Bayesian Inference for a Time-Fractional HIV Model with Nonlinear Diffusion

Abstract: This study investigates an inverse problem associated with a time-fractional HIV infection model incorporating nonlinear diffusion. The model describes the dynamics of uninfected target cells, infected cells, and free virus particles, where the diffusion terms are nonlinear density functions. The primary objective is to recover the unknown diffusion functions by utilizing final-time measurement data. Due to the inherent ill-posedness of the inverse problem and the presence of measurement noise, we employ a Bayesian inference framework to obtain stable and reliable estimates while quantifying uncertainty. To solve the inverse problem efficiently, we develop an Iterative Regularizing Ensemble Kalman Method (IREKM), which enables the simultaneous estimation of multiple diffusion terms without requiring gradient information. Numerical experiments validate the effectiveness of the proposed method in reconstructing the unknown diffusion terms under different noise levels, demonstrating its robustness and accuracy. These findings contribute to a deeper understanding of HIV infection dynamics and provide a computational approach for parameter estimation in fractional diffusion models.