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2014, 11(3): 599-619. doi: 10.3934/mbe.2014.11.599

Modeling and analysis of the San Francisco City Clinic Cohort (SFCCC) HIV-epidemic including treatment

Brandy Rapatski 1, and  Juan Tolosa 1,

1. 

Department of Natural Sciences and Mathematics, Richard Stockton College of New Jersey, 101 Vera King Farris Drive, Galloway, NJ 08205-9441, United States, United States

Received  August 2012 Revised  August 2013 Published  January 2014

We investigate two HIV/AIDS epidemic models. The first model represents the early San Francisco men having sex with men (MSM) epidemic. We use data from the San Francisco City Clinic Cohort Study (SFCCC), documenting the onset of HIV in San Francisco (1978-1984). The second model is a ``what-if'' scenario model including testing and treatment in the SFCCC epidemic. We use compartmental, population-level models, described by systems of ordinary differential equations. We find the basic reproductive number $R_0$ for each system, and we prove that if $R_0<1$, the system has only the disease-free equilibrium (DFE) which is locally and globally stable, whereas if $R_0>1$, the DFE is unstable. In addition, when $R_0>1$, both systems have a unique endemic equilibrium (EE). We show that treatment alone would not have stopped the San Francisco MSM epidemic, but would have significantly reduced its impact.
Keywords: mathematical model, Lyapunov function., reproduction number, stability, Human Immunodeficiency Virus (HIV).
Mathematics Subject Classification: Primary: 92D30, 34A34, 34D23; Secondary: 93D0.
Citation: Brandy Rapatski, Juan Tolosa. Modeling and analysis of the San Francisco City Clinic Cohort (SFCCC) HIV-epidemic including treatment. Mathematical Biosciences & Engineering, 2014, 11 (3) : 599-619. doi: 10.3934/mbe.2014.11.599
References:
[1]

D. J. Ahlgren, M. K. Gorny and A. C. Stein, Model-based optimization of infectivity parameters: A study of the early epidemic in San Francisco, J. Acqr. Immune. Defic. Syndr., 3 (1990), 631-643.

[2]

H. de Arazoza and R. Lounes, A non-linear model for a sexually transmitted disease with contact tracing, IMA Journ of Math. Appl. in Medicine and Biology, 19 (2002), 221-234.

[3]

R. B. Bapat and T. E. S. Raghavan, Non-negative Matrices and Applications, Cambridge University Press, New York, 1997. doi: 10.1017/CBO9780511529979.

[4]

F. Brauer, Some simple epidemic models, Math Biosci and Engineering, 3 (2006), 1-15. doi: 10.3934/mbe.2006.3.1.

[5]

F. Brauer and J. Watmough, Age of infection epidemic models with heterogeneous mixing, J. Biological Dynamics, 3 (2009), 324-330. doi: 10.1080/17513750802415822.

[6]

D. Brown, HIV Drugs Sharply Cut Risk of Transmission, Study Finds, The Washington Post, 2011.

[7]

CDC, Update: Acquired immunodeficiency syndrome in the San Francisco cohort study, 1978-1985, MMWR, 34 (1985), 573-575.

[8]

J. W. Curran, et al., The epidemiology of AIDS: Current status and future prospects, Science, 229 (1985), 1352-1357. doi: 10.1126/science.2994217.

[9]

C. F. Gilks, S. Crowley and R. Ekpini, et. al., The WHO public-health approach to antiretroviral treatment against HIV in resource-limited settings, Lancet, 368 (2006), 505-510. doi: 10.1016/S0140-6736(06)69158-7.

[10]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmenatl models of disease transmission, Math Biosci, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[11]

R. M. Granich, C. F. Gilks, C. Dye and K. M. De Cock and B. G. Williams, Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: A mathematical model, Lancet, 373 (2009), 48-57. doi: 10.1016/S0140-6736(08)61697-9.

[12]

P. Hartman, Ordinary Differential Equations, Baltimore, 1973.

[13]

H. W. Hethcote and J. W. Van Ark, Modeling HIV Transmission and AIDS in the United States, Berlin: New York, Springer-Verlag, 1992. doi: 10.1007/978-3-642-51477-7.

