A $ (Q,\tau)$-FRACTIONAL AGING MODEL WITH MEMORY EFFECTS AND ADAPTIVE HEALING DYNAMICS

Comparison of exponential decay ($ \alpha = 1 $) and fractional memory-driven decay with $ \alpha = 0.7 $ and $ \alpha = 0.5 $. While the exponential model rapidly declines without memory, the fractional decay exhibits slower convergence, encoding long-term memory effects characteristic of biological systems.

Plot of the $(q, \tau)$-Gamma function $\Gamma_{q, \tau}(x)$ for various values of deformation parameters $q \in \{0.5, 0.7, 0.9\}$ and $\tau \in \{0.5, 1.0, 1.5\}$. The classical Gamma behavior is recovered as $q \to 1^-$ and $\tau \to 1$. For lower $q$, the function exhibits stronger deformation from the classical shape, especially as $\tau$ increases. These curves highlight how memory and scaling parameters affect growth and singularity behavior in fractional models.

Decay of physiological state $ A(t) $ under various $(q, \tau, \alpha)$ parameter combinations. Lower $ \alpha $ results in slower decay, while larger $ q $ or $ \tau $ accelerates aging.

Comparison of aging trajectories with and without external healing term $ f(t) = 0.3 e^{-0.05 t} $. The healing input delays or softens the decay due to physiological repair.

Comparison of $(q, \tau)$-fractional aging dynamics under three scenarios: (1) No healing, (2) exponentially decaying healing input, and (3) periodic pulse healing (weekly). Periodic interventions help maintain vitality more effectively in long term due to memory retention.

Simulated trajectories of physiological state $ A(t) $ under composite healing. The combination of exponential and periodic therapies yields the most robust aging delay.

Time evolution of physiological state $ A(t) $ under composite healing input for different combinations of $(\alpha, q, \tau)$. Higher values of $\alpha$ and moderate deformation parameters $ q, \tau $ yield stronger long-term health preservation due to enhanced memory effects and time-slowing deformation.

Comparison of observed (blue dots), true (dashed), and fitted (solid) physiological state $ A(t) $ in the $(q, \tau)$-fractional aging model. The observed data include Gaussian noise, while the fitted model is recovered via optimization using the mean squared error criterion.

Residuals of the fitted $(q, \tau)$-fractional aging model. The plot shows the difference between observed and fitted values: $ A_{\mathrm{obs}}(t) - A_{\mathrm{fit}}(t) $. Ideally, residuals should center around zero with no visible trend, indicating a well-fitting model.

Simulation of recovery dynamics under different memory parameters $ \alpha $ following a healing pulse at $ t = 20 $. Higher memory ($ \alpha = 0.5 $) sustains recovery longer, while lower memory ($ \alpha = 0.9 $) leads to faster decay post-intervention.

Recovery dynamics under the $(q, \tau)$-fractional model with a healing pulse at $ t = 20 $. Different memory strengths are tested via fractional orders $ \alpha \in \{0.9, 0.7, 0.5\} $, using fixed deformation parameters $ q = 0.7 $, $ \tau = 1.2 $. Lower $ \alpha $ yields more persistent recovery and slower decay.