A review of fractional calculus models for avian influenza: Dynamics, memory effects, and control perspectives

Avian influenza is a serious global public health threat, and both humans and animals can be infected. Classical integer-order compartmental models are generally not capable of capturing the complex temporal dynamics of epidemic systems, especially memory-dependent phenomena such as delayed immune response and environmental virus persistence. In this review, we have surveyed the use of fractional calculus in the modeling of avian influenza epidemics with emphasis on disease transmission, memory effects, and intervention strategies. In detail, we consider four fractional operators, namely, the Caputo, Caputo-Fabrizio, Atangana-Baleanu, and fractal-fractional operators, and compare the structures of their kernels and their memory properties. We review progress in data-driven modeling, including calibration to real epidemic surveillance data, seasonal transmission patterns, and fractional optimal control formulations in a One Health framework. Based on the review of 16 studies combining fractional calculus with avian influenza or influenza transmission modeling, consistent evidence is provided that fractional-order models reproduce the real outbreak dynamics better than the classical ones. As far as surveillance data is concerned, the Atangana-Baleanu-Caputo operator yields the fewest prediction errors. Key research gaps are identified, and future directions are proposed, including hybrid fractional-machine learning frameworks for real-time adaptive forecasting.