The Caputo fractional-order differential operator is used in epidemiological models, but its accuracy benefits are typically ignored. We validated the suggested fractional epidemiological seasonal influenza model of the SVEIHR type to demonstrate the Caputo operator’s relevance. We analysed the model using fractional calculus, revealing its basic properties and enhancing our understanding of disease progression. Furthermore, the positivity, bounds, and symmetry of the numerical scheme were examined. Adjusting the Caputo fractional-order parameter $\mathsf{\alpha }$ = 0.99 provided the best fit for epidemiological data on infection rates. We compared the suggested model with the Caputo fractional-order system and the integer-order equivalent model. The fractional-order model had lower absolute mean errors, suggesting that it could better represent sickness transmission and development. The results underline the relevance of using the Caputo fractional-order operator to improve epidemiological models’ precision and forecasting. Integrating fractional calculus within the framework of symmetry helps us build more reliable models that improve public health interventions and policies.

Transport of pollutants is a serious environmental concern, where accurate and effective mathematical models are essential for developing viable mitigation programs. In this work, this study proposes new formulation of advection dispersion equations of fractional order and employ them to model the highly complex advection dispersion phenomena. The derivative using the Modified Atangana–Baleanu–Caputo (MABC) fractional derivative is an advanced extension of the classical Atangana Baleanu derivative and provides greater flexibility in describing memory and nonlocal effects. To solve the resulting problem numerically, we utilize the framework of physics informed neural networks (PINNs), in which the governing physical laws serve as the building blocks of a deep learning model. This approach enables the derivation of highly accurate and fast convergent semi-analytical solutions. The main contributions of this work are threefold: (1) the development of specific PINNs algorithm to solve fractional differential equations in the MABC sense; (2) an extensive performance analysis demonstrating higher precision and computational efficiency compared to conventional numerical and perturbative methods; and (3) validation through a variety of case studies, confirming the robustness and applicability of the proposed approach in different contexts. Several numerical examples are provided to illustrate the effectiveness of the approach, and the results are compared with existing methods to justify both the efficiency and feasibility of the proposed scheme.