Bulletin of Mathematical Biology

, Volume 68, Issue 3, pp 525–550

Modeling the HIV/AIDS Epidemic Among Injecting Drug Users and Sex Workers in Kunming, China

Original Article

Abstract

This paper presents a mathematical model of the HIV/AIDS epidemic in Kunming,the provincial capital of Yunnan, China. The population is divided into several groups, with individuals possibly changing group. Two transmission routes of HIV are considered: needle sharing betweeninjecting drug users (IDUs) and commercial sex between female sex worker(FSWs) and clients. The model includes male IDUs who are also clients and female IDUs who are also FSWs. Groups are split in two—risky and safe—according to condom use and needle sharing. A system of partialdifferential equations is derived to describe the spread of the disease. For the simulation, parameters are chosen to fit as much as possibledata publicly available for Kunming. Some mathematical properties of the model—in particular the epidemic threshold R0 which determines the goal of public health interventions—are also presented. Though the model couples two transmission routes of HIV, the approximation \(R_0\simeq \max\{R_0^{{\rm IDU}},R_0^{{\rm sex}}\}\), with closed formulas for \(R_0^{{\rm IDU}}\) and \(R_0^{{\rm sex}}\), appears to be quite good. The critical levels of condom use and clean needle use necessary to stop both the sexual transmission and the transmission among IDUs can therefore be determined independently.

Keywords

HIV/AIDS Kermack-McKendrick epidemic model R0 Female sex workers Injecting drug users China 