[14]

H. W. Hethcote and J. W. Van Ark, Modeling HIV Transmission and AIDS in the United States, Springer-Verlag, 1992. doi: 10.1007/978-3-642-51477-7.

[15]

H. W. Jaffe, et al., The acquired immunodeficiency syndrome in a cohort of homosexual men: a six-year follow-up study, Ann. Intern. Med., 103 (1985), 210-214. doi: 10.7326/0003-4819-103-2-210.

[16]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a non-homogeneous population, Math Biosci, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.

[17]

B. McKay, Scientists See Breakthrough in the Global AIDS Battle, The Washington Post, 2011.

[18]

A. Nold, Heterogeinity in disease transmission modeling, Math Biosci, 52 (1980), 227-240. doi: 10.1016/0025-5564(80)90069-3.

[19]

NIH, Guidelines for the Use of Antiretroviral Agents in HIV-1-Infected Adults and Adolescents, http://aidsinfo.nih.gov/contentfiles/lvguidelines/AdultandAdolescentGL.pdf (2012).

[20]

S. M. Osnaga, On rank one matrices and invariant subspaces, Balkan J. of Geometry and Its Applications, 10 (2005), 145-148.

[21]

J. Price, Study: Early HIV Treatment Slows Spread of Disease, Lexington Herald Chapel Hill, The Washington Post, 2011.

[22]

B. L. Rapatski, P. Klepak, S. Dueck, M. Liu, and L. I. Weiss, Mathematical epidemiology of HIV-AIDS in Cuba during the period 1986-2000, Math Biosci and Engineering, 3 (2006), 545-556. doi: 10.3934/mbe.2006.3.545.

[23]

B. L. Rapatski, F. Suppe and J. A. Yorke, HIV epidemics driven by late disease-stage transmission, JAIDS, 38 (2005), 241-253.

[24]

B. L. Rapatski and J. Tolosa, What would have stopped the San Francisco gay HIV/AIDS epidemic, paper submitted for publication.

[25]

J. Sterne, M. May and D. Costagliola, et. al., Timing of initiation of antiretroviral therapy in AIDS-free HIV-1-infected patients: A collaborative analysis of 18 HIV cohort studies, Lancet, 373 (2009), 1352-1363.

[26]

C. K. Yang and F. Brauer, Calculation of $R_0$ for age-of-infection models, Math Biosci and Engineering, 5 (2008), 585-599. doi: 10.3934/mbe.2008.5.585.

[27]

W. Winkelstein, D. M. Lyman, N. Padian, R. Grant and M. Samuel, J. A. Wiley, R. E. Anderson, W. Lang, J. Riggs and J. A. Levy, Sexual practices and risk of infection by the human immunodeficiency virus: the San Francisco men's health study, JAMA, 257 (1987), 321-325. doi: 10.1001/jama.1987.03390030051019.

show all references

References:
[1]

D. J. Ahlgren, M. K. Gorny and A. C. Stein, Model-based optimization of infectivity parameters: A study of the early epidemic in San Francisco, J. Acqr. Immune. Defic. Syndr., 3 (1990), 631-643.

[2]

H. de Arazoza and R. Lounes, A non-linear model for a sexually transmitted disease with contact tracing, IMA Journ of Math. Appl. in Medicine and Biology, 19 (2002), 221-234.

[3]

R. B. Bapat and T. E. S. Raghavan, Non-negative Matrices and Applications, Cambridge University Press, New York, 1997. doi: 10.1017/CBO9780511529979.

[4]

F. Brauer, Some simple epidemic models, Math Biosci and Engineering, 3 (2006), 1-15. doi: 10.3934/mbe.2006.3.1.

[5]

F. Brauer and J. Watmough, Age of infection epidemic models with heterogeneous mixing, J. Biological Dynamics, 3 (2009), 324-330. doi: 10.1080/17513750802415822.

[6]

D. Brown, HIV Drugs Sharply Cut Risk of Transmission, Study Finds, The Washington Post, 2011.

[7]

CDC, Update: Acquired immunodeficiency syndrome in the San Francisco cohort study, 1978-1985, MMWR, 34 (1985), 573-575.

