Niaz Ali Shah | Searcharxiv

arXiv2026

Fractal-Fractional HIV Dynamics with Mittag-Leffler Kernel: Analysis, Stability, and Numerical Simulations

In this paper, a fractal--fractional HIV model with the Mittag--Leffler kernel is proposed using the Atangana--Baleanu--Caputo operator to capture the memory and hereditary properties of the disease dynamics. The existence and uniqueness of the solutions are investigated using suitable analytical techniques, and the Hyers--Ulam stability analysis is carried out to verify the stability behavior of the proposed system. For the numerical simulations, the Newton polynomial approximation method together with the Atangana--Toufik numerical scheme is employed to obtain approximate solutions for different parameter settings. Furthermore, several visualization techniques, including sensitivity heatmap representation and tornado diagram analysis, are utilized to study the influence of model parameters on the HIV dynamics. The obtained numerical results demonstrate that the proposed fractal--fractional framework provides an effective and reliable approach for analyzing the transient and long-term behavior of HIV transmission dynamics.

math.NA↗

arXiv2026

State dependent delay differential equations with distributed memory

We investigate state dependent delay differential equations with distributed memory, combining discrete state dependent delays and a convolution type memory operator. Under Lipschitz type assumptions on the delay, kernel, and nonlinear term, we establish local well posedness using a fixed point argument in a Banach space of histories. The distributed memory operator is shown to be locally Lipschitz, ensuring existence and uniqueness. As a benchmark, we analyze a logistic model with a Gamma distributed memory kernel of order two. The convolution term is reduced to a finite dimensional form, yielding an explicit characteristic equation and closed form Hopf bifurcation threshold and frequency. The results show that higher order memory kernels may induce oscillatory dynamics, unlike purely exponential memory, and provide a rigorous framework for stability and bifurcation analysis in delay systems with state dependent memory.

math.DS↗