A deep neural network with optimization approach for a nonlinear fractional bird flu model with vaccination and treatment via spectral method

This study presents a fractional-order mathematical model for the transmission dynamics of bird flu, incorporating memory effects through fractional derivatives to capture realistic disease progression. A comprehensive dynamical analysis is performed, including the derivation of the basic reproduction number and the investigation of Ulam-Hyers stability. The Shifted Vieta-Fibonacci Operational Matrix (SVFOM) method is employed for numerical approximation, supported by convergence analysis and validated using the Adams-Bashforth-Moulton (ABM) scheme. Sensitivity analysis is carried out to identify the most influential parameters governing disease spread. Additionally, a Deep Neural Network (DNN) is trained using ABM-generated data to evaluate its predictive capability relative to the numerical method. Finally, the Average Subtraction-Based Optimization (ASBO) technique is used to minimize the basic reproduction number by optimizing key epidemiological parameters. The combined analytical, numerical, and machine-learning framework provides a robust approach for understanding and controlling bird flu dynamics.