A Fractional‐Order Alcoholism Model Incorporating Hypothetical Social Influence: A Theoretical and Numerical Study

This study introduces a fractional‐order mathematical model for alcoholism dynamics using the Hilfer derivative to capture memory effects and hereditary properties within a unified framework. The model incorporates hypothetical social influence through sentiment‐based variables to represent positive and negative social interactions. No real social media data is analyzed in this work; instead, a conceptual framework for future integration of Twitter sentiment analysis using tools such as VADER, TextBlob, or BERT is proposed. Existence, uniqueness, and Ulam–Hyers stability of the model are rigorously established using fixed‐point theory. Numerical simulations are performed via the fractional Adams–Bashforth method, and artificial neural networks (ANNs) are employed solely as a surrogate approximation tool for fractional dynamics, reducing computational cost for future large‐scale simulations. Sensitivity analysis reveals that parameters ζ and δ strongly influence alcoholism prevalence, while the fractional order β governs the persistence of behavioral patterns. Comparative results demonstrate the qualitative advantages of the Hilfer derivative over integer‐order and Caputo models in capturing long‐term dependencies. This work provides a flexible theoretical framework for future empirical validation and real‐time predictive modeling.