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

• Nicolas Bacaër
• Xamxinur Abdurahman
• Jianli Ye
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 R 0 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

92C60 35Q80

## 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.
2. Bacaër, N., 2003. The asymptotic behavior of the McKendrick equation with immigration. Math. Popul. Stud. 10, 1–20.
3. Berman, A., Plemmons, R.J., 1994. Nonnegative Matrices in the Mathematical Sciences. Society for Industrial and Applied Mathematics, Philadelphia.
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.
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.
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.
12. Cohen, J., 2004b. HIV/AIDS in China—Changing course to break the HIV-heroin connection. Science 304, 1434–1435.
13. Cohen, J., 2004c. HIV/AIDS in Asia—Asia and Africa: On different trajectories? Science 304, 1932–1938.
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.
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.
17. Hethcote, H.W., Thieme, H.R., 1985. Stability of the endemic equilibrium in epidemic models with subpopulations. Math. Biosci. 75, 205–227.
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.
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.
22. Hyman, J.M., Stanley, E.A., 1988. Using mathematical models to understand the AIDS epidemic. Math. Biosci. 90, 415–473.
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.
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.
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.
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.
38. Thieme, H., 2003. Mathematics in Population Biology. Princeton University Press, Princeton.
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.
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.

## Authors and Affiliations

• Nicolas Bacaër
• 1
• 4
Email author
• 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