Modeling the joint epidemics of TB and HIV in a South African township

  • Nicolas BacaërEmail author
  • Rachid Ouifki
  • Carel Pretorius
  • Robin Wood
  • Brian Williams


We present a simple mathematical model with six compartments for the interaction between HIV and TB epidemics. Using data from a township near Cape Town, South Africa, where the prevalence of HIV is above 20% and where the TB notification rate is close to 2,000 per 100,000 per year, we estimate some of the model parameters and study how various control measures might change the course of these epidemics. Condom promotion, increased TB detection and TB preventive therapy have a clear positive effect. The impact of antiretroviral therapy on the incidence of HIV is unclear and depends on the extent to which it reduces sexual transmission. However, our analysis suggests that it will greatly reduce the TB notification rate.


HIV TB Epidemic model Bifurcation diagram 

Mathematics Subject Classification (2000)

34C60 92D30 


  1. 1.
    Atun, R.A., Lebcir, R., Drobniewski, F., Coker, R.J.: Impact of an effective multidrug-resistant tuberculosis control programme in the setting of an immature HIV epidemic: system dynamics simulation model. Int. J. STD AIDS 16, 560–570 (2005)CrossRefGoogle Scholar
  2. 2.
    Atun, R.A., Lebcir, R.M., Drobniewski, F., McKee, M., Coker, R.J.: High coverage with HAART is required to substantially reduce the number of deaths from tuberculosis: system dynamics simulation. Int. J. STD AIDS 18, 267–273 (2007)CrossRefGoogle Scholar
  3. 3.
    Badri, M., Wilson, D., Wood, R.: Effect of highly active antiretroviral therapy on incidence of tuberculosis in South Africa: a cohort study. Lancet 359, 2059–2064 (2002)CrossRefGoogle Scholar
  4. 4.
    Bermejo, A., Veeken, H., Berra, A.: Tuberculosis incidence in developing countries with high prevalence of HIV infection. AIDS 6, 1203–1206 (1992)CrossRefGoogle Scholar
  5. 5.
    Blyuss, K.B., Kyrychko, Yu.N.: On a basic model of a two-disease epidemic. Appl. Math. Comput. 160, 177–187 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borgdorff, M.W.: Annual risk of tuberculosis infection—time for an update?. Bull. WHO 80, 501–502 (2002)Google Scholar
  7. 7.
    Breban, R., Blower, S.: The reinfection threshold does not exist. J. Theor. Biol. 235, 151–152 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Brewer, T.F., Heymann, S.J., Colditz, G.A., Wilson, M.E., Auerbach, K., Kane, D., Fineberg, H.V.: Evaluation of tuberculosis control policies using computer simulation. J. Am. Med. Assoc. 276, 1898–1903 (1996)CrossRefGoogle Scholar
  9. 9.
    Castillo-Chavez, C., Song, B.: Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. 1, 361–404 (2004)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Cohen, T., Lipsitch, M., Walensky, R.P., Murray, M.: Beneficial and perverse effects of isoniazid preventive therapy for latent tuberculosis infection in HIV-tuberculosis coinfected populations. Proc. Natl. Acad. Sci. USA 103, 7042–7047 (2006)CrossRefGoogle Scholar
  11. 11.
    Corbett, E.L., Charalambous, S., Fielding, K., Clayton, T., Hayes, R.J., De Cock, K.M., Churchyard, G.J.: Stable incidence rates of tuberculosis (TB) among human immunodeficiency virus (HIV)-negative South African gold miners during a decade of epidemic HIV-associated TB. J. Infect. Dis. 188, 1156–1163 (2003)CrossRefGoogle Scholar
  12. 12.
    Corbett, E.L., Charalambous, S., Moloi, V.M. et al.: Human immunodeficiency virus and the prevalence of undiagnosed tuberculosis in African gold miners. Am. J. Respir. Care Med. 170, 673–679 (2004)CrossRefGoogle Scholar
  13. 13.
    Corbett, E.L., Watt, C.J., Walker, N., Maher, D., Williams, B.G., Raviglione, M.C., Dye, C.: The growing burden of tuberculosis: global trends and interactions with the HIV epidemic. Arch. Intern. Med. 163, 1009–1021 (2003)CrossRefGoogle Scholar
  14. 14.
