Poliomyelitis dynamics with fractional order derivatives and deep neural networks.

This paper presents a comprehensive study of poliomyelitis transmission dynamics using two fractional-order models that incorporate the Atangana--Baleanu derivatives in the Caputo sense (ABC). The model includes critical epidemiological features, including vaccination and a post-paralytic population class. By utilizing the Mittag-Leffler kernel, the fractional framework captures memory and hereditary properties in disease progression. The existence and uniqueness of the model's solution are established using fixed-point theory. To assess the model's robustness, Ulam-Hyers stability analysis is conducted through nonlinear techniques. For numerical approximation, the iterative Adams-Bashforth scheme tailored for fractional orders is employed. Simulations are performed for a range of fractional orders and control strategies. The results indicate that all compartments achieve convergence and dynamic stability over time, with lower fractional orders exhibiting faster stabilization. These findings underscore the effectiveness of fractional modeling in capturing the complex behaviors of diseases. To enhance predictive capabilities, deep neural network (DNN) techniques are integrated into the framework. The dataset is partitioned into training, testing, and validation sets. The DNN is then used for classification, forecasting, and data-driven simulation of disease dynamics. The DNN-based results closely align with numerical simulations, demonstrating high accuracy and validating the proposed hybrid modeling approach. This study presents a novel integration of fractional-order modeling and machine learning for infectious disease analysis, providing a powerful tool for understanding and predicting poliomyelitis spread under vaccination and post-paralytic effects.