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REVIEW

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

Sharaf Aldeen O. Albteiha1 ,  Farah Aini Abdullah1,2 ,  Iqbal M. Batiha1,3* ,  Bushra M. Al-Smadi4 ,  Shaher Momani2,4
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1 School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang , Malaysia
2 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman , United Arab Emirates
3 Department of Mathematics, Al Zaytoonah University of Jordan, Amman , Jordan
4 Department of Mathematics, Faculty of Science, The University of Jordan, Amman , Jordan
IJOCTA, 026240116 https://doi.org/10.36922/IJOCTA026240116
Received: 11 June 2026 | Revised: 18 August 2026 | Accepted: 18 August 2026 | Published online: 14 October 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

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.

Keywords
Avian influenza
Zoonotic transmission
Fractional-order models
Memory effects
Optimal control
One Health
Funding
None.
Conflict of interest
The authors declare that they have no conflict of interest.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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