Research Article

Microbiology and Infectious Disease

A mechanistic model for long-term immunological outcomes in South African HIV-infected children and adults receiving ART

Department of Science and Technology and National Research Foundation, South African Centre for Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, South Africa
African Institute for Mathematical Sciences (AIMS), Next Einstein Initiative, Rwanda
Harriet Shezi Children’s Clinic, Wits Reproductive Health and HIV Institute, Faculty of Health Sciences, University of the Witwatersrand, South Africa
Kheth’Impilo AIDS Free Living, South Africa
Division of Epidemiology and Biostatistics, Department of Global Health, Faculty of Medicine and Health Sciences, Stellenbosch University, South Africa
Institute of Social and Preventive Medicine, University of Bern, Switzerland
Centre for Infectious Disease Epidemiology and Research, School of Public Health and Family Medicine, University of Cape Town, South Africa
Institute for Biomedical Engineering (IBE), Stellenbosch University, South Africa

Jan 14, 2021

https://doi.org/10.7554/eLife.42390

Open access
Copyright information

Figures
Tables
Additional files

6 figures, 7 tables and 7 additional files

Figures

Figure 1

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Data chart explaining the exclusion and inclusion criteria.

Figure 2

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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
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Figure 3 with 2 supplements

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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 3 – Table 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
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Figure 3—figure supplement 1

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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

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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
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Figure 4 with 2 supplements

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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 3 – Table 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
Download elife-42390-fig4-data1-v2.txt

Figure 4—figure supplement 1

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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

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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
Download elife-42390-fig4-figsupp2-data1-v2.txt

Appendix 1—figure 1

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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
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Appendix 1—figure 2

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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.

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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Tables

Table 1

Patient demographics.

Category	Variable, unit	Category	Full data set	Sample for scenario 1
Children
Demographic	Number of patients	All	19,060	1312
	Median age, years (IQR)	All	4.4 (1.1,8.8)	4.5 (1.4,7.9)
	Baseline WHO stage (% relative to all)	Stage I	693 (3.6%)	31 (2.3%)
		Stage II	1232 (6.4%)	108 (8.2%)
		Stage III	3968 (20.8%)	614 (46.7%)
		Stage IV	2756 (14.4%)	431 (32.8%)
	Median BMI z-scores	All	−0.85 (-2.2,0.2)	−0.76 (-1.98,0.26)
	Gender	Female	9606 (50.4%)	674 (51.3%)
	Gender	Male	9454 (49.6%)	638 (48.6%)
	Median time on ART, years (IQR)	All	2.0 (0.0,4.0)	4.0 (3.0,5.0)
	Year of ART initiation (IQR)		2004–2012	2004–2012
Clinical characteristics	CD4+ 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)	Yes	1673 (26%)	479 (36.5%)
		No	4764 (74%)	833 (63.5%)
Adults
Demographic	Number of patients	All	189,647	12,238
	Median age, years (IQR)	All	35 (29,42)	36 (30.6,42.9)
	Baseline WHO stage (% relative to all)	Stage I	20,711 (10.9%)	1588 (12.9%)
		Stage II	15,815 (8.3%)	817 (6.6%)
		Stage III	42,393 (22.3%)	4050 (33%)
		Stage IV	11,466 (6%)	1336 (10.9%)
	Gender	Female	124,006 (65.4%)	6944 (56.7%)
	Gender	Male	65,641 (34.6%)	5294 (43.2%)
	Median time on ART, years (IQR)	All	1.0 (0.0,3.0)	3.0 (2.0,5.0)
	Year of ART Initiation IQR		2004–2012	2004–2012
Clinical characteristics	Median 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)	Yes	10,746 (36.2%)	5011 (40.9%)
		No	18,945 (63.8%)	7227 (59.0%)

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 $z_{0}$ .

		Scenario 1: $z_{0}$ estimated		Scenario 2: $z_{0}$ as a predictor
		Ratio model	Asymptotic model	Ratio model	Asymptotic model
Adults	Sample size	12,238		14,542
Adults	BIC	−134,016.7	−126,716.2	−178,702.3	−186,244.2
Children	Sample size	1312		1616
Children	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%.

	Children		Adults
Model	Scenario 1: $z_{0}$ estimated (1312 subjects) BIC = −2,457.019	Scenario 2: $z_{0}$ as a predictor (1616 subjects) BIC = −6,156.762	Scenario 1: $z_{0}$ estimated (12,238 subjects) BIC = −136,284.7	Scenario 2: $z_{0}$ as a predictor (14,542 subjects) BIC = −180,557.8
Variable	Estimate (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, month	0.005 (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 load	0.06 (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 ( $z_{0}$ )	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)***	-
BMI	0.15 (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, month	0.008 (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 load	0.03 (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.

		K	Q	$z_{0}$	s	r
Scaled carrying capacity post ART – K	Adults	Scenario 2	0.40	-	0.48	−0.28
Scaled carrying capacity post ART – K	Children	Scenario 1	−0.03	-	0.57	0.01
Scaled carrying capacity healthy ind – Q	Adults	0.30	Scenario 2	-	0.37	−0.41
Scaled carrying capacity healthy ind – Q	Children	0.10	Scenario 1	-	−0.47	−0.24
Baseline scaled CD4+ T-cell count - $z_{0}$	Adults	−0.67	0.27	Scenario 2	-	-
Baseline scaled CD4+ T-cell count - $z_{0}$	Children	−0.85	0.38	Scenario 1	-	-
Rate of growth of CD4+ healthy ind – s	Adults	0.36	0.61	−0.30	Scenario 2	0.15
Rate of growth of CD4+ healthy ind – s	Children	0.45	−0.13	−0.52	Scenario 1	−0.22
Rate of growth of CD4+ post ART – r	Adults	−0.23	−0.33	−0.30	0.23	Scenario 2
Rate of growth of CD4+ post ART – r	Children	−0.03	−0.40	−0.25	0.32	Scenario 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/µL		Age (years)/CD4+ T-cell counts/µL		Age (years)/CD4+ T-cell counts/µL		Age (years)/CD4+ T-cell counts/µL
0	3244	1	2554	5	1392	9	1108	13	1085
3	3046	2	2083	6	1290	10	1118	14	1080
6	2866	3	1761	7	1220	11	1102	15	1077
9	2703	4	1557	8	1172	12	1095	16	1075

Appendix 1—table 2

Studies that evaluated normal CD4+ T-cells references ranges normal ranges in healthy African adult populations.

Reference	Value (range) by gender		Country	Age-related comments
Reference	Male	Female	Country	Age-related comments
Malaza et al., 2013, IQR	683 (542–849)	833 (660–1038)	Durban, South Africa	Median 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 Africa	Average age of 41 years for all
Ngowi et al., 2009, mean ± sd	665.6 ± 246.8	802 ± 250.2	Tanzania	Mean age for women is 30.9 years vs 35.2 for men
Lugada et al., 2004, 5th and 95th percentile	754 (362–1376)	894 (454–1485)	Uganda	Decrease 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 ± sd	838 ± 193	818 ± 213	Nigeria	No age effect found
Oladepo et al., 2009, median	746 (351–1455)	892 (383–1654)	Nigeria	Decreasing with age

Author response table 1

	Asymptote	Log of the rate of increase	Intercept	BIC
Min 3 observations# 17241 adults	0.632CV 0.70%	0.071CV 1.88%	0.142CV 0.77%	-148917
Min 4 observations14,542	0.58CV 0.63%	0.11CV 2.0%	0.14CV 0.81%	-136244
Min 5 observations12,238	0.66CV 0.77%	0.06CV 2.11%	0.14CV 0.91%	-126716