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Malaria Attributable to the HIV-1 Epidemic, Sub-Saharan Africa

Eline L. Korenromp,*† Brian G. Williams,* Sake J. de Vlas,† Eleanor Gouws,‡ Charles F. Gilks,* Peter D. Ghys,‡ and Bernard L. Nahlen*
*World Health Organization, Geneva, Switzerland; †Erasmus University Medical Center, Rotterdam, the Netherlands; and ‡Joint United Nations Programme on HIV/AIDS, Geneva, Switzerland

Appendix. Estimating Distribution of CD4 Cell Counts among HIV Patients over the Course of the Epidemic

When HIV is first introduced into a population, most of the infected persons must have been recently infected and still have high CD4 cell counts. The CD4 cell count distribution among HIV-positive persons later in the epidemic is determined by, first, the frequency distribution of the times since infection among HIV-positive people at that time, and second, the frequency distribution of CD4 cell counts among HIV-positive people at different times since infection.

We estimate the frequency distribution of the times since infection by modeling HIV incidence and deaths from its prevalence, as a function of calendar time. Trends over time in prevalence of HIV infection were estimated by fitting double logistic curves to the data of the prevalence of HIV among women attending antenatal clinics. To convert prevalence to incidence, we assumed that the survival distribution of HIV-1 infected adults is a Weibull function with median of 9.0 years (1) and a shape parameter of 2.28 (2).

If no one died of AIDS, the incidence, I₀(t), would be equal to the time derivative of the prevalence curve, P(t), so that

     1

To correct for the effect of AIDS-related mortality, if the proportion of persons who survive for t years after infection is W(t), the probability that persons die t years after they are infected is

         2

Since deaths will decrease prevalence, we add deaths due to all previous incident infections to our estimate of incidence so that

      3

Equation 3 can be solved using Fourier transforms (indicated with curly-brackets) so that

      4

The probability that a person who is alive and HIV positive at the present time t was infected with HIV at time t − τ is

   5

enabling us to calculate the probability density of times since infection as a function of calendar time.

For the density function of CD4 cell counts as a function of time since HIV-1 infection, data from a variety of populations not receiving antiretroviral therapy suggest that the decline in CD4 counts is approximately linear throughout the survival with HIV-1 (2,3). If the CD4 cell counts, c, start from a nominal value of 1 and if τ is the time of death, the expected value of c is

          6

where the probability density function of τ is

     7

and

           8

Equation 8 is easily evaluated analytically. Scaling the CD4 cell decline curve to observed mean cell counts (in this study, 825 cells/μL at seroconversion, the median value found in African HIV-1 uninfected adults [3–7], and 20/μL at death from AIDS [8]), this gives the probability density function of the CD4 cell counts as a function of time since infection for different calendar times.

Appendix References

1. Improved methods and assumptions for estimation of the HIV/AIDS epidemic and its impact: Recommendations of the UNAIDS Reference Group on Estimates, Modelling and Projections. AIDS. 2002;16:W1–14.
2. Williams BG, Dye C. Antiretroviral drugs for tuberculosis control in the era of HIV/AIDS. Science. 2003;301:1535–7.
3. Urassa W, Bakari M, Sandstrom E, Swai A, Pallangyo K, Mbena E, et al. Rate of decline of absolute number and percentage of CD4 T lymphocytes among HIV-1-infected adults in Dar es Salaam, Tanzania. AIDS. 2004;18:433–8.
4. Kelly P, Zulu I, Amadi B, Munkanta M, Banda J, Rodrigues LC, et al. Morbidity and nutritional impairment in relation to CD4 count in a Zambian population with high HIV prevalence. Acta Trop. 2002;83:151–8.
5. Levin A, Brubaker G, Shao JS, Kumby D, O'Brien TR, Goedert JJ, et al. Determination of T-lymphocyte subsets on site in rural Tanzania: results in HIV-1 infected and non-infected individuals. Int J STD AIDS. 1996;7:288–91.
6. Tsegaye A, Messele T, Tilahun T, Hailu E, Sahlu T, Doorly R, et al. Immunohematological reference ranges for adult Ethiopians. Clin Diagn Lab Immunol. 1999;6:410–4.
7. Tsegaye A, Messele T, Tilahun T, Hailu E, Sahlu T, Doorly R, et al. Immunohematological reference ranges for adult Ethiopians. Clin Diagn Lab Immunol. 1999;6:410–4.
8. Morgan D, Mahe C, Mayanja B, Okongo JM, Lubega R, Whitworth JA. HIV-1 infection in rural Africa: is there a difference in median time to AIDS and survival compared with that in industrialized countries? AIDS. 2002;16:597–603.

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This page posted August 18, 2005
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