Modeling the impact of antiretroviral therapy on HIV and related kidney diseases in Tanzania

Abstract

This work presents a mathematical model for the dynamics of HIV-related kidney diseases. The study examines two cases, considering the effects of the absence of treatment and the effects of Highly Active Antiretroviral Therapy (HAART) on different infection groups. Studying these cases is important because many developing countries implement HAART late, and not all HIV-infected individuals receive this therapy. Kidney diseases in HIV individuals are modeled as arising from both HIV infection itself and the use of nephrotoxic drugs. In the analysis of the mathematical model, it is shown that the state variables of the model are non-negative and bounded. Furthermore, we derived a formula for control reproduction number [Formula: see text] which was used to compare the cases considered. The sensitivity analysis of the model reveals that the effect of HAART in reducing the progression from HIV to HIV-related kidney diseases is more significant compared to other effects of HAART on disease dynamics, which is also confirmed through numerical simulations. The Markov Chain Monte Carlo (MCMC) method was used to estimate parameters and evaluate the model using real data of the HIV population from Tanzania from 1990 to 2022. Numerical simulations demonstrated that while HAART reduces HIV progression to the AIDS stage, the population of individuals with HIV-related kidney diseases is increasing and is projected to continue increasing. Additionally, the results show that improving the effectiveness of HAART by 90% in preventing the progression from HIV to HIV-related kidney diseases can significantly decrease the prevalence of HIV-related kidney diseases. This study addresses a gap in understanding how population-level HAART availability influences the dynamics of HIV-related kidney disease, particularly in settings with delayed or inconsistent treatment access. By analyzing disease progression under these conditions, our findings provide insights that can inform public health strategies for improving HIV care in resource-limited settings and other contexts where access disparities persist. In addition, the work investigated scenarios related to data quality in which the model parameters can be well identified, which can serve as a guide for obtaining informative real data.

Keywords: HIV-related kidney diseases; Markov chain Monte Carlo; Mathematical model; Reproduction number; Sensitivity analysis.

Fig. 1

Flow diagram of HIV/AIDS-related kidney diseases.

Fig. 2

Synthetics data and model with different levels of noise by considering two compartments (Susceptible and HIV); (a) for (b) for .

Fig. 3

Pairwise chain parameters; red for four compartments, green for two compartments (Susceptible and HIV) and blue for one compartment (HIV).

Fig. 4

Pairwise chain parameters for two compartments (Susceptible and HIV) with different data set; red for 1000 data points and blue for 33 data points.

Fig. 5

Pairwise chain parameters for 33 data sample; red for four compartments, green for two compartments, and blue for one compartment (HIV).

Fig. 6

Pairwise chain parameters for 1000 data points; red for four compartments, green for two compartments, blue for one compartment (HIV).

Fig. 7

Pairwise chain parameters for two compartments with different data; red for 1000 data and blue for 33 data.

Fig. 8

Plot of the synthetics data-fitted model solutions together with the uncertainties. Circles represent data points for susceptible and HIV in stages 1 and 2. Solid lines represent a posterior mean solution and grey areas represent solutions from different parameter uncertainties.

Fig. 9

(a) Trace plot of mean value of the parameters. (b) Two-dimensional marginal posterior distributions of the model parameters. (c) Histogram of the chain for posterior error standard deviation with prior as a line.

Fig. 10

Fitted curve for HIV in stage one and two from 1990 to 2022. The squares denote the data and the solid lines show the median fits (the posterior means of the model parameters), while the lighter areas present the same uncertainty level in predicting new observations.

Fig. 11

Results of parameter estimation for Case II parameters: (a) Trace plot (b) Pairwise correlation.

Fig. 12

Predictive envelopes of the model with seven parameters from . (a) represents fitted HIV data with a posterior range of confidence interval, (b) represents a population with different compartments by using posterior mean, and (c) predictive graph with random posterior value for susceptible, HIV, AIDS, and HIV/AIDS-related kidney diseases.

Fig. 13

Projection of the total population considering various therapy effects, also including a scenario where the therapy is completely effective (effect of therapy is 1), from 2023 to 2050.

Fig. 14

Projection of HIV-related kidney diseases with different effects of therapy based on the confidence interval and posterior mean in Table 4. (a) effect of HAART to block the progression from HIV to AIDS varies, (b) effect of HAART to reduce the progression from AIDS to HIV to HIV+ kidney diseases varies (c) effect of HAART to reduce mortality due to HIV+ kidney diseases varies (d) effect of HAART to reduce mortality due to AIDS varies. For all cases, the posterior means of the non-varied parameters were used as fixed values for the simulations.

Fig. 15

Plots of projection of infectious groups, when the effect of therapy in reducing progression from HIV to HIV-related kidney diseases () varies within the confidence interval values provided in Table 4, other parameters are assigned their posterior mean values. Additionally, the analysis extends to consider the scenario where therapy is fully effective ().

Fig. 16

Comparison plots of infection groups over 33 years for two scenarios using the posterior mean values from Tables 3 and 4. (a) HIV without symptom population, (b) AIDS population, (c) population of HIV/AIDS with kidney diseases.

Fig. 17

Plots of projection of comparison of infectious groups for two scenarios using the posterior mean values from Tables 3 and 4. (a) HIV without symptom population, (b) AIDS population, (c) population of HIV/AIDS with kidney diseases.

Fig. 18

Comparison plots of infection groups for two scenarios: one with no HAART and the other where the effect of HAART is greater than 0.9. (a) HIV without symptom population, (b) AIDS population, (c) HIV/AIDS related kidney diseases population.

Fig. 19

Sensitivity indices by using the posterior mean value from Table 5.