This study presents a new, generalized family of special polynomials. These polynomials are designated as the Apostol–Hermite–Kampé de Fériet–Bell–Bernoulli-type polynomials. Based on their generating function, the corresponding series expansions and summation formulas are obtained. Furthermore, determinant representation and several differential and integral representations of these polynomials are derived. The fractional extension of this polynomial family is explored, resulting in the formulation of several associated identities. Finally, the study utilizes Mathematica to deliver zero distributions and graphical representations.

In this study, we proposed a modified SEVIR-S (susceptible, exposed, vaccinated, infected, recovered) model for the transmission dynamics of adenovirus by incorporating the effects of immunity waning and reinfection. Unlike the classical SEVIR framework, the extended model accounted for the possibility that recovered individuals may lose immunity over time and become susceptible again — a critical feature for accurately modeling diseases like adenovirus. To better capture the disease's memory effects and temporal dynamics, the model used the fractal-fractional Caputo-Fabrizio derivative with a power-law kernel. The paper analyzed the model's existence and stability using fixed point theory and Hyers-Ulam (H-U) stability. Furthermore, both the disease-free and endemic equilibrium points and their stability were analyzed. Also, the basic reproduction number was provided. The findings were validated through numerical simulations using an extended Adams-Bashforth method.

Modeling the long-term dynamics of the COVID-19 pandemic is challenged by evolving public behavior and interventions. We propose a novel piecewise fractional-order (SVIR) model incorporating vaccination and education controls. The model uniquely employs a classical derivative for the initial, memoryless phase of the epidemic. It then transitions to a Caputo-Fabrizio fractional derivative to capture long-term collective memory effects on transmission. We establish the model's mathematical well-posedness and derive the basic reproduction number ( $R_0$). Under our baseline parameterization, the reproduction number is $R_0\simeq 4.95$. An optimal control problem is formulated to determine the ideal implementation of time-varying vaccination and education. Numerical simulations validate the distinct crossover dynamics produced by our piecewise approach. Results demonstrate that a synergistic strategy combining vaccination and education is highly effective, reducing the peak of infected individuals by over 90% compared to the uncontrolled scenario, and significantly outperforms isolated interventions. This study offers a flexible tool for understanding and controlling epidemics.

In this work, we extended the classical Susceptible–Exposed–Vaccinated–Infected–Recovered (SEVIR) framework by incorporating an asymptomatic class, leading to the formulation of a new Susceptible–Exposed–Vaccinated–Asymptomatic–Infected–Recovered (SEVAIR) model that more accurately reflects the transmission characteristics of adenovirus. To account for memory effects and capture complex temporal behavior, the model was developed using the fractal-fractional Caputo-Fabrizio derivative with a power-law kernel. Existence and stability of solutions based on fixed point theory and Hyers-Ulam stability criteria were derived. Both the disease-free and endemic equilibrium states were derived, and their local stability properties were examined. Additionally, the basic reproduction number was computed to understand the disease's spread threshold. The theoretical results were supported by numerical simulations, which were performed using a modified Adams-Bashforth approach tailored for the fractional framework.

In this study, we perform a comparative symmetry analysis and investigate the qualitative properties of a nonlinear impulsive fractional differential system with multiple delays and nonlocal boundary conditions. By utilizing the generalized power Caputo fractional derivative, we present a unified theoretical framework that encompasses several operators—including the Atangana-Baleanu, Caputo-Fabrizio, and weighted Hattaf derivatives—as special cases. This generality guarantees that our findings are relevant to a range of fractional kernels, emphasizing the inherent symmetry characteristics of these operators. We establish adequate criteria for the existence and uniqueness of solutions through fixed-point theory. We also show that the system is Ulam-Hyers stable, an important property for maintaining its strength in the face of change. A convergent numerical scheme confirms the theoretical results, and a sensitivity analysis demonstrates the effect of kernel symmetry on stability margins. The networked control system application shows how useful the framework is in real life. The results show that the framework can include complex genetic phenomena and spontaneous interactions that are often ignored in traditional models.

This research investigated a tri-trophic food chain model, incorporating carrying capacity and Holling-type predation, formulated using the fractal-fractional Caputo derivative. The four equilibrium states---trivial, prey-only, prey-predator, and coexistence---presented and their stability was discussed. Existence and uniqueness of solutions were established using Schaefer's and Banach's fixed point theorems. Also, stability requirements in the sense of Hyers-Ulam (H-U) were investigated. Numerical simulations were performed using the extended numerical method of Adams-Bashforth-Moulton (ABM), and comparative results were graphically presented to demonstrate the impact of varying fractal-fractional orders. A sensitivity analysis revealed how perturbations in individual parameters influence the model's outcome. The model accounts for memory and hereditary effects in ecological interactions. The proposed method enhances accuracy, stability, and convergence for long-time simulations compared to classical models.