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.

翻译：本研究探讨了包含非线性扩散项的时间分数阶HIV感染模型的逆问题。该模型描述了未感染靶细胞、感染细胞和游离病毒颗粒的动力学行为，其中扩散项采用非线性密度函数。主要目标是通过利用终时刻测量数据来重构未知的扩散函数。鉴于逆问题固有的不适定性以及测量噪声的存在，我们采用贝叶斯推断框架以获得稳定可靠的估计并量化不确定性。为高效求解该逆问题，我们提出了一种迭代正则化集成卡尔曼方法（IREKM），该方法无需梯度信息即可实现多个扩散项的同时估计。数值实验验证了所提方法在不同噪声水平下重构未知扩散项的有效性，证明了其鲁棒性与准确性。这些发现有助于深化对HIV感染动力学的理解，并为分数阶扩散模型中的参数估计提供了一种计算方法。