1. Microbiology and Infectious Disease
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A mechanistic model for long-term immunological outcomes in South African HIV-infected children and adults receiving ART

  1. Eva Liliane Ujeneza  Is a corresponding author
  2. Wilfred Ndifon
  3. Shobna Sawry
  4. Geoffrey Fatti
  5. Julien Riou
  6. Mary-Ann Davies
  7. Martin Nieuwoudt
  8. IeDEA-Southern Africa collaboration
  1. Department of Science and Technology and National Research Foundation, South African Centre for Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, South Africa
  2. African Institute for Mathematical Sciences (AIMS), Next Einstein Initiative, Rwanda
  3. Harriet Shezi Children’s Clinic, Wits Reproductive Health and HIV Institute, Faculty of Health Sciences, University of the Witwatersrand, South Africa
  4. Kheth’Impilo AIDS Free Living, South Africa
  5. Division of Epidemiology and Biostatistics, Department of Global Health, Faculty of Medicine and Health Sciences, Stellenbosch University, South Africa
  6. Institute of Social and Preventive Medicine, University of Bern, Switzerland
  7. Centre for Infectious Disease Epidemiology and Research, School of Public Health and Family Medicine, University of Cape Town, South Africa
  8. Institute for Biomedical Engineering (IBE), Stellenbosch University, South Africa
Research Article
Cite this article as: eLife 2021;10:e42390 doi: 10.7554/eLife.42390
6 figures, 7 tables and 7 additional files

Figures

Data chart explaining the exclusion and inclusion criteria.
Plot of the logistic growth model.

The dotted red line represents that carrying capacity k, while the dotted blue line is at the inflexion point k/2.

Figure 2—source data 1

Data source to reproduce the plot for the logistic growth model.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig2-data1-v2.txt
Figure 3 with 2 supplements
Children population-level CD4 trajectory, as estimated by the unadjusted ratio and asymptotic models, and the adjusted ratio model.

Simulation of population-level CD4+ count trajectory for children, from unadjusted fixed estimates of the asymptotic model (AM) in blue and the ratio model (RM) in green. The red line represents simulation from the adjusted population-level RM estimates. Parameters used for the AM are presented in Supplementary file 3Table 1, scenario 1. Those used for the RM are estimated fixed effect for the null model (not shown in the paper): K = 3.4, Q = 0.9, r = 0.35, s = 0.017, z0 = 0.18. Fixed effect presented in Table 3 (scenario 1) are used for the adjusted ratio model (Adj RM), for baseline covariates z-score BMI, age, log viral load; and sex and suppression of viral load within 12 months of starting therapy. Convergence plots for the Adj RM are given in Figure 3—figure supplement 1, and simulation of individual fits in Figure 3—figure supplement 2.

Figure 3—source data 1

Data source to reproduce the population-level CD4 trajectories plot for children.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig3-data1-v2.txt
Figure 3—figure supplement 1
Convergence plots for children adjusted ratio model.

The x-axis represents the number of iterations, while the y-axis is the parameter value. The red vertical line indicates the end of the 300 iterations, where the algorithm explores freely the parameter space. The second phase has 150 iterations, where the step size is gradually decreased in order to ensure convergence.

Figure 3—figure supplement 2
Sample of individual children plots for the adjusted ratio model.

TOFU on the x-axis stands for the time of follow-up since ART initiation. The points are individual scaled CD4+ counts. The dashed line and the solid line represent the population-level and individual-level fits respectively.

Figure 3—figure supplement 2—source data 1

Data source to reproduce the individual-level CD4 trajectories plot for children.

Data source to reproduce the individual-level CD4 trajectories plot for adults.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig3-figsupp2-data1-v2.txt
Figure 4 with 2 supplements
Adults population-level CD4 trajectory, as estimated by the unadjusted ratio and asymptotic models, and the adjusted ratio model.

Simulation of population-level CD4+ count trajectory for adults, from unadjusted fixed estimates of the asymptotic model (AM) in blue, and the ratio model (RM) in green. The red line represents simulation from the adjusted population-level RM estimates. Parameters used for the AM are presented in Supplementary file 3Table 1, scenario 1. Those used for the RM are estimated fixed effect for the null model (not shown in the paper): K = 2.54, Q = 0.38, r = 1.23, s = 0.01, z0 = 0.13. Fixed effect presented in Table 3 (scenario 1) are used for the adjusted ratio model (Adj RM), for baseline covariates sex, age, log viral load, and suppression of viral load within 12 months of starting therapy. Convergence plots for the Adj RM are given in Figure 4—figure supplement 1, and simulation of individual fits in Figure 4—figure supplement 2.

