This study investigates the complex dynamics and control mechanisms of fractional-order glucose–insulin regulatory systems, incorporating memory-dependent properties through fractional derivatives. Employing the Laplace–Adomian Decomposition Method (LADM) and the Generalized Euler Method (GEM), the research models glucose–insulin interactions with time-varying fractional orders to simulate long-term physiological processes. Key aspects include the derivation of Lyapunov exponents, bifurcation diagrams, and phase diagrams to explore system stability and chaotic behavior. A novel control strategy using simple linear controllers is introduced to stabilize chaotic oscillations. The effectiveness of this approach is validated through numerical simulations, where Lyapunov exponents are reduced from positive values (${\lambda }_1=0.123$) in the uncontrolled system to negative values (${\lambda }_1=-0.045$) post-control application, indicating successful stabilization. Additionally, bifurcation analysis demonstrates a shift from chaotic to periodic behavior when control is applied, and time-series plots confirm a significant reduction in glucose–insulin fluctuations. These findings underscore the importance of fractional calculus in accurately modeling nonlinear and memory-dependent glucose–insulin dynamics, paving the way for improved predictive models and therapeutic strategies. The proposed framework provides a foundation for personalized diabetes management, real-time glucose monitoring, and intelligent insulin delivery systems.

This study presents a comprehensive analysis of zoonotic disease transmission dynamics between baboon and human populations using both deterministic and stochastic modeling approaches. The model is constructed with a symmetric compartmental structure for each species—susceptible, infected, and recovered—which reflects a biological and mathematical symmetry between the two interacting populations. Public health control strategies such as sterilization, restricted food access, and reduced human–baboon interaction are incorporated symmetrically, allowing for a balanced evaluation of their effectiveness across species. The basic reproduction number (${\mathcal{R}}_0$) is derived analytically and examined through sensitivity indices to identify critical epidemiological parameters. Numerical simulations, implemented via the Euler–Maruyama method, explore the influence of stochastic perturbations on disease trajectories. Statistical tools including Maximum Likelihood Estimation (MLE), Mean Squared Error (MSE), and Power Spectral Density (PSD) analysis validate model predictions and assess variability across noise levels. The results provide probabilistic confidence intervals and highlight the robustness of the proposed control strategies. This symmetry-aware, dual-framework modeling approach offers novel insights into zoonotic disease management, particularly in ecologically dynamic regions with frequent human–wildlife interactions.

This study presents a novel fractional-order mathematical model to investigate zoonotic disease transmission between humans and baboons, incorporating the Generalized Euler Method and highlighting key control strategies such as sterilization, restricted food access, and reduced human–baboon interaction. The model’s structure exhibits an inherent symmetry in the transmission dynamics between baboon and human populations, reflecting balanced interaction patterns. This symmetry is further analyzed through the stability of infection-free and endemic equilibrium points, guided by the basic reproduction number ${\mathcal{R}}_0$. Theoretical analyses confirmed the existence, uniqueness, and boundedness of solutions, while sensitivity analysis identified critical parameters influencing disease spread. Numerical simulations validated the effectiveness of intervention strategies, demonstrating the impact of symmetrical measures on minimizing zoonotic disease risks and promoting balanced population health outcomes. This work contributes to epidemiological modeling by illustrating how symmetry in control interventions can optimize zoonotic disease management.

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings.