmitted Diseases , New York : Springer-Verlag . (Lecture Notes in Biomathematics 83)
   Google Scholar
• 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. Math. Appl. Med. Biol. , 8 : 1 – 29 .
   Google Scholar
• Castillo-Chavez , C. 1989 . Mathematical and Statistical Approaches to AIDS Epidemiology (Lecture Notes in Biomathematics 83) , Edited by: Castillo-Chavez , C. New York : Springer-Verlag .
   Google Scholar
• Castillo-Chavez , C. and Busenberg , S. On the solution of the two-sex mixing problem . Proceedings of the International Conference on Differential Equations and Applications to Biology and Population Dynamics . Edited by: Busenberg , S. and Martelli , M. pp. 80 – 98 . Berlin-Heidelberg, New York : Springer-Verlag . Lecture Notes in Biomathematics 92
   Google Scholar
• Castillo-Chavez , C. and Hsu Schmitz , S.-F. 1997 . “ The evolution of age-structured marriage functions: It takes two to tango ” . In Structured-Population Models Marine, Terrestrial, and Freshwater Systems , Edited by: Tuljapurkar , S. and Caswell , H. 533 – 550 . New York : Chapman & Hall .
   Google Scholar
• Castillo-Chavez , C. , Busenberg , S. and Gerow , K. 1990 . “ Pair formation in structured populations ” . In Differential Equations with Applications in Biology, Physics and Engineering , Edited by: Goldstein , J. , Kappel , F. and Schappacher , W. 4765 New York : Marcel Dekker .
   Google Scholar
• Castillo-Chavez , C. , Fridman , S. and Luo , X. Stochastic and deterministic models in epidemiology . Proceedings of the First World Congress on Nonlinear Analysts . August 19–26 1992 , Tampa, Florida.
   Google Scholar
• Castillo-Chavez , C. , Velasco-Hernandez , J. X. and Fridman , S. 1993 . Modeling Contact Structures in Biology (Lecture Notes in Biomathematics 100) , New York : Springer-Varlag .
   Google Scholar
• Castillo-Chavez , C. , Huang , W. and Li , J. 1996 . On the existence of stable pair distributions . J. Math. Biol. , 34 : 413 – 441 .
   Google Scholar
• Castillo-Chavez , C. , Yakubu , A.-A. , Thieme , H. and Martcheva , M. 2002 . “ Nonlinear mating models for populations with discrete generations ” . In Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction , Edited by: Castillo-Chavez , C. , van den Driessche , P. , Kirschner , D. and Yakubu , A.-A. Vol. 125 , 251 – 268 . Berlin–Heidelberg–New York : Springer-Verlag . IMA
   Google Scholar
• Caswell , H. September 2000 . Matrix Population Models: Construction, Analysis and Interpretation , 2 , September , Sunderland, Massachusetts : Sinauer Associates .
   Google Scholar
• Chowell , G. and Castillo-Chavez , C. 2003 . Worst Case Scenarios and Epidemics. In (Banks model building and evaluation . Can. J. Vet. Res. , 67 : 307 – 311 .
   Google Scholar
• Chowell , G. , Cintron-Arias , A. , Del Valle , S. , Sanchez , F. , Song , B. , Hyman , M. and Castillo-Chavez , C. 2006 . “ Mathematical applications associated to the deliberate release of infectious agents ” . In Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges , Edited by: Gumel , A. , Castillo-Chavez , C. , Clemence , D. P. and Mickens , R. E. Vol. 410 , 51 – 72 . Rhode Island, Providence : American Mathematical Society .
   Google Scholar
• Chowell , G. , Hyman , J. M. , Bettencourt , L. M.A. and Castillo-Chavez , C. 2009 . Mathematical and Statistical Estimation Approaches in Epidemiology , Edited by: Chowell , G. , Hyman , J. M. , Bettencourt , L. M.A. and Castillo-Chavez , C. 370 Springer .
   Google Scholar
• Crawford , C. M. , Schwager , S. J. and Castillo-Chavez , C. 1990 . A methodology for asking sensitive questions among college undergraduates , Ithaca, NY : Cornell University . Tech. Rep. # BU-1105-M in the Biometrics Unit series
   Google Scholar
• Crow , J. F. and Kimura , M. 1970 . An Introduction to Population Genetics Theory , New York : Harper & Row .
   Google Scholar
• Dietz , K. 1988 . On the transmission dynamics of HIV . Math. Biosci. , 90 : 397 – 414 .
   Web of Science ®Google Scholar
• Dietz , K. and Hadeler , K. P. 1988 . Epidemiological models for sexually transmitted diseases . J. Math. Biol. , 26 : 1 – 25 .
   PubMed Web of Science ®Google Scholar
• Elsner , L. and Hadeler , K. P. October 1 2009 . The Spectral Radius and the Largest Basic Reproduction Number in Multi-type Epidemic Models with Mixing October 1 , Unpublished Manuscript
   Google Scholar
• Fredrickson , A. G. 1971 . A mathematical theory of age structure in sexual populations: Random mating and monogamous marriage models . Math. Biosci. , 20 : 117 – 143 .
   Google Scholar
• Goodreau , S. , Kitts , J. and Morris , M. 2009 . Birds of a feather, or friend of a friend? Using statistical network analysis to investigate adolescent social networks . Demography , 46 ( 1 ) : 103 – 125 .
   PubMed Web of Science ®Google Scholar
• Gumel , A. , Castillo-Chavez , C. , Clemence , D. P. and Mickens , R. E. 2006 . Modeling the dynamics of human diseases: Emerging paradigms and challenges . Am. Math. Soc. , 410 : 389
   Google Scholar
• Hadeler , K. P. 1989 . Pair formation in age-structured populations . Acta Appl. Math. , 14 : 91 – 102 .
   PubMed Web of Science ®Google Scholar
• Hadeler , K. P. 47th Session of the International Statistical Institute . August/September , Paris. Modeling AIDS in structured populations , pp. 83 – 99 . Conf. Proc., C1–2
   Google Scholar
• Hethcote , H. W. 2000 . The mathematics of infectious diseases . SIAM Rev. , 42 ( 4 ) : 599 – 653 .
   Web of Science ®Google Scholar
• Hadeler , K. P. and Nagoma , K. 1990 . Homogeneous models for sexually transmitted diseases . Rocky Mountain J. Math. , 20 : 967 – 986 .
   Web of Science ®Google Scholar
• Hethcote , H. W. and Van Ark , J. W. 1992 . Modeling HIV Transmission and AIDS in the United States (Lecture Notes in Biomathematics 95) , New York : Springer-Verlag .
   Google Scholar
• Hethcote , H. W. and Yorke , J. A. 1984 . Gonorrhea Transmission Dynamics and Control , Berlin, Heiderlberg, New York, London, Paris, Tokyo, , Hon Kong : Springer-Verlag .
   Google Scholar
• Hethcote , H. W. and Yorke , J. A. 1984 . Gonorrhea Transmission Dynamics and Control , New York : Springer-Verlag . (Lecture Notes in Biomathematics 56
   Google Scholar
• Hsu Schmitz , S.-F. 1993 . Some theories, estimation methods and applications of marriage functions and two-sex mixing functions in demography and epidemiology Unpublished doctoral dissertation, Cornell University, Ithaca, NY, USA
   Google Scholar
• Hsu Schmitz , S.-F. and Castillo-Chavez , C. 1994 . “ Parameter estimation in nonclosed social networks related to dynamics of sexually transmitted diseases ” . In Modelling the AIDS Epidemic: Planning, Policy, and Prediction , Edited by: Kaplan , E. H. and Brandeau , M. L. 533 – 559 . New York : Raven Press Ltd. .
   Google Scholar
• Hsu Schmitz , S.-F. and Castillo-Chavez , C. 2000 . A note on pair-formation functions . Math. Comput. Model. , 31 : 83 – 91 .