    Currie, C.S.M., Williams, B.G., Cheng, R.C.H., Dye, C.: Tuberculosis epidemics driven by HIV: is prevention better than cure?. AIDS 17, 2501–2508 (2003)CrossRefGoogle Scholar
  15. 15.
    Currie, C.S.M., Floyd, K., Williams, B.G., Dye, C.: Cost, affordability and cost-effectiveness of strategies to control tuberculosis in countries with high HIV prevalence. BMC Public Health 5, 130 (2005). doi: 10.1186/1471-2458-5-130 CrossRefGoogle Scholar
  16. 16.
    Daley, C.L., Small, P.M., Schecter, G.F., Schoolnik, G.K., McAdam, R.A., Jacobs, W.R. Jr, Hopewell, P.C.: An outbreak of tuberculosis with accelerated progression among persons infected with the human immunodeficiency virus: An analysis using restriction-fragment-length polymorphisms. N. Engl. J. Med. 326(4), 231–235 (1992)Google Scholar
  17. 17.
    Debanne, A.M., Bielefeld, R.A., Cauthen, G.M., Daniel, T.M., Rowland, D.Y.: Multivariate markovian modeling of tuberculosis: forecasts for the United States. Emerg. Infect. Dis. 6, 148–157 (2000)Google Scholar
  18. 18.
    Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases—Model Building, Analysis and Interpretation. Wiley, Chichester (2000)Google Scholar
  19. 19.
    Di Perri, G., Cruciani, M., Danzi, M.C., Luzzati, R., De Checchi, G., Malena, M. et al.: Nosocomial epidemic of active tuberculosis among HIV-infected patients. Lancet 2, 1502–1504 (1989)Google Scholar
  20. 20.
    Dowdy, D.W., Chaisson, R.E., Moulton, L.H., Dorman, S.E.: The potential impact of enhanced diagnostic techniques for tuberculosis driven by HIV: a mathematical model. AIDS 20, 751–762 (2006)Google Scholar
  21. 21.
    Dye, C., Garnett, G.P., Sleeman, K., Williams, B.G.: Prospects for worldwide tuberculosis control under the WHO DOTS strategy. Lancet 352, 1886–1891 (1998)CrossRefGoogle Scholar
  22. 22.
    Egwaga, S.M., Cobelens, F.G., Muwinge, H., Verhage, C., Kalisvaart, N., Borgdorff, M.W.: The impact of the HIV epidemic on tuberculosis transmission in Tanzania. AIDS 20, 915–921 (2006)CrossRefGoogle Scholar
  23. 23.
    Elliott, A.M., Halwiindi, B., Hayes, R.J., Luo, N., Mwinga, A.G., Tembo, G. et al.: The impact of human immunodeficiency virus on mortality of patients treated for tuberculosis in a cohort study in Zambia. Trans. R. Soc. Trop. Med. Hyg. 89, 78–82 (1995)CrossRefGoogle Scholar
  24. 24.
    Feng, Z., Castillo-Chavez, C., Capurro, A.F.: A model for tuberculosis with exogenous reinfection. Theor. Popul. Biol. 57, 235–247 (2000)zbMATHCrossRefGoogle Scholar
  25. 25.
    Feng, Z., Huang, W., Castillo-Chavez, C.: On the role of variable latent periods in mathematical models for tuberculosis. J. Dyn. Differ. Equ. 13, 425–452 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Gomes, M.G.M., Franco, A.O., Gomes, M.C., Medley, G.F.: The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy. Proc. R. Soc. Lond. B 271, 617–623 (2004)CrossRefGoogle Scholar
  27. 27.
    Gomes, M.G.M., White, L.J., Medley, G.F.: The reinfection threshold. J. Theor. Biol. 236, 111–113 (2005)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Gomes, M.G.M., Rodrigues, P., Hilker, F.M., Mantilla-Beniers, N.B., Muehlen, M., Paulo, A.C., Medley, G.F.: Implications of partial immunity on the prospects for tuberculosis control by post-exposure interventions. J. Theor. Biol. 248, 608–617 (2007)CrossRefGoogle Scholar
  29. 29.
    Guwatudde, D., Debanne, S.M., Diaz, M., King, C., Whalen, C.C.: A re-examination of the potential impact of preventive therapy on the public health problem of tuberculosis in contemporary sub-Saharan Africa. Prev. Med. 39, 1036–1046 (2004)CrossRefGoogle Scholar
  30. 30.
