AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA026240116
Submit to IJOCTA
Cite this article
17
Download
92
Views
Related Info Links
Google Scholar
More by Authors Links
Iqbal M. Batiha
Journal Browser
Volume | Year
Issue
Search
News and Announcements
View All
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
Show Less
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/ )
Download PDF
XML
Cite
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.
References
  1. Hutchinson EC. Influenza virus. Trends Microbiol. 2018;26(9):809-810. https://doi.org/10.1016/j.tim.2018.05.013
  2. Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599-653. https://doi.org/10.1137/S0036144500371907
  3. Javanian M, Barary M, Ghebrehewet S, Koppolu V, Vasigala VKR, Ebrahimpour S. A brief review of influenza virus infection. J Med Virol. 2021;93(8):4638-4646. https://doi.org/10.1002/jmv.26990
  4. Iwami S, Takeuchi Y, Liu X. Avian-human influenza epidemic model. Math Biosci. 2007;207(1):1-25. https://doi.org/10.1016/j.mbs.2006.08.001
  5. Pantin-Jackwood MJ, Miller PJ, Spackman E, et al. Role of poultry in the spread of novel H7N9 influenza virus in China. J Virol. 2014;88(10):5381-5390. https://doi.org/10.1128/jvi.03689-13
  6. Iqbal M, Yaqub T, Reddy K, McCauley JW. Novel genotypes of H9N2 influenza A viruses isolated from poultry in Pakistan containing NS genes similar to highly pathogenic H7N3 and H5N1 viruses. PLoS One. 2009;4(6):e5788. https://doi.org/10.1371/journal.pone.0005788
  7. Gonzales JL, Elbers ARW. Effective thresholds for reporting suspicions and improve early detection of avian influenza outbreaks in layer chickens. Sci Rep. 2018;8(1):8533. https://doi.org/10.1038/s41598-018-26954-9
  8. Gonzales JL, Pritz-Verschuren S, Bouwstra R, Wiegel J, Elbers ARW, Beerens N. Seasonal risk of low pathogenic avian influenza virus introductions into free-range layer farms in the Netherlands. Transbound Emerg Dis. 2021;68(1):127-136. https://doi.org/10.1111/tbed.13649
  9. Dietze K, Graaf A, Homeier-Bachmann T, et al. From low to high pathogenicity---characterization of H7N7 avian influenza viruses in two epidemiologically linked outbreaks. Transbound Emerg Dis. 2018;65(6):1576-1587. https://doi.org/10.1111/tbed.12906
  10. Alexander DJ. An overview of the epidemiology of avian influenza. 2007;25(30):5637-5644. https://doi.org/10.1016/j.vaccine.2006.10.051
  11. Webster RG, Govorkova EA. H5N1 influenza---continuing evolution and spread. N Engl J Med. 2006;355(21):2174-2177. https://doi.org/10.1056/NEJMp068205
  12. Chowell G, Sattenspiel L, Bansal S, Viboud C. Mathematical models to characterize early epidemic growth: A review. Phys Life Rev. 2016;18:66-97. https://doi.org/10.1016/j.plrev.2016.07.005
  13. Ferguson N. Capturing human behaviour. 2007;446(7137):733. https://doi.org/10.1038/446733a
  14. González-Parra G, Arenas AJ, Chen-Charpentier BM. A fractional order epidemic model for the simulation of outbreaks of influenza A (H1N1). Math Methods Appl Sci. 2014;37(15):2218-2226. https://doi.org/10.1002/mma.2968
  15. Abdoon MA, Alzahrani ABM. Comparative analysis of influenza modeling using novel fractional operators with real data. 2024;16(9):1126. https://doi.org/10.3390/sym16091126
  16. Magin R. Fractional calculus in bioengineering, part 1. Crit Rev Biomed Eng. 2004;32(1):1-104. https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10
  17. Ebrahimzadeh A. Fractional optimal control of avian-human influenza with environmental and behavioral interactions. Int J Dyn Control. 2026;14:101. https://doi.org/10.1007/s40435-026-02038-9
  18. Khan MA, Farhan M, Islam S, Bonyah E. Modeling the transmission dynamics of avian influenza with saturation and psychological effect. Discrete Contin Dyn Syst Ser S. 2019;12(3):455-474. https://doi.org/10.3934/dcdss.2019030
  19. Khan MA, DarAssi MH, Ahmad I, Alzahrani E, Mat Isa N. Existence, stability, and data-driven analysis of a fractional-order avian influenza model in poultry farms. Eng Rep. 2026;8(3):e70711. https://doi.org/10.1002/eng2.70711
  20. Rezapour S, Mohammadi H. A study on the AH1N1/09 influenza transmission model with the fractional Caputo-Fabrizio derivative. Adv Differ Equ. 2020;2020:488. https://doi.org/10.1186/s13662-020-02945-x
  21. Caputo M. Linear models of dissipation whose Q is almost frequency independent. Geophys J Int. 1967;13(5):529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