MSC

92C60 35Q80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, R.M., Medley, G.F., May, R.M., and Johnson, A.M., 1986. A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS. IMA J. Math. Appl. Med. Biol. 3, 229–263.PubMedMathSciNetCrossRefMATHGoogle Scholar
  2. Bacaër, N., 2003. The asymptotic behavior of the McKendrick equation with immigration. Math. Popul. Stud. 10, 1–20.CrossRefMathSciNetMATHGoogle Scholar
  3. Berman, A., Plemmons, R.J., 1994. Nonnegative Matrices in the Mathematical Sciences. Society for Industrial and Applied Mathematics, Philadelphia.MATHGoogle Scholar
  4. Bignami-Van Assche, S., 2004. Estimates and projections of HIV/AIDS for Yunnan province, China. Popul. Rev. 43, No. 2, Section 1.Google Scholar
  5. Brown, T., Peerapatanapokin, W., 2004. The Asian epidemic model: a process model for exploring HIV policy and programme alternatives in Asia. Sex. Transm. Infect. 80, i19–i24.CrossRefPubMedGoogle Scholar
  6. Capasso, V., Morale, D., Di Somma, M., Villa, M., Nicolisi A., Sicurello, F., 1997. Multistage models of HIV transmission among injecting drug users via shared drug injection equipment, in Arino O. et al. (Eds.), Advances in Mathematical Population Dynamics–Molecules, Cells and Man. World Scientific, Singapore, pp. 511–527.Google Scholar
  7. Castillo-Chavez, C., Cooke, K., Huang, W., Levin, S.A., 1989a. On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS)-part 1: Single population models. J. Math. Biol. 27, 373–398.CrossRefMathSciNetMATHGoogle Scholar
  8. Castillo-Chavez, C.,Cooke, K., Huang, W., Levin, S.A., 1989b. On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS)-part 2: multiple group models. In: Castillo-Chavez, C., (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology. Springer-Verlag, New York, pp. 200–217.Google Scholar
  9. Cheng, H., Zhang, J., Kou, J., Zhang, Y., Zhang, X., Jia, M., Bi, X., Ma, Y., Liang, Y., Yang, Z., Pan, S., An, X., 1996. HIV infection tends to spread to whole Yunnan province. Chin. J. STD/AIDS Prev. Cont. 2, 54–57.Google Scholar
  10. China AIDS Info, 2004. Yunnan passes new AIDS policy, will provide clean needles to drug users. Online:http://www.china-aids.org.
  11. Cohen, J., 2004a. HIV/AIDS in China—Poised for takeoff? Science 304, 1430–1432.CrossRefGoogle Scholar
  12. Cohen, J., 2004b. HIV/AIDS in China—Changing course to break the HIV-heroin connection. Science 304, 1434–1435.CrossRefGoogle Scholar
  13. Cohen, J., 2004c. HIV/AIDS in Asia—Asia and Africa: On different trajectories? Science 304, 1932–1938.CrossRefGoogle Scholar
  14. Diekmann, O., Heesterbeek, J.A.P., 2000. Mathematical Epidemiology of Infectious Diseases. John Wiley Chichester.Google Scholar
  15. Duan, Y., Zheng, X., Zhen, C., 1995. Investigation of mortality among HIV-infected intravenous drug users in Ruili region of Yunnan province. Zhonghua Liu Xing Bing Xue Za Zhi 16, 71–73.PubMedGoogle Scholar
  16. Greenhalgh, D., Lewis, F., 2002. The general mixing of addicts and needles in a variable-infectivity needle-sharing environment. J. Math. Biol. 44, 561–598.CrossRefPubMedMathSciNetMATHGoogle Scholar
  17. Hethcote, H.W., Thieme, H.R., 1985. Stability of the endemic equilibrium in epidemic models with subpopulations. Math. Biosci. 75, 205–227.CrossRefMathSciNetMATHGoogle Scholar
  18. Horizon Market Research and Futures Group Europe, 2002a. 2001 behavioural surveillance survey in Yunnan and Sichuan—adult male report. Online:http://www.futures-group.com/Documents/2001BSSadultmale.pdf.
  19. Horizon Market Research and Futures Group Europe, 2002b. 2001 behavioural surveillance survey in Yunnan and Sichuan—sex worker report. Online:http://www.fu-turesgroup.com/Documents/2001BSSsexworker.pdf.
  20. Horn, R.A., Johnson, C.R., 1985. Matrix Analysis. Cambridge University Press, Cambridge.MATHGoogle Scholar
  21. Huang, W., Cooke, K.L., Castillo-Chavez, C., 1992. Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission. SIAM J. Appl. Math. 52, 833–854.CrossRefMathSciNetGoogle Scholar
  22. Hyman, J.M., Stanley, E.A., 1988. Using mathematical models to understand the AIDS epidemic. Math. Biosci. 90, 415–473.CrossRefMathSciNetMATHGoogle Scholar
  23. Iannelli, M., Milner, F.A., Pugliese, A., Gonzo, M. 1997. The HIV/AIDS epidemics among drug injectors: A study of contact structure through a mathematical model. Math. Biosci. 139, 25–58.CrossRefPubMedMATHGoogle Scholar
  24. Ka, K., 2004. Voluntary counseling and testing among injecting drug users in Kunming city, Yunnan Province. 15th International AIDS Conference, Bangkok, abstract no. WePeC5999.Google Scholar
  25. Kermack, W.O., McKendrick, A.G., 1933. Contributions to the mathematical theory of epidemics - III. Further studies of the problem of endemicity. Proc. Roy. Soc. London Ser. A 141, 94–122. (Reprinted in Bull. Math. Biol. 53 (1991), 89–118.Google Scholar
  26. Lu, L., Jia, M., Zhang, X., Luo, H., Ma, Y., Fu. L., Lu, J., 2004. Analysis for epidemic trend of acquired immunodeficiency syndrome in Yunnan Province of China. Chin. J. Prev. Med. 38, 309–312.CrossRefGoogle Scholar
  27. Mastro, T.D., Satten, G.A., Nopkesorn, T., Sangkharomya, S., Longini, I.M., 1994. Probability of female-to-male transmision of HIV-1 in Thailand. Lancet 434, 204–207.CrossRefGoogle Scholar
  28. McCoy, C.B., Lai, S., Metsch, L.R., Wang, X., Li, C., Yang, M., Yulong, L., 1997. No pain no gain, establishing the Kunming, China, drug rehabilitation center. Journal of Drug Issues 27, 73–85.Google Scholar
