Sensitivity Study of Parameters in a Mathematical Model of HIV Infection Using Clinical Data

Abstract

Modeling the early stages of HIV infection is essential for understanding virus dynamics and improving treatment strategies. In this project, we solve a system of three ordinary differential equations (ODEs) that describe the interactions between uninfected immune cells, infected cells, and the viral load. Our goal is to reconstruct key parameters and optimize the model using real clinical data from four patients. The study is divided into two phases. In the first phase, we focus on estimating the viral load V (t) by solving an inverse problem with a time-adaptive optimization method. Two test cases, Test 1 and Test 2, are evaluated, and we find that Test 2 provides better accuracy across all four patients. The results show a good fit between the computed and measured viral load up to day 50, confirming the effectiveness of Test 2 in capturing early infection dynamics. In the second phase, we extend the model by including the total immune response Σ = T + I, which represents the sum of uninfected and infected CD4+ T cells. We continue using Test 2, as it provided the best accuracy in the first phase. Additionally, we modify the adjoint equations by incorporating an extra term derived from the adjoint problem to improve the optimization process. The model is applied to clinical data from four patients to analyze how well it reconstructs their immune response and viral load dynamics. An interesting result emerges from the numerical experiments. In many cases, the optimization improves the fit for both the viral load V (t) and the immune response Σ = T+I at the same time. This suggests that the model and the conjugate gradient algorithm can reconstruct both variables in parallel with good accuracy. However, in more challenging situations such as for Patient 1 and 4, it is still difficult to achieve a perfect match for both quantities. These findings highlight the importance of having a good initial guess and confirm the reliability of the optimization method, even though further refinements may still be necessary. From a theoretical point of view, we implement a time-adaptive reconstruction method that improves numerical stability and allows better control of the parameters. The optimization is based on minimizing a Tikhonov functional, where the regularization term is chosen to balance data fitting and smoothness. Adding extra terms from the adjoint equations improves the reconstruction but also makes it harder to achieve good results for both state variables at the same time. Overall, our results show that choosing test cases and optimization strategies carefully is important in order to reconstruct both V (t) and Σ = T +I effectively. Time adaptivity improves numerical performance, but more improvements are needed to consistently match both outputs. This study contributes to building better mathematical models for HIV infection, which can be useful in clinical prediction and treatment planning.