Figure 4—source data 1

Data source to reproduce the population-level CD4 trajectories plot for adults.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig4-data1-v2.txt
Figure 4—figure supplement 1
Convergence plots for adults adjusted ratio model.

The x-axis represents the number of iterations, while the y-axis is the parameter value. The red vertical line indicates the end of the 300 iterations, where the algorithm explores freely the parameter space. The second phase has 150 iterations, where the step size is gradually decreased in order to ensure convergence.

Figure 4—figure supplement 2
Sample of individual adult plots for the adjusted ratio model.

TOFU on the x-axis stands for the time of follow-up since ART initiation. The points are an individual scaled CD4+ counts. The dashed line and the solid line represent the population-level and individual-level fits, respectively. The table below indicates the clinical details about the patients represented above, in their respective index location.

Figure 4—figure supplement 2—source data 1

Data source to reproduce the individual—level CD4 trajectories plot for adults.

https://cdn.elifesciences.org/articles/42390/elife-42390-fig4-figsupp2-data1-v2.txt
Appendix 1—figure 1
Plot of the simulated reference values for children.

The dots represent the cross-sectional data for healthy children. The fitted red line shows the age-dependent reference values used in the scaling of CD4+ counts of HIV-infected children.

Appendix 1—figure 1—source data 1

Simulated reference values for children.

https://cdn.elifesciences.org/articles/42390/elife-42390-app1-fig1-data1-v2.csv
Appendix 1—figure 2
Plot of the simulated reference values for adults.

The points represent the published median values. The red line shows the CD4+ count for women, blue line is for men. CD4+ reference values were simulated yearly, for ages ranging between 17 and 95.

Tables

Table 1
Patient demographics.
CategoryVariable, unitCategoryFull data setSample for scenario 1
Children
DemographicNumber of patientsAll19,0601312
Median age, years (IQR)All4.4 (1.1,8.8)4.5 (1.4,7.9)
Baseline WHO stage
(% relative to all)
Stage I693 (3.6%)31 (2.3%)
Stage II1232 (6.4%)108 (8.2%)
Stage III3968 (20.8%)614 (46.7%)
Stage IV2756 (14.4%)431 (32.8%)
Median BMI z-scoresAll−0.85 (-2.2,0.2)−0.76 (-1.98,0.26)
GenderFemale9606 (50.4%)674 (51.3%)
Male9454 (49.6%)638 (48.6%)
Median time on ART, years (IQR)All2.0 (0.0,4.0)4.0 (3.0,5.0)
Year of ART initiation (IQR)2004–20122004–2012
Clinical characteristicsCD4+ T-cell count at baseline, count/µL (IQR)493 (229,890)404 (159,706)
Median scaled CD4+ T-cell count at baseline, (IQR)0.30 (0.15,1.50)0.24 (0.09,0.42)
Median viral load at baseline, per 1000 copies/mL (IQR)150 (20,7342)155 (29,670)
Median log viral load at baseline, copies/mL (IQR)5.1 (4.3,6.8)5.1 (4.4,5.8)
Number of patients that suppressed viral load within 12 months of treatment initiation (% relative to non-missing)Yes1673
(26%)
479
(36.5%)
 No4764
(74%)
833
(63.5%)
Adults
DemographicNumber of patientsAll189,64712,238
Median age, years (IQR)All35 (29,42)36 (30.6,42.9)
Baseline WHO stage
(% relative to all)
Stage I20,711 (10.9%)1588 (12.9%)
Stage II15,815 (8.3%)817 (6.6%)
Stage III42,393 (22.3%)4050 (33%)
Stage IV11,466 (6%)1336 (10.9%)
GenderFemale124,006 (65.4%)6944 (56.7%)
Male65,641 (34.6%)5294 (43.2%)
Median time on ART, years (IQR)All1.0 (0.0,3.0)3.0 (2.0,5.0)
Year of ART Initiation IQR2004–20122004–2012
Clinical characteristicsMedian CD4+ T-cell count at baseline, count/µL (IQR)140 (67,206)128 (62,196)
Median scaled CD4+ T-cell count at baseline (IQR)0.18 (0.08,0.26)0.16 (0.07,0.24)
Median viral load at baseline, per 1000 copies/mL (IQR)27 (0.8,132.3)39 (2.3, 151)
Median log viral load at baseline, per CD4+ category, copies/mL (IQR)4.4 (2.9,5.1)4.5 (3.3,5.1)
Number of patients that suppressed viral load within 12 months of treatment initiation (% relative to relative non missing)Yes10,746
(36.2%)
5011
(40.9%)
 No18,945
(63.8%)
7227
(59.0%)
  1. Note: A z-score BMI of −1 indicates that the child’s body mass index is at one standard deviation below the body mass of a healthy child, while a z-score BMI of 0 means that the child has normal body mass for his/her age.