   Google Scholar
• Hyman , J. M. and Stanley , E. A. 1988 . Using mathematical models to understand the AIDS epidemic . Math. Biosci. , 90
   Google Scholar
• Hyman , J. M. and Stanley , E. A. 1989 . “ The effects of social mixing patterns on the spread of AIDS ” . In Mathematical Approaches to Problems in Resource Management and Epidemiology (Ithaca, NY, 1987) , Edited by: Castillo-Chavez , C. , Levin , S. A. and Shoemaker , C. A. 190 – 219 . Berlin : Springer . Lecture notes in Biomathematics, 81
   Google Scholar
• Hyman , J. M. and Stanley , E. A. 1994 . “ A risk-based heterosexual model for the AIDS epidemic with biased sexual partner selection ” . In Modeling the AIDS Epidemic , Edited by: Kaplan , E. and Brandeau , M. 331 – 364 . New York : Raven Press .
   Google Scholar
• Hyman , J. M. , Li , J. and Stanley , E. A. 2001 . The initialization and sensitivity of multigroup models for the transmission of HIV . J. Theor. Bio. , 208 : 227 – 249 .
   Google Scholar
• Hyman , J. M. , Li , J. and Stanley , E. A. 1999 . The Differential Infectivity and Staged Progression Models for the Transmission of HIV . Math. Biosci. , 155 ( 2 ) : 77 – 109 .
   PubMed Web of Science ®Google Scholar
• Jacquez , J. A. , Simon , C. P. , Koopman , J. , Sattenspiel , L. and Perry , T. 1988 . Modeling and the analysis of HIV transmission: The effect of contact patterns . Math. Biosci. , 92 : 119 – 199 .
   Web of Science ®Google Scholar
• Jewell , N. P. , Dietz , K. and Farewell , V. T. 1991 . AIDS Epidemiology: Methodological Issues , Boston, Basel, Berlin : Birkhäuser .
   Google Scholar
• Kaplan , E. H. 1989 . What Are the Risks of Risky Sex? . Oper. Res. , 37 : 198 – 209 .
   Web of Science ®Google Scholar
• Kaplan , E. H. and Brookmeyer , R. 1999 . Snapshot estimators of recent HIV incidence rates . Oper. Res. , 47 : 29 – 37 .
   Web of Science ®Google Scholar
• Kasseem , G. T. , Roudenko , S. , Tennenbaum , S. and Castillo-Chavez , C. 2006 . “ The role of transactional sex in spreading HIV/AIDS in Nigeria: A modeling perspective ” . In Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges , Edited by: Gumel , A. , Castillo-Chavez , C. , Clemence , D. P. and Mickens , R. E. Vol. 410 , 367 – 389 . Rhode Island, Providence : American Mathematical Society .
   Google Scholar
• Kendall , D. G. 1949 . Stochastic processes and population growth . R. Statist. Soc. Ser. B , 2 : 230 – 264 .
   Google Scholar
• Keyfitz , N. The Mathematics of Sex and Marriage . Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability . Vol. 4 , pp. 89 – 108 . Berkeley: California : University of California Press .
   Google Scholar
• Kirschner , D. 1996 . A diffusion model for AIDS in a closed, heterosexual populations: Estimating rates of infection . SIAM J. Appl. Math. , 56 : 143 – 166 .
   Google Scholar
• Koopman , J. , Simon , C. P. , Jacquez , J. A. , Joseph , J. , Sattenspiel , L. and Park , T. 1988 . Sexual partner selectiveness effects on homosexual HIV transmission dynamics . J. AIDS , 1 : 486 – 504 .
   PubMed Web of Science ®Google Scholar
• Koopman , J. S. , Jacquez , J. A. , Welch , G. W. , Simon , C. P. , Foxman , B. , Pollock , S. M. , Barth-Jones , D. , Adams , A. L. and Lange , K. 1997 . The role of primary HIV infection in the spread of HIV . J. AIDS , 14 : 249 – 258 .
   Web of Science ®Google Scholar
• Kretzschmar , M. 1995 . Deterministic and stochastic pair formation models for the spread of sexually transmitted diseases . J. Biol. Syst. , 3 : 789 – 801 .