    Hethcote, H.W., van den Driessche, P.: Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271–287 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Heymann, S.J.: Modelling the efficacy of prophylactic and curative therapies for preventing the spread of tuberculosis in Africa. Trans. R. Soc. Trop. Med. Hyg. 87, 406–411 (1993)CrossRefGoogle Scholar
  32. 32.
    Hughes, G.R., Currie, C.S.M., Corbett, E.L.: Modeling tuberculosis in areas of high HIV prevalence. In: Perrone, L.F., Wieland, F.P., Liu, J., Lawson, B.G., Nicol, D.M., Fujimoto, R.M. (eds.) Proceedings of the 2006 Winter Simulation Conference, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 459–465 (2006)Google Scholar
  33. 33.
    Kline, S.E., Hedemark, L.L., Davies, S.F.: Outbreak of tuberculosis among regular patrons of a neighborhood bar. N. Engl. J. Med. 333, 222–227 (1995)CrossRefGoogle Scholar
  34. 34.
    Lawn, S.D., Badri, M., Wood, R.: Tuberculosis among HIV-infected patients receiving HAART: long term incidence and risk factors in a South African cohort. AIDS 19, 2109–2116 (2005)Google Scholar
  35. 35.
    Lawn, S.D., Bekker, L.-G., Middelkoop, K., Myer, L., Wood, R.: Impact of HIV infection on the epidemiology of tuberculosis in a peri-urban community in South Africa: The need for age-specific interventions. Clin. Infect. Dis. 42, 1040–1047 (2006)CrossRefGoogle Scholar
  36. 36.
    Lawn, S.D., Bekker, L.-G., Wood, R.: How effectively does HAART restore immune responses to Mycobacterium tuberculosis? Implications for tuberculosis control. AIDS 19, 1113–1124 (2005)Google Scholar
  37. 37.
    Lawn, S.D., Myer, L., Bekker, L.-G., Wood, R.: Burden of tuberculosis in an antiretroviral treatment programme in sub-Saharan Africa: impact on treatment outcomes and implications for tuberculosis control. AIDS 20, 1605–1612 (2006)Google Scholar
  38. 38.
    Lawn, S.D., Wood, R.: The epidemic of HIV-associated tuberculosis in sub-Saharan Africa: does this also impact non-HIV-infected individuals?. AIDS 20, 1787–1788 (2006)CrossRefGoogle Scholar
  39. 39.
    Lawn, S.D., Wood, R.: Tuberculosis control in South Africa—will HAART help?. S. Afr. Med. J. 96, 502–504 (2006)Google Scholar
  40. 40.
    Lawn, S.D., Wood, R.: When should antiretroviral treatment be started in patients with HIV-associated tuberculosis in South Africa?. S. Afr. Med. J. 97, 412–414 (2007)Google Scholar
  41. 41.
    Lawn, S.D., Wilkinson, R.J., Lipman, M.C.I., Wood, R.: Immune reconstitution and ‘unmasking’ of tuberculosis during antiretroviral therapy. Am. J. Respir. Crit. Care Med. (2008). doi: 10.1164/rccm.200709-1311PP zbMATHGoogle Scholar
  42. 42.
    Lipsitch, M., Murray, M.B.: Multiple equilibria: Tuberculosis transmission require unrealistic assumptions. Theor. Popul. Biol. 63, 169–170 (2003)CrossRefGoogle Scholar
  43. 43.
    Lungu, E.: Anti-tuberculosis resistance in patients co-infected with HIV and TB. Abstract presented at CMS-MITACS Joint Conference, Winnipeg, MB, Canada, May 31–June 3, 2007.
  44. 44.
    Massad, E., Burattini, M.N., Coutinho, F.A.B., Yang, H.M., Raimundo, S.M.: Modeling the interaction between AIDS and tuberculosis. Math. Comput. Modell. 17, 7–21 (1993)zbMATHCrossRefGoogle Scholar
  45. 45.
    Miranda, A., Morgan, M., Jamal, L. et al.: Impact of antiretroviral therapy on the incidence of tuberculosis: the Brazilian experience, 1995–2001. PLoS ONE 2, e826 (2007)CrossRefGoogle Scholar
  46. 46.