Table 2
BIC comparison of the unadjusted ratio and asymptotic models, under different scenarios for baseline scaled CD4+ T-cell counts z0.
Scenario 1: z0 estimatedScenario 2: z0 as a predictor
Ratio modelAsymptotic modelRatio modelAsymptotic model
AdultsSample size12,23814,542
 BIC−134,016.7−126,716.2−178,702.3−186,244.2
ChildrenSample size13121616
 BIC−2,137.1−1,523.1−6,028.8−7,969.7
Table 3
Ratio model estimated parameters for children and adults.

*** means significant at 99%, and ** significant at 95%.

ChildrenAdults
ModelScenario 1: z0 estimated
(1312 subjects)
BIC = −2,457.019
Scenario 2: z0 as a predictor
(1616 subjects)
BIC = −6,156.762
Scenario 1: z0 estimated
(12,238 subjects)
BIC = −136,284.7
Scenario 2: z0 as a predictor
(14,542 subjects)
BIC = −180,557.8
VariableEstimate (95% CI)Estimate (95% CI)Estimate (95% CI)Estimate (95% CI)
Scaled carrying capacity post ART (K)1.03 (0.73,1.34)1.13 (0.84, 1.41)1.75 (1.59, 1.90)1.69 (1.57, 1.80)
Sex, ref is male--−0.06 (−0.09,–0.03)***-
Age, month0.005 (0.004, 0.007)***0.0039 (0.002, 0.004)***−0.0007 (−0.008,–0.0006)***−0.0006 (−0.0007,–0.0004)***
BMI−0.12 (−0.15,–0.08)***−0.09 (−0.12,–0.06)***--
Log viral load0.06 (0.04, 0.08)***0.07 (0.05, 0.09)***0.069 (0.065, 0.073)***0.061 (0.058, 0.065)***
Suppress, ref is no-0.001 (−0.096, 0.092)**0.11 (0.08, 0.14)***0.12 (0.10, 0.15)***
Scaled carrying capacity healthy individuals (Q)0.68 (0.57, 0.78)2.02 (1.59, 2.44)0.49 (0.47, 0.52)0.46 (0.36,0.56)
Sex, ref is male--−0.55 (−0.63,–0.47)***−0.54 (−0.63,–0.46)***
Age, month−0.005 (0.003, 0.007)***--0.0002 (−0.0001, 0.0006)
 Scaled CD4+ T-cell count at ART initiation (z0)0.74 (0.49, 1.00)-0.20 (0.18, 0.22)-
Sex, ref is male--0.08 (0.05, 0.12)***-
Age, month−0.007 (−0.009,–0.006)***-0.0006 (0.0004,0.0008)***-
BMI0.15 (0.11, 0.19)***---
Log viral load−0.070 (−0.096,–0.044)***-−0.081 (−0.087,–0.076)***-
Rate of growth of CD4+ healthy individuals, cells per μL per day (s)0.011 (0.007, 0.016)0.002 (0.001, 0.003)0.022 (0.016, 0.028)0.02 (0.01,0.03)
Sex, ref is male--−0.40 (−0.53,–0.28)***−0.50 (−0.62,–0.37)***
Age, month0.008 (0.004, 0.017)***0.015 (0.010, 0.020)***−0.0012 (−0.0017,–0.0007)***−0.0016 (−0.0022,–0.0010)***
Rate of growth of CD4+ post ART, cells per μL per day (r)0.27 (0.160,0.32)0.17 (0.12, 0.21)0.49 (0.39, 0.60)0.79 (0.60, 0.97)
Sex, ref is male-−0.19 (−0.33,–0.04)**
Age, month−0.001 (−0.003, 0.0001)**---
BMI−0.08 (−0.12,–0.04)***−0.07 (−0.11,–0.03)***--
Log viral load0.03 (0.00, 0.62)**0.05 (0.03, 0.08)***0.11 (0.09, 0.13)***0.11 (0.09, 0.13)***
Suppress, ref is no−0.12 (-−0.27, 0.02)−0.15 (−0.28,–0.03)***−0.57 (−0.70,–0.44)***−0.62 (−0.76,–0.47)***
Table 4
Correlations of individual random effects.

Scenario 1 below the diagonal. Scenario 2 above.