   Google Scholar
• Leslie , P. H. 1945 . On the use of matrices in certain population mathematics . Biometrika , 33 : 183 – 212 .
   PubMed Web of Science ®Google Scholar
• Lotka , A. J. 1922 . The stability of the normal age distribution . Proc. Natl Acad. Sci. , 8 : 339 – 345 .
   PubMedGoogle Scholar
• Lubkin , S. and Castillo-Chavez , C. A Pair Formation Approach to Modeling Inheritance of Social Traits . Proceedings of the First World Congress on Nonlinear Analysts . August , Tampa, FL. pp. 19 – 26 . Hawthorne, NJ : Walter de Gruyter & Co .
   Google Scholar
• Luo , X. and Castillo-Chavez , C. 1991 . Limit behavior of pair formation for a large dissolution rate . J. Math. Syst. Estimation Control , 3 ( 2 ) : 247 – 264 .
   Google Scholar
• Luo , X. , Castillo-Chavez , C. and Cheng , M. 1991 . A probabilistic model of pair formation , Ithaca, NY : Cornell University . Biometrics Unit Tech. Rep., BU-1145-M
   Google Scholar
• MacKendrick , A. G. 1926 . Applications of mathematics to medical problems . Proc. Edinb. Math. Soc. , 44 : 98 – 130 .
   Google Scholar
• McFarland , D. D. 1972 . “ Comparison of alternative marriage models ” . In Population Dynamics , Edited by: Greville , T. N.E. 89 – 106 . New York, London : Academic Press .
   Google Scholar
• Mubayi , A. , Greenwood , P. E. , Castillo-Chavez , C. , Gruenewald , P. and Gorman , D. M. 2010 . Impact of Relative Residence Times on the Distribution of Heavy Drinkers in Highly Distinct Environments . Socio-Econ. Plann. Sci. , 43 ( 1 ) : 1 – 12 .
   Google Scholar
• Newman , M. E.J. 2003 . The structure and function of complex networks . SIAM Rev. , 45 : 167 – 256 .
   Web of Science ®Google Scholar
• Parlett , B. 1972 . “ Can there be a marriage function? ” . In Population Dynamics , Edited by: Greville , T. N.E. 107 – 135 . New York, London : Academic Press .
   Google Scholar
• Pollard , J. H. 1973 . Mathematical Models for the Growth of Human Populations , Cambridge : Cambridge University Press .
   Google Scholar
• Ross , R. 1911 . The Prevention of Malaria , 2 , London : John Murray . with addendum
   Google Scholar
• Rubin , G. , Umbauch , D. , Shyu , S.-F. and Castillo-Chavez , C. 1992 . Application of capture-recapture methodology to estimation of size of population at risk of AIDS and/or other sexually-transmitted diseases . Stat. Med. , 11 : 1533 – 1549 .
   Google Scholar
• Sattenspiel , L. 1989 . “ The structure and context of social interactions and the spread of HIV ” . In Mathematical and Statistical Approaches to AIDS Epidemiology , Edited by: Castillo-Chavez , C. 242 – 259 . Berlin : Springer-Verlag . Lecture Notes in Biomathematics 83
   Google Scholar
• Sattenspiel , L. , Koopman , J. , Simon , C. P. and Jacquez , J. A. 1990 . The effects of population subdivision on the spread of the HIV infection . Am. J. Phys. Anthropol. , 82 : 421 – 429 .
   PubMed Web of Science ®Google Scholar
• Sattenspiel , L. and Castillo-Chavez , C. 1990 . Environmental context, social interactions, and the spread of HIV . Am. J. Hum. Biol. , 2 : 397 – 417 .
   PubMed Web of Science ®Google Scholar
• Tennenbaum , S. 2006 . A two-sex, age-structured population model in discrete time , Ph.D. thesis, Cornell University
   Google Scholar
• Waldstätter , R. 1989 . Pair Formation in Sexually Transmitted Diseases (Lecture Notes in Biomathematics) , Vol. 83 , 260 – 274 . New York : Springer-Verlag .
   Google Scholar