    Moghadas, S.M., Gumel, A.B.: Analysis of a model for transmission dynamics of TB. Can. Appl. Math. Q. 10, 411–428 (2002)zbMATHMathSciNetGoogle Scholar
  47. 47.
    Moghadas, S.M., Gumel, A.B.: An epidemic model for the transmission dynamics of HIV and another pathogen. ANZIAM J. 45, 1–13 (2003)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Moghadas, S.M., Alexander, M.E.: Exogenous reinfection and resurgence of tuberculosis: a theoretical framework. J. Biol. Syst. 12, 231–247 (2004)zbMATHCrossRefGoogle Scholar
  49. 49.
    Murphy, B.M., Singer, B.H., Kirschner, D.: On the treatment of TB in heterogeneous populations. J. Theor. Biol. 223, 391–404 (2003)CrossRefGoogle Scholar
  50. 50.
    Murray, C.J.L., Styblo, K., Rouillon, A.: Tuberculosis in developing countries: burden, intervention, and cost. Bull. Int. Union Tuberc. Lung Dis. 65, 6–24 (1990)Google Scholar
  51. 51.
    Murray, C.J.L., Salomon, J.A.: Modeling the impact of global tuberculosis control strategies. Proc. Natl. Acad. Sci. USA 95, 13881–13886 (1998)CrossRefGoogle Scholar
  52. 52.
    Naresh, R., Tripathi, A.: Modelling and analysis of HIV–TB coinfection in a variable size population. Math. Model. Anal. 10, 275–286 (2005)zbMATHMathSciNetGoogle Scholar
  53. 53.
    Nunn, A.J., Mulder, D.W., Kamali, A., Ruberantwari, A., Kengeya-Kayondo, J.F., Whitworth, J.: Mortality associated with HIV-1 infection over five years in a rural Ugandan population: cohort study. Br. Med. J. 315(7111), 767–771 (1997)Google Scholar
  54. 54.
    Porco, T.C., Blower, S.M.: Quantifying the intrinsic transmission dynamics of tuberculosis. Theor. Popul. Biol. 54, 117–132 (1998)zbMATHCrossRefGoogle Scholar
  55. 55.
    Porco, T.C., Small, P.M., Blower, S.M.: Amplification dynamics: predicting the effect of HIV on tuberculosis outbreaks. JAIDS 28, 437–444 (2001)Google Scholar
  56. 56.
    Raimundo, S.M., Yang, H.M., Bassanezi, R.C., Ferreira, M.A.C.: The attracting basins and the assessment of the transmission coefficients for HIV and M. Tuberculosis infections among women inmates. J. Biol. Syst. 10, 61–83 (2002)zbMATHCrossRefGoogle Scholar
  57. 57.
    Raimundo, S.M., Engel, A.B., Yang, H.M., Bassanezi, R.C.: An approach to estimating the transmission coefficients for AIDS and for tuberculosis using mathematical models. Syst. Anal. Model. Simul. 43, 423–442 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Schinazi, R.B.: Can HIV invade a population which is already sick?. Bull. Braz. Math. Soc. 34, 479–488 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Schulzer, M., Fitzgerald, J.M., Enarson, D.A., Grzybowski, S.: An estimate of the future size of the tuberculosis problem in sub-Saharan Africa resulting from HIV infection. Tuber. Lung. Dis. 73, 52–58 (1992)CrossRefGoogle Scholar
  60. 60.
    Schulzer, M., Radhamani, M.P., Grzybowski, S., Mak, E., Fitzgerald, J.M.: A mathematical model for the prediction of the impact of HIV infection on tuberculosis. Int. J. Epidemiol. 23, 400–407 (1994)CrossRefGoogle Scholar
  61. 61.
    Selwyn, P.A., Hartel, D., Lewis, V.A., Schoenbaum, E.E., Vermund, S.H., Klein, R.S., Walker, A.T., Friedland, G.H.: A prospective study of the risk of tuberculosis among intravenous drug users with human immunodeficiency virus infection. N. Engl. J. Med. 320, 545–550 (1989)Google Scholar
  62. 62.