KQz0sr
Scaled carrying capacity post ART – KAdultsScenario 20.40-0.480.28
ChildrenScenario 10.03-0.570.01
Scaled carrying capacity healthy ind – QAdults0.30Scenario 2-0.37−0.41
Children0.10Scenario 1-−0.47−0.24
Baseline scaled CD4+ T-cell count - z0Adults−0.670.27Scenario 2--
Children−0.850.38Scenario 1--
Rate of growth of CD4+ healthy ind – sAdults0.360.61−0.30Scenario 20.15
Children0.45−0.13−0.52Scenario 1−0.22
Rate of growth of CD4+ post ART – rAdults0.23−0.33−0.300.23Scenario 2
Children0.03−0.40−0.250.32Scenario 1
  1. In red: Opposite direction.

    Underlined: Difference of correlation between children and adults.

Appendix 1—table 1
Sample of simulated healthy children’s CD4+ count values.
Age (months)/
CD4+ T-cell counts/µL
Age (years)/CD4+ T-cell counts/µLAge (years)/CD4+ T-cell counts/µLAge (years)/CD4+ T-cell counts/µLAge (years)/CD4+ T-cell counts/µL
03244125545139291108131085
330462208361290101118141080
628663176171220111102151077
927034155781172121095161075
Appendix 1—table 2
Studies that evaluated normal CD4+ T-cells references ranges normal ranges in healthy African adult populations.
ReferenceValue (range) by genderCountryAge-related comments
 MaleFemale
Malaza et al., 2013, IQR683 (542–849)833 (660–1038)Durban, South AfricaMedian age for women is 35 years vs 23 years for men. CD4+ count increase slightly with age till 64 years old
Lawrie et al., 2009, 2.5th and 97.5th percentile(503–1807)(561–2051)Gauteng, South AfricaAverage age of 41 years for all
Ngowi et al., 2009, mean ± sd665.6 ± 246.8802 ± 250.2TanzaniaMean age for women is 30.9 years vs 35.2 for men
Lugada et al., 2004, 5th and 95th percentile754 (362–1376)894 (454–1485)UgandaDecrease with age from birth to 18 years. All aged 24 were lumped together
Institute of Human Virology/Plateau State Specialist Hospital AIDS Prevention in Nigeria Study Team et al., 2005, mean ± sd838 ± 193818 ± 213NigeriaNo age effect found
Oladepo et al., 2009, median746 (351–1455)892 (383–1654)NigeriaDecreasing with age
Author response table 1
AsymptoteLog of the rate of increaseInterceptBIC
Min 3 observations# 17241 adults0.632CV 0.70%0.071CV 1.88%0.142CV 0.77%-148917
Min 4 observations14,5420.58CV 0.63%0.11CV 2.0%0.14CV 0.81%-136244
Min 5 observations12,2380.66CV 0.77%0.06CV 2.11%0.14CV 0.91%-126716

Data availability

Data used are from the International Epidemiologic Databases to Evaluate AIDS Southern Africa collaboration. They maintain a database of routinely collected data from various clinics, mostly located in South Africa. We recommend that interested readers contact Dr Morna Cornell, Project Manager IeDEA-SA in Cape Town (morna.cornell@uct.ac.za) to establish a data-sharing agreement. A research proposal highlighting how the data will be used is required. Source data for figures and figure supplements are provided, and the source code is available at https://github.com/EvaLiliane/RM_Code_eLife copy archived at https://archive.softwareheritage.org/swh:1:rev:624ff31c5fc969885f29b7291ee06886d24c64f7/.

Additional files

Supplementary file 1

Range of the ratio model parameters.

https://cdn.elifesciences.org/articles/42390/elife-42390-supp1-v2.docx
Supplementary file 2

Comparison of parameter estimates for adults, obtained using different fitting scenarios.

https://cdn.elifesciences.org/articles/42390/elife-42390-supp2-v2.docx
Supplementary file 3

Evaluation of the variance–covariance matrix for the ratio and asymptotic models.

https://cdn.elifesciences.org/articles/42390/elife-42390-supp3-v2.docx
Supplementary file 4

Additional estimated parameters.

https://cdn.elifesciences.org/articles/42390/elife-42390-supp4-v2.docx
Transparent reporting form
https://cdn.elifesciences.org/articles/42390/elife-42390-transrepform-v2.docx
Appendix 1—figure 1—source data 1

Simulated reference values for children.

https://cdn.elifesciences.org/articles/42390/elife-42390-app1-fig1-data1-v2.csv
Appendix 1—figure 2—source data 1

Simulated reference values for adults.

https://cdn.elifesciences.org/articles/42390/elife-42390-app1-fig2-data1-v2.csv

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