Deep learning investigation of discrete fractional order Shingles disease model with neural networking

This study investigates the discrete fractional-order Shingles disease (DFOS) mathematical model. The considered model is framed with five compartments over time. The existence and uniqueness of solutions are determined via fixed point theory, confirming well-defined solutions. Three scenarios, each with various fractional orders, were used to study the effects of memory and fractional dynamics on disease propagation. To solve the model, we operate a discrete numerical iterative method (DNIM) linked to neural networks, specifically employing the Levenberg-Marquardt (LM) algorithm, denoted DNIM-LM. This hybrid technique increases the accuracy of numerical results from neural networks. The data sets are distributed into 70% for training, 15% for validation, and 15% for testing, confirming a balanced calculation. The performance of the discrete fractional mathematical modeling is evaluated through artificial intelligence (AI), which involves training progress, error distribution, regression metrics, and overall model fit. The outcomes are thoroughly visualized via graphical illustrations, providing insights into the dynamics of each part and the impact of various fractional orders on the spread of shingles. This research addresses a novel approach to demonstrate infectious diseases with fractional dynamics and enhanced computational methods, improving our understanding of shingle transmission as process innovation.