This work enhanced the mathematical modeling of vector-borne infections involving vertical transmission and treatment effects within a targeted population by incorporating fractional calculus techniques that account for nonlocal properties and non-singular fading memory behavior. This work investigated a fractional-order mathematical model for poliomyelitis governed by the Mittag-Leffler kernel. By employing fixed point theory, we qualitatively analyzed the model and confirmd the existence and uniqueness of solutions. Additionally, Ulam's type stability is examined through nonlinear analytical methods. To approximate the solution, a fractional Adams-Bashforth numerical scheme is utilized. The model was simulated under various fractional orders and different control scenarios. Results indicate that all compartments exhibit convergence and long-term stability. Notably, lower fractional orders tend to reach stability more rapidly. Furthermore, Artificial Neural Networks were applied, with the dataset partitioned into training, validation, and testing subsets. A comprehensive assessment was carried out for each dataset partition.

Water pollution significantly threatens public health and environmental sustainability, particularly in developing nations. This study introduced an innovative fractional-order mathematical model for analyzing water pollution dynamics, incorporating four distinct compartments to represent the interactions between polluted water sources, susceptible water bodies, contamination processes, and restoration mechanisms. The model used the Atangana-Baleanu fractional derivative in the Caputo sense, offering a more precise representation of memory effects and complex pollutant transport mechanisms. The proposed model underwent rigorous qualitative validation, ensuring the existence and uniqueness of solutions via fixed-point theory, while stability analysis was conducted using the Ulam-Hyers approach. The Adams-Bashforth numerical method was employed to obtain approximate solutions, enabling a more accurate simulation of pollution dynamics. Numerical simulations further highlighted the impact of treatment strategies in reducing contamination levels and restoring water quality. Additionally, artificial neural networks (ANN) were integrated into the framework to enhance predictive capabilities. The dataset used for ANN training was derived from simulated pollution levels based on model parameters calibrated with empirical studies on water contamination dynamics. This combined fractional-ANN methodology established a robust foundation for effective water quality management, aiding in decision-making for pollution control policies and remediation strategies.