In this paper, we investigate the existence-uniqueness, and Ulam Hyers stability (UHS) of solutions to a fractional-order pantograph differential equation (FOPDE) with two Caputo operators. Banach's fixed point (BFP) and Leray-alternative Schauder's are used to prove the existence- uniqueness of solutions. In addition, we discuss and demonstrate various types of Ulam-stability for our problem. Finally, an example is provided for clarity.

The goal of this study is to see if there is a solution for the fuzzy delay predator-prey system (FDPPS) with Caputo derivative. To begin, we use Schaefer's fixed point theorem to obtain results for the existence theorem of at least one solution in a Caputo FDPPS where the initial condition is also represented by a fuzzy number on fuzzy number space. We also determine the necessary and sufficient conditions of solutions for the system. Several examples are also presented to explain the oscillatory property and the existence of a solution.

Rabies remains a significant public health challenge, particularly in areas with substantial dog populations, necessitating a deeper understanding of its transmission dynamics for effective control strategies. This study addressed the complexity of rabies spread by integrating two critical delay effects—vaccination efficacy and incubation duration—into a delay differential equations model, capturing more realistic infection patterns between dogs and humans. To explore the multifaceted drivers of transmission, we applied a novel framework using piecewise derivatives that incorporated singular and non-singular kernels, allowing for nuanced insights into crossover dynamics. The existence and uniqueness of solutions was demonstrated using fixed-point theory within the context of piecewise derivatives and integrals. We employed a piecewise numerical scheme grounded in Newton interpolation polynomials to approximate solutions tailored to handle singular and non-singular kernels. Additionally, we leveraged artificial neural networks to split the dataset into training, testing, and validation sets, conducting an in-depth analysis across these subsets. This approach aimed to expand our understanding of rabies transmission, illustrating the potential of advanced mathematical tools and machine learning in epidemiological modeling.

This paper introduces, analyzes, and numerically investigates a fractional-order SIQR epidemic model with the normalized Caputo–Fabrizio derivative. The model captures memory effects and the impact of quarantine or isolation interventions, offering a more realistic description of epidemic dynamics. We establish the existence, uniqueness, positivity, and population conservation properties, and then propose a robust numerical scheme. The influence of the memory parameter and kernel normalization is illustrated via simulations, with a discussion on their implications for epidemic forecasting and real-world control strategies. Furthermore, artificial neural networks are applied, with the dataset partitioned into training, validation, and testing subsets. A comprehensive assessment is carried out for each dataset partition.

This paper introduces, analyzes, and numerically investigates a fractional-order SEIR epidemic model employing the normalized Caputo-Fabrizio (NCF) derivative. The model captures memory effects and the role of an exposed (latent) compartment, allowing for more realistic epidemic dynamics. We establish existence, uniqueness, positivity, and population conservation, then propose a robust numerical scheme. The impact of the memory parameter and kernel normalization is illustrated via simulations, with a discussion on their significance for epidemic forecasting and potential real-world applications.

This research uses Susceptible–Exposed–Infectious–Recovered (SEIR) and Susceptible–Exposed–Infectious–Vaccinated (SEIV) models to analyze rabies transmission in human and canine populations. The framework includes eight epidemiological compartments to evaluate intervention strategies. A fractional-order model is employed using the Atangana-Baleanu derivative in the Caputo sense to capture memory and the system's complexity. The model's validity is established through qualitative analysis. Existence and uniqueness are confirmed via fixed-point theory, and Ulam-Hyers criteria assess robustness. Numerical solutions are obtained using the iterative Adams and Adams-Bashforth methods for accurate time-series simulations. Numerical experiments evaluate vaccination effects under a constant rate for a subset of the population. The results show that vaccination effectively reduces disease prevalence, emphasizing its critical role in rabies control. Deep neural network (DNN) techniques are applied for training, validation, and testing. The DNN has three hidden layers (10,100, 10 neurons) and is trained over 1000 epochs using the Levenberg-Marquardt algorithm. The model achieves high predictive accuracy, with mean square errors as low as 0.00027 and root mean square errors under 0.17 across compartments. Overall, combining fractional calculus with deep learning provides a robust framework for modeling complex disease dynamics and offers valuable insights for public health strategies in regions with significant dog populations.