  29. Merli, M.G., Hertog, S., Wang, B., Li, J., 2004. Modelling the course of the HIV/AIDS epidemic in China: An application of a bio-behavioral macrosimulation model of the spread of HIV/AIDS. Center for Demography and Ecology Working Paper No. 2004–14, University of Wisconsin-Madison.Google Scholar
  30. Motulsky, H.J., Christopoulos, A., 2003. Fitting models to biological data using linear and nonlinear regression. A practical guide to curve fitting. GraphPad Software Inc., San Diego, CA, Online:http://www.graphpad.com.
  31. Monitoring the AIDS Pandemic Network 2004. AIDS in Asia: face the facts—a comprehensive analysis of the AIDS epidemics in Asia. Online:http://www.mapnet-work.org/docs/MAP_AIDSinAsia2004.pdf.
  32. National Bureau of Statistics of China, 1994–2003. China Population Statistics Yearbook. China Statistic Press, Beijing.Google Scholar
  33. Pan, S., Cheng, H., Zhang, J., Jia, M., Bi, X., Zhang, Y., Zhang, X., An, J., Ma, Y., Yang, Z., Liang, Y., Kou, J., 1997. Survey of the current situation of the HIV infection epidemic in Yunnan province. Chin. J. STD/AIDS Prev. Cont. 3, 244–247.Google Scholar
  34. Pugliese, A., 1992. Stationary solutions of a multigroup model for AIDS with distributed incubation and variable infectiousness. In Da Prato, G. (Ed.), Mathematical aspects of human diseases, Giardini, Pisa, pp. 110–125.Google Scholar
  35. Razak, M.H., 2002. Situation assessment of injection drug users in Yunnan province - People's Republic of China. Futures Group Europe Report.Google Scholar
  36. Thieme, H., Castillo-Chavez, C., 1989. On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic. In Castillo-Chavez, C. (Ed.), Mathematical and statistical approaches to AIDS epidemiology, Springer-Verlag, New York, pp. 157–177.Google Scholar
  37. Thieme, H., Castillo-Chavez, C., 1993. How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS? SIAM J. Appl. Math. 53, 1447–1479.MathSciNetGoogle Scholar
  38. Thieme, H., 2003. Mathematics in Population Biology. Princeton University Press, Princeton.MATHGoogle Scholar
  39. UN Theme Group on HIV/AIDS in China, 2002. HIV/AIDS: China's Titanic Peril. Online:http://www.youandaids.org/unfiles/chinastitanicperillast.pdf.
  40. UN Theme Group on HIV/AIDS in China and State Council AIDS Working Committee Office, 2004. A Joint Assessment of HIV/AIDS Prevention, Treatment and Care in China. Online: http://www.unchina.org/unaids/JAREng04.pdf.
  41. Wang, S., Li, L., Zhang, L., Ni, Z., Yang, L., 2001. Surveillance and analysis of HIV/AIDS in Kunming in 1999. Guangxi Prev. Med. 7, 157–158.Google Scholar
  42. Wang, S., 2004. Surveillance and control of HIV/AIDS in Kunming. Chin. J. STD/AIDS Prev. Cont. 10, 216–217.Google Scholar
  43. WHO Regional Office for the Western Pacific and Ministry of Health of the People's Republic of China, 2001. Prevalence survey of sexually transmitted infections among female sex workers and truck drivers in China 1999–2000. Online: http://www.wpro.who.int.
  44. Xinhua News Agency, 2004. Yunnan declares last-ditch war against AIDS. Online:http://news.xinhuanet.com/english/2004-02/23/content_1326621.htm.
  45. Xinhua News Agency, 2004. Yunnan province reports progress in HIV/AIDS prevention. Online:http://english.sina.com/china/1/2004/-1201/12161.html.
  46. Xinhua News Agency, 2005. Kunming conducts AIDS, venereal disease test in service sector. www.sina.com.Google Scholar
  47. Yu, H., An, X., Jia, M., Pan, S., Ma, Y., Zhang, G., Li, H., Zhang, X., Zhang, Y., Liang, Y., Li, J., Zhang, J., Cheng, H., 2001. Report on HIV/AIDS surveillance in Yunnan province in 1999. Chin. J. STD/AIDS Prev. Cont. 7, 74–76.Google Scholar
  48. Yuan, J., Ionita, G., Xu, Y., Jiang, T., Li, J., Zhang, J. 2002. The HIV/AIDS projection in Yunnan. Chin. J. STD/AIDS Prev. Cont. 8, 78–81.Google Scholar
  49. Zhang, J., Cheng, H., Duan, S., 1994. Survey of the current situation of the HIV infection epidemic in Yunnan province. Chin. J. Epidemiol. 5, 259–262.Google Scholar
  50. Zhang, J., Cheng, H., Jia, M., Zhang, Y., 1999. Ten years of experience on AIDS control in Yunnan (1989∼1998). Chin. J. Epidemiol. 20, 377–380.MATHGoogle Scholar
  51. Zhang, X., Ma, Y., Yu, H., An, J., Liang, Y., Zhang, G., Zhang, Z., Li, H., Wang, W., Pan, S., Zhang, Y., Jia, M., 2002. Analysis on the surveillance result of HIV/AIDS in Yunnan in 2001. Ji Bing Jian Ce 17, 327–330.Google Scholar
  52. Zhang, X., Lu, J., Fu, L., Luo, H., Jia, M., 2004. Analysis on survey results of HIV/AIDS in Yunnan Province in 2003. Ji Bing Jian Ce 19, 409–412.Google Scholar
  53. Zheng, X., Zhang, J., Wang, X., Duan, S., Qu, S., Duan, Y., Zhang, G., 2000. The natural history of HIV infection among IDUs in Ruili, Yunnan province, China. Chin. J. Epidemiol. 21, 17–18.Google Scholar
  54. Zhong, W., 2000. A close look at China's “sex industry.” Lianhe Zaobao, Singapore, Online:http://www.usembassy-china.org.cn/sandt/sex-industry.html.

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Nicolas Bacaër
    • 1
    • 4
  • Xamxinur Abdurahman
    • 2
  • Jianli Ye
    • 3
  1. 1.Institut de Recherche pour le Développement (IRD)BondyFrance
  2. 2.College of Mathematics and System SciencesXinjiang UniversityUrumqiP.R. China
  3. 3.Center for Public Health Surveillance and Information ServicesChinese Center for Disease Control and Prevention (CCDC)BeijingP.R. China
  4. 4.IRDBondy CedexFrance

Personalised recommendations