  22. Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl. 2015;1(2):73-85. https://doi.org/10.12785/pfda/010201
  23. Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm Sci. 2016;20(2):763-769. https://doi.org/10.2298/TSCI160111018A
  24. Khan H, Rajpar AH, Alzabut J, Aslam M, Etemad S, Rezapour S. On a fractal-fractional-based modeling for influenza and its analytical results. Qual Theory Dyn Syst. 2024;23(2):70. https://doi.org/10.1007/s12346-023-00918-5
  25. Baleanu D, Mohammadi H, Rezapour S. A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative. Adv Differ Equ. 2020;2020:299. https://doi.org/10.1186/s13662-020-02762-2
  26. Ghanbari B, Gómez-Aguilar JF. Analysis of two avian influenza epidemic models involving fractal-fractional derivatives with power and Mittag-Leffler memories. 2019;29(12):123113. https://doi.org/10.1063/1.5117285
  27. Diekmann O, Heesterbeek JAP, Metz JAJ. On the definition and the computation of the basic reproduction ratio R_0 in models for infectious diseases in heterogeneous populations. J Math Biol. 1990;28(4):365-382. https://doi.org/10.1007/BF00178324
  28. Van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci. 2002;180(1-2):29-48. https://doi.org/10.1016/S0025-5564(02)00108-6
  29. Ahmed E, El-Sayed AMA, El-Saka HAA. Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J Math Anal Appl. 2007;325(1):542-553. https://doi.org/10.1016/j.jmaa.2006.01.087
  30. Ye X, Lin S, Xu C. Dynamical analysis of a fractional-order avian-human influenza epidemic model with logistic growth for avian population. J Algorithms Comput Technol. 2020;14:1748302620966704. https://doi.org/10.1177/1748302620966704
  31. Diethelm K. A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn. 2013;71(4):613-619. https://doi.org/10.1007/s11071-012-0475-2
  32. Matignon D. Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications; 1996; Lille, France. 963-968. https://www.researchgate.net/profile/Denis-Matignon/publication/2581881_Stability_Results_For_Fractional_Differential_Equations_With_Applications_To_Control_Processing/links/00b7d52dd1c17e4b1a000000/Stability-Results-For-Fractional-Differential-Equations-With-Applications-To-Control-Processing.pdf
  33. Asif M, Popa IL, Ismail EAA, Awwad FA, Ishtiaq U. Insights from Atangana-Baleanu fractional derivatives modeling of influenza epidemics and sensitivity analysis. Sci Rep. 2025;15:34887. https://doi.org/10.1038/s41598-025-18512-x
  34. Al-Qureshi M, Rashid S, Jarad F, Alharthi MS. Dynamical behavior of a stochastic highly pathogenic avian influenza A (HPAI) epidemic model via piecewise fractional differential technique. AIMS Math. 2023;8(1):1737-1756. https://doi.org/10.3934/math.2023089
  35. Yaagoub Z, Farah EM, Ahmad S. Three-strain epidemic model for influenza virus involving fractional derivative and treatment. J Appl Math Comput. 2025;71(1):1247-1266. https://doi.org/10.1007/s12190-024-02284-0
  36. Yasin F, Afzal Z, Saleem MS, Jahangir N, Shang Y. Hermite-Hadamard type inequality for non-convex functions employing the Caputo-Fabrizio fractional integral. Res Math. 2024;11(1):2366164. https://doi.org/10.1080/27684830.2024.2366164
  37. Shang Y. Vulnerability of networks: Fractional percolation on random graphs. Phys Rev E. 2014;89(1):012813. https://doi.org/10.1103/PhysRevE.89.012813
  38. Ye X, Xu C. A fractional order epidemic model and simulation for avian influenza dynamics. Math Methods Appl Sci. 2019;42(14):4765-4779. https://doi.org/10.1002/mma.5690
  39. González-Parra G, Arenas AJ, Aranda DF, Segovia L. Modeling the epidemic waves of AH1N1/09 influenza around the world. Spat Spatiotemporal Epidemiol. 2011;2(4):219-226. https://doi.org/10.1016/j.sste.2011.05.002
  40. Evirgen F, Uçar E, Uçar S, Özdemir N. Modelling influenza A disease dynamics under Caputo-Fabrizio fractional derivative with distinct contact rates. Math Model Numer Simul Appl. 2023;3(1):58-73. https://doi.org/10.53391/mmnsa.1274004
  41. Atangana A, Alkahtani BST. Modeling the spread of rubella disease using the concept of local derivative with fractional parameter: Beta-derivative. 2016;21(6):442-451. https://doi.org/10.1002/cplx.21704
  42. Rashid S, Sultana S, Karaca Y, Khalid A, Chu YM. Some further extensions considering discrete proportional fractional operators. 2022;30(1):2240026. https://doi.org/10.1142/S0218348X22400266
