This study presents a novel numerical framework for simulating glucose-insulin regulatory dynamics using the Caputo-Fabrizio (CF) fractal-fractional operator with both constant and variable fractional orders. The model incorporates an exponential decay kernel to capture memory and hereditary effects in metabolic regulation. A Newton interpolation-based numerical scheme is developed to approximate the CF-FF derivatives, ensuring computational stability and accuracy. For the variable-order formulation, the fractional order β(t) dynamically evolves with time, reflecting physiological variability typically observed during intravenous glucose tolerance tests (IVGTT). Numerical experiments reproduce physiologically realistic glucose-insulin oscillations and demonstrate how feedback control stabilizes chaotic metabolic behavior. The results are based entirely on simulation evidence calibrated within clinically reported parameter ranges, providing conceptual validation rather than direct patient-data comparison. The proposed approach bridges mathematical fractional calculus with biomedical applications, offering new insights for personalized diabetes management and adaptive glucose control strategies. • Fractal-fractional model formulation capturing glucose-insulin memory and adaptation • Stable numerical scheme using Newton interpolation for accurate fractional integration • Linear feedback control applied to regulate chaotic glucose-insulin dynamics • Numerical Methodology for glucose-insulin dynamics. Our investigation of the fractal-fractional glucose-insulin system employs the following analytical framework: • Model Development: We formulate a fractal-fractional-order extension of the minimal glucose insulin model, incorporating an exponential decay type kernel to capture the system's memory effects and anomalous diffusion characteristics inherent in metabolic processes. The model accounts for both insulin-dependent and independent glucose utilization dynamics. • Computational Implementation: We develop a novel numerical solver based on Newton's interpolation polynomials, implementing the Atangana-Seda fractal-fractional derivative formulation. This method provides an efficient computational framework for solving the coupled nonlinear fractional differential equations while maintaining numerical stability across different fractional orders. • The purpose of this section is to define a mathematical model to study the dynamic behavior of glucose-insulin physiology. • With the Adams-Bashforth-Moulton numerical scheme, we compute the Lyapunov exponent of the system, which is useful for studying dissipative. • In a generalized numerical method, we simulate the solutions of the system using the time-fractal fractional derivative of Atangana-Seda.

This paper presents a novel fractional-order model for glucose-insulin dynamics using the Caputo-Fabrizio (CF) derivative, which accounts for memory effects through its nonsingular exponential kernel. Existence and uniqueness of the solution are established via fixed point theory, and infinite series solutions are obtained using the Sumudu transform. Hyers-Ulam stability is analyzed to assess the system’s robustness against perturbations. A linear control strategy is introduced to regulate glucose levels, demonstrating potential for integration with real-time insulin delivery systems. Compared to classical integer-order models, the proposed approach provides improved accuracy, enhanced stability, and deeper insight into the chaotic behavior of glucose-insulin interactions. This framework supports the development of personalized diabetes treatment and adaptive control strategies.

In this study, we develop a fractal-fractional pneumonia transmission model using the Atangana-Baleanu derivative to capture long-memory effects. We analyze the existence and uniqueness of the model’s solutions and examine its stability using Hyers-Ulam criteria. Numerical simulations are conducted to explore the influence of different fractional orders on disease dynamics. The results indicate that fractional-order modeling provides a more flexible and accurate framework than classical integer-order models, particularly in representing population heterogeneities and intervention strategies such as vaccination and treatment. This approach enhances our understanding of pneumonia transmission and offers valuable insights for public health decision-making.

This study develops a mathematical model to explore zoonotic disease transmission dynamics between baboons and humans in the Al-Baha region. The model incorporates key intervention strategies—sterilization, restricted food access, and reduced human-baboon interactions—to mitigate disease spread. Theoretical analysis establishes the existence, uniqueness, non-negativity, and boundedness of solutions, ensuring biological validity. Stability analysis identifies infection-free and endemic equilibria, with the basic reproduction number \( R_0 \) characterizing disease persistence or eradication. A stochastic version of the model, implemented using the Milstein method, captures the impact of environmental and social variability on disease dynamics. Sensitivity analysis highlights critical parameters influencing transmission dynamics, while numerical simulations emphasize the importance of combining control measures. This comprehensive approach offers actionable insights for effective public health and wildlife management strategies.

The study of the $\mathfrak{q}$-analogue of the classical results of geometric function theory is currently of great interest to scholars. In this article, we define generalized classes of close-to-convex functions and quasi-convex functions with the help of the $\mathfrak{q}$-difference operator. Moreover, by using the $\mathfrak{q}$-analogues of a certain family of linear operators, the classes $K_{\mathfrak{q},b}^s\left(\mathfrak{h}\right)$, ${\tilde{K}}_{\mathfrak{q},s}^b\left(\mathfrak{h}\right)$, $Q_{\mathfrak{q},b}^s\left(\mathfrak{h}\right)$, and ${\tilde{Q}}_{\mathfrak{q},s}^b\left(\mathfrak{h}\right)$ are introduced. Several interesting inclusion relationships between these newly defined classes are discussed, and the invariance of these classes under the $\mathfrak{q}$-Bernadi integral operator was examined. Furthermore, some special cases and useful consequences of these investigations were taken into consideration.