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

Mohamed BenSalah, Salih Tatar, Suleyman Ulusoy

TL;DR

An Iterative Regularizing Ensemble Kalman Method (IREKM) is developed, which enables the simultaneous estimation of multiple diffusion terms without requiring gradient information, and provides a computational approach for parameter estimation in fractional diffusion models.

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.

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

TL;DR

An Iterative Regularizing Ensemble Kalman Method (IREKM) is developed, which enables the simultaneous estimation of multiple diffusion terms without requiring gradient information, and provides a computational approach for parameter estimation in fractional diffusion models.

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.

Paper Structure

This paper contains 14 sections, 3 theorems, 45 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The direct and the inverse problems
  3. Bayesian framework
  4. Numerical experiments
  5. Numerical solutions of the forward problem
  6. Approximation method
  7. Validation test
  8. Numerical solutions of the inverse problem
  9. Reconstruction results
  10. Example 1.
  11. Example 2.
  12. Example 3.
  13. Stability and convergence
  14. Concluding Remarks

Key Result

Theorem 2.1

Let $D_i, i = 1, 2, 3 \in \mathbb{D}$ and $(u_1, u_2, u_3) \in L^2\left( 0, T; H_0^1(\Omega) \right) \cap W_2^\beta \left( 0, T; L^2(\Omega) \right)^3$ be the unique weak solution to the direct problem (eq1). Then the following estimates holds:

Figures (5)

  • Figure 1: Comparison of Exact and Approximated Solutions for $\alpha=0.5$ with $h_x=h_t=0.001$. The top row depicts the exact solutions, while the bottom row illustrates the approximated solutions.
  • Figure 2: The numerical results of $D_1(\cdot)$ (left), $D_2(\cdot)$ (middle) and $D_3(\cdot)$ (right) for different noises $\sigma$ with $\alpha=0.7$ in Example 1.
  • Figure 3: The numerical results of $D_1(\cdot)$ (left), $D_2(\cdot)$ (middle) and $D_3(\cdot)$ (right) for different noises $\sigma$ with $\alpha=0.5$ in Example 2.
  • Figure 4: The numerical results of $D_1(\cdot)$ (left), $D_2(\cdot)$ (middle) and $D_3(\cdot)$ (right) for different noises $\sigma$ with $\alpha=0.2$ in Example 3.
  • Figure 5: Logarithmic relative errors ($E_n$, top row) and logarithmic residuals ($R_n$, bottom row) for different noise levels: $\sigma = 0.0001$, $\sigma = 0.001$, and $\sigma = 0.01$. From left to right: results for Example 1, Example 2 and Example 3.

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3