  43. Rashid S, Khalid A, Sultana S, Jarad F, Abualnaja KM, Hamed YS. Novel numerical investigation of the fractional oncolytic effectiveness model with M1 virus via generalized fractional derivative with optimal criterion. Results Phys. 2022;37:105553. https://doi.org/10.1016/j.rinp.2022.105553
  44. Gao W, Baskonus HM, Shi L. New investigation of bats-hosts-reservoir-people coronavirus model and application to 2019-nCoV system. Adv Differ Equ. 2020;2020(1):391. https://doi.org/10.1186/s13662-020-02831-6
  45. Doungmo Goufo EF, Khan Y, Chaudhry QA. HIV and shifting epicenters for COVID-19, an alert for some countries. Chaos Solitons Fractals. 2020;139:110030. https://doi.org/10.1016/j.chaos.2020.110030
  46. Ismail H, Hariharan S, Jeyaraj M, Shangerganesh L, Rong L. Hopf bifurcation and optimal control studies for avian-influenza virus with multi-delay model. Adv Contin Discret Models. 2025;2025:77. https://doi.org/10.1186/s13662-025-03936-6
  47. Atangana A, Araz Sİ. Modeling third waves of COVID-19 spread with piecewise differential and integral operators: Turkey, Spain and Czechia. Results Phys. 2021;29:104694. https://doi.org/10.1016/j.rinp.2021.104694
  48. Hristov J. On the Atangana-Baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation. In: Fractional Derivatives with Mittag-Leffler Kernel: Trends and Applications in Science and Engineering. Springer; 2019:175-193. https://doi.org/10.1007/978-3-030-11662-0_11
  49. Althubyani M, Adam HDS, Alalyani A, et al. Understanding zoonotic disease spread with a fractional order epidemic model. Sci Rep. 2025;15:13921. https://doi.org/10.1038/s41598-025-95943-6
  50. von Waldburg-Zeil CG, van Staaveren N, Harlander-Matauschek A. Do laying hens eat and forage in excreta from other hens? 2019;13(2):367-373. https://doi.org/10.1017/S1751731118001143
  51. Zahra F, Li Z, Al-Ahmari A. Analysis and optimal control of a fractional order influenza epidemic model. Sci Rep. 2025;16:38. https://doi.org/10.1038/s41598-025-28790-0
  52. Gao HN, Lu HZ, Cao B, et al. Clinical findings in 111 cases of influenza A (H7N9) virus infection. N Engl J Med. 2013;368(24):2277-2285. https://doi.org/10.1056/NEJMoa1305584
  53. Alsubaie NE, El Guma F, Boulehmi K, Al-kuleab N, Abdoon MA. Improving influenza epidemiological models under Caputo fractional-order calculus. 2024;16(7):929. https://doi.org/10.3390/sym16070929
  54. Xing Y, Song L, Sun GQ, Jin Z, Zhang J. Assessing reappearance factors of H7N9 avian influenza in China. Appl Math Comput. 2017;309:192-204. https://doi.org/10.1016/j.amc.2017.04.007
  55. World Health Organization (WHO), Food and Agriculture Organization of the United Nations (FAO), World Organisation for Animal Health (OIE). Taking a Multisectoral, One Health Approach: A Tripartite Guide to Addressing Zoonotic Diseases in Countries. Geneva: World Health Organization; 2019. ISBN: 978-92-4-151493-4. https://openknowledge.fao.org/items/870b5a92-2fc4-430e-87a9-f8086335acc6
  56. Evirgen F, Uçar S, Özdemir N. Mathematical analysis and optimal control of a Caputo fractional diabetes system with parameter identification. J Comput Appl Math. 2026;477:117151. https://doi.org/10.1016/j.cam.2025.117151
  57. Uçar S. Tuberculosis disease dynamics with fractal-fractional derivative under the use of real data. J Comput Appl Math. 2026;473:116889. https://doi.org/10.1016/j.cam.2025.116889
  58. Uçar E. Examining of a tumor system with Caputo derivative. J Balıkesir Univ Inst Sci Technol. 2023;25(1):37-48. https://doi.org/10.25092/baunfbed.1113646
  59. Evirgen F, Uçar S, Özdemir N, Hammouch Z. System response of an alcoholism model under the effect of immigration via non-singular kernel derivative. Discrete Contin Dyn Syst Ser S. 2021;14(7):2199-2212. https://doi.org/10.3934/dcdss.2020145
  60. Uçar S, Evirgen F, Özdemir N, Hammouch Z. Mathematical analysis and simulation of a giving up smoking model within the scope of non-singular derivative. Proc Inst Math Mech Natl Acad Sci Azerb. 2022;48:84-99. https://doi.org/10.30546/2409-4994.48.2022.8499
  61. Dharmalingam KM, Jeeva N, Ali N, et al. Mathematical analysis of Zika virus transmission: Exploring semi-analytical solutions and effective controls. Commun Math Biol Neurosci. 2024;2024:112. https://doi.org/10.28919/cmbn/8719
  62. Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol. 2008;254(1):178-196. https://doi.org/10.1016/j.jtbi.2008.04.011
Share
Back to top
An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept