We proposed an original hybrid approach that combined a continuous modeling framework based on Caputo fractional differential equations with an Earliest Deadline First–Ant Colony Optimization (EDF–ACO) algorithm for solving the parallel machine scheduling problem with precedence constraints. Unlike most existing works, where the fractional order is usually fixed at its classical value, we investigated the influence of the fractional order $\alpha$ in the interval $(0,1]$ and analyzed its impact on the optimization process. The results showed that intermediate values of $\alpha$ allowed the effective incorporation of memory effects, leading to improved numerical stability, smoother convergence, and a reduction of oscillatory behavior. The fractional evaluation mechanism was coupled with the pheromone update strategy of the EDF–ACO algorithm, providing a more stable guidance for the search process. The existence and uniqueness of the solution to the fractional model were established using Banach's fixed point theorem, ensuring the consistency of the proposed continuous evaluation framework. Numerical experiments confirmed the effectiveness of the approach in terms of solution quality and convergence stability across different scheduling configurations.

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.