[8]

J. W. Curran, et al., The epidemiology of AIDS: Current status and future prospects, Science, 229 (1985), 1352-1357. doi: 10.1126/science.2994217.

[9]

C. F. Gilks, S. Crowley and R. Ekpini, et. al., The WHO public-health approach to antiretroviral treatment against HIV in resource-limited settings, Lancet, 368 (2006), 505-510. doi: 10.1016/S0140-6736(06)69158-7.

[10]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmenatl models of disease transmission, Math Biosci, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[11]

R. M. Granich, C. F. Gilks, C. Dye and K. M. De Cock and B. G. Williams, Universal voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission: A mathematical model, Lancet, 373 (2009), 48-57. doi: 10.1016/S0140-6736(08)61697-9.

[12]

P. Hartman, Ordinary Differential Equations, Baltimore, 1973.

[13]

H. W. Hethcote and J. W. Van Ark, Modeling HIV Transmission and AIDS in the United States, Berlin: New York, Springer-Verlag, 1992. doi: 10.1007/978-3-642-51477-7.

[14]

H. W. Hethcote and J. W. Van Ark, Modeling HIV Transmission and AIDS in the United States, Springer-Verlag, 1992. doi: 10.1007/978-3-642-51477-7.

[15]

H. W. Jaffe, et al., The acquired immunodeficiency syndrome in a cohort of homosexual men: a six-year follow-up study, Ann. Intern. Med., 103 (1985), 210-214. doi: 10.7326/0003-4819-103-2-210.

[16]

A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a non-homogeneous population, Math Biosci, 28 (1976), 221-236. doi: 10.1016/0025-5564(76)90125-5.

[17]

B. McKay, Scientists See Breakthrough in the Global AIDS Battle, The Washington Post, 2011.

[18]

A. Nold, Heterogeinity in disease transmission modeling, Math Biosci, 52 (1980), 227-240. doi: 10.1016/0025-5564(80)90069-3.

[19]

NIH, Guidelines for the Use of Antiretroviral Agents in HIV-1-Infected Adults and Adolescents, http://aidsinfo.nih.gov/contentfiles/lvguidelines/AdultandAdolescentGL.pdf (2012).

[20]

S. M. Osnaga, On rank one matrices and invariant subspaces, Balkan J. of Geometry and Its Applications, 10 (2005), 145-148.

[21]

J. Price, Study: Early HIV Treatment Slows Spread of Disease, Lexington Herald Chapel Hill, The Washington Post, 2011.

[22]

B. L. Rapatski, P. Klepak, S. Dueck, M. Liu, and L. I. Weiss, Mathematical epidemiology of HIV-AIDS in Cuba during the period 1986-2000, Math Biosci and Engineering, 3 (2006), 545-556. doi: 10.3934/mbe.2006.3.545.

[23]

B. L. Rapatski, F. Suppe and J. A. Yorke, HIV epidemics driven by late disease-stage transmission, JAIDS, 38 (2005), 241-253.

[24]

B. L. Rapatski and J. Tolosa, What would have stopped the San Francisco gay HIV/AIDS epidemic, paper submitted for publication.

[25]

J. Sterne, M. May and D. Costagliola, et. al., Timing of initiation of antiretroviral therapy in AIDS-free HIV-1-infected patients: A collaborative analysis of 18 HIV cohort studies, Lancet, 373 (2009), 1352-1363.

[26]

C. K. Yang and F. Brauer, Calculation of $R_0$ for age-of-infection models, Math Biosci and Engineering, 5 (2008), 585-599. doi: 10.3934/mbe.2008.5.585.

[27]

W. Winkelstein, D. M. Lyman, N. Padian, R. Grant and M. Samuel, J. A. Wiley, R. E. Anderson, W. Lang, J. Riggs and J. A. Levy, Sexual practices and risk of infection by the human immunodeficiency virus: the San Francisco men's health study, JAMA, 257 (1987), 321-325. doi: 10.1001/jama.1987.03390030051019.

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