    Selwyn, P.A., Sckell, B.M., Alcabes, P., Friedland, G.H., Klein, R.S., Schoenbaum, E.E.: High risk of active tuberculosis in HIV-infected drug users with cutaneous anergy. JAMA 268, 504–509 (1992)CrossRefGoogle Scholar
  63. 63.
    Sharomi, O., Podder, C.N., Gumel, A.B.: Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment. Math. Biosci. Eng. 5, 145–174 (2008)zbMATHMathSciNetGoogle Scholar
  64. 64.
    Singer, B.H., Kirschner, D.E.: Influence of backward bifurcation on interpretation of R 0 in a model of epidemic tuberculosis with reinfection. Math. Biosci. Eng. 1, 81–93 (2004)zbMATHMathSciNetGoogle Scholar
  65. 65.
    Sutherland, I., Svandova, E., Radhakrishna, S.: The development of clinical tuberculosis following infection with tubercle bacilli. Tubercle 63, 255–268 (1982)CrossRefGoogle Scholar
  66. 66.
    UNAIDS: 2006 Report on the global AIDS epidemic. UNAIDS, Geneva (2006)Google Scholar
  67. 67.
    Veening, G.J.: Long term isoniazid prophylaxis: controlled trial on INH prophylaxis after recent tuberculin conversion in young adults. Bull. Int. Union Tuberc. 41, 169–171 (1968)Google Scholar
  68. 68.
    Vynnycky, E., Fine, P.E.M.: The natural history of tuberculosis: the implications of age-dependent risks of disease and the role of reinfection. Epidemiol. Infect. 119, 183–201 (1997)CrossRefGoogle Scholar
  69. 69.
    Wang, W.: Epidemic models with nonlinear infection forces. Math. Biosci. Eng. 3, 267–279 (2006)zbMATHMathSciNetGoogle Scholar
  70. 70.
    West, R.W., Thompson, J.R.: Modeling the impact of HIV on the spread of tuberculosis in the United States. Math. Biosci. 143, 35–60 (1997)zbMATHCrossRefGoogle Scholar
  71. 71.
    Williams, B.G., Gouws, E.: The epidemiology of human immunodeficiency virus in South Africa. Philos. Trans. R. Soc. Lond. B Biol. Sci. 356, 1077–1086 (2001)Google Scholar
  72. 72.
    Williams, B.G., Granich, R., Chauhan, L.S., Dharmshaktu, N.S., Dye, C.: The impact of HIV/AIDS on the control of tuberculosis in India. Proc. Natl. Acad. Sci. USA 102, 9619–9624 (2005)CrossRefGoogle Scholar
  73. 73.
    Williams, B.G., Lloyd-Smith, J.O., Gouws, E., Hankins, C., Getz, W.M., Hargrove, J., de Zoysa, I., Dye, C., Auvert, B.: The potential impact of male circumcision on HIV in sub-Saharan Africa. PLoS Med. 3(7), e262 (2006)CrossRefGoogle Scholar
  74. 74.
    Williams, B.G., Maher, D.: Tuberculosis fueled by HIV: Putting out the flames. Am. J. Resp. Crit. Care 175, 6–7 (2007)CrossRefGoogle Scholar
  75. 75.
    Wood, R., Middelkoop, K., Myer, L., Grant, A.D., Whitelaw, A., Lawn, S.D., Kaplan, G., Huebner, R., McIntyre, J., Bekker, L.-G.: Undiagnosed tuberculosis in a community with high HIV-prevalence: implications for TB control. Am. J. Respir. Crit. Care 175, 87–93 (2007)CrossRefGoogle Scholar
  76. 76.
    World Health Organization: Global tuberculosis control: surveillance, planning, financing. WHO, Geneva (2007)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Nicolas Bacaër
    • 1
    Email author
  • Rachid Ouifki
    • 2
  • Carel Pretorius
    • 2
  • Robin Wood
    • 3
  • Brian Williams
    • 4
  1. 1.Institut de Recherche pour le Développement (IRD)BondyFrance
  2. 2.SACEMA, DST/NRF Centre of Excellence in Epidemiological Modelling and AnalysisStellenbosch UniversityStellenboschSouth Africa
  3. 3.Desmond Tutu HIV Centre, Institute of Infectious Disease and Molecular MedicineUniversity of Cape TownCape TownSouth Africa
  4. 4.Stop TB DepartmentWorld Health OrganizationGenevaSwitzerland

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