Fractal-Fractional Modeling and Analysis of Monkeypox Disease Using Atangana-Baleanu Derivative

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-2
Published: Jan 6, 2026
Cite as:
Tharmalingam Gunasekar, Shanmugam Manikandan, Murgan Suba, Irshad Ayoob, Nabil Mlaiki, Fractal-Fractional Modeling and Analysis of Monkeypox Disease Using Atangana-Baleanu Derivative, Int. J. Anal. Appl., 24 (2026), 2.

Main Article Content

Tharmalingam Gunasekar, Shanmugam Manikandan, Murgan Suba, Irshad Ayoob, Nabil Mlaiki

Abstract

In this study, we formulate a deterministic mathematical model to describe the transmission dynamics of the monkeypox virus using fractal and fractional-order differential equations. The model incorporates all possible interactions influencing disease propagation within the population. Our analysis primarily focuses on the stability of fractal–fractional derivatives, aiming to establish the existence and uniqueness of solutions through the fixed-point theorem. Additionally, we examine Ulam-Hyers stability and other significant findings related to the proposed model. To enhance numerical accuracy, we employ Lagrange polynomial interpolation for computational approximations. Finally, graphical simulations for various fractal–fractional orders are presented to validate the model’s effectiveness and demonstrate its practical relevance.

Article Details

References

  1. I.D. Ladnyj, P. Ziegler, E. Kima, A Human Infection Caused by Monkeypox Virus in Basankusu Territory, Democratic Republic of the Congo, Bull. World Health Organ. 46 (1972), 593–597.
  2. I. Arita, D.A. Henderson, Smallpox and Monkeypox in Non-Human Primates, Bull. World Health Organ. 39 (1968), 277.
  3. D.L. Heymann, M. Szczeniowski, K. Esteves, Re-emergence of Monkeypox in Africa: a Review of the Past Six Years, Br. Med. Bull. 54 (1998), 693–702. https://doi.org/10.1093/oxfordjournals.bmb.a011720.
  4. WHO, Monkeypox Fact Sheet, World Health Organization. https://www.who.int/news-room/fact-sheets/detail/monkeypox.
  5. C. Bhunu, S. Mushayabasa, J. Hyman, Modelling Hiv/aids and Monkeypox Co-Infection, Appl. Math. Comput. 218 (2012), 9504–9518. https://doi.org/10.1016/j.amc.2012.03.042.
  6. A.P. Lemos-Paião, C.J. Silva, D.F. Torres, An Epidemic Model for Cholera with Optimal Control Treatment, J. Comput. Appl. Math. 318 (2017), 168–180. https://doi.org/10.1016/j.cam.2016.11.002.
  7. E. Bonyah, M. Khan, K. Okosun, J. Gómez‐Aguilar, On the Co‐infection of Dengue Fever and Zika Virus, Optim. Control. Appl. Methods 40 (2019), 394–421. https://doi.org/10.1002/oca.2483.
  8. M. Suba, A. Venkatesan, T. Gunasekar, S. Manikandan, P. Raghavendran, S.S. Santra, S. Karmakar, Advancing Zika Virus Control with Fractional Order Optimal Control with Wolbachia-Infected Aedes Aegypti Mosquitoes, in: J. Singh, G.A. Anastassiou, D. Baleanu, D. Kumar (Eds.), Advances in Mathematical Modelling, Applied Analysis and Computation, Springer Nature Switzerland, Cham, 2025: pp. 160–183. https://doi.org/10.1007/978-3-031-90917-7_11.
  9. O. Makinde, K. Okosun, Impact of Chemo-Therapy on Optimal Control of Malaria Disease with Infected Immigrants, Biosystems 104 (2011), 32–41. https://doi.org/10.1016/j.biosystems.2010.12.010.
  10. E. Bonyah, K. Badu, S.K. Asiedu-Addo, Optimal Control Application to an Ebola Model, Asian Pac. J. Trop. Biomed. 6 (2016), 283–289. https://doi.org/10.1016/j.apjtb.2016.01.012.
  11. P.A. Naik, J. Zu, K.M. Owolabi, Global Dynamics of a Fractional Order Model for the Transmission of Hiv Epidemic with Optimal Control, Chaos, Solitons Fractals 138 (2020), 109826. https://doi.org/10.1016/j.chaos.2020.109826.
  12. K.S. Nisar, K. Logeswari, V. Vijayaraj, H.M. Baskonus, C. Ravichandran, Fractional Order Modeling the Gemini Virus in Capsicum Annuum with Optimal Control, Fractal Fract. 6 (2022), 61. https://doi.org/10.3390/fractalfract6020061.
  13. A. Atangana, Modelling the Spread of Covid-19 with New Fractal-Fractional Operators: Can the Lockdown Save Mankind Before Vaccination?, Chaos Solitons Fractals 136 (2020), 109860. https://doi.org/10.1016/j.chaos.2020.109860.
  14. X. Liu, S. Ahmad, M.U. Rahman, Y. Nadeem, A. Akgül, Analysis of a Tb and Hiv Co-Infection Model Under Mittag-Leffler Fractal-Fractional Derivative, Phys. Scr. 97 (2022), 054011. https://doi.org/10.1088/1402-4896/ac645e.
  15. S. Ahmad, A. Ullah, A. Akgül, M. De la Sen, Study of Hiv Disease and Its Association with Immune Cells Under Nonsingular and Nonlocal Fractal‐fractional Operator, Complexity 2021 (2021), 1904067. https://doi.org/10.1155/2021/1904067.
  16. S. Etemad, B. Tellab, A. Zeb, S. Ahmad, A. Zada, S. Rezapour, H. Ahmad, T. Botmart, A Mathematical Model of Transmission Cycle of Cc-Hemorrhagic Fever Via Fractal–fractional Operators and Numerical Simulations, Results Phys. 40 (2022), 105800. https://doi.org/10.1016/j.rinp.2022.105800.
  17. Z. Ali, F. Rabiei, K. Shah, T. Khodadadi, Fractal-fractional Order Dynamical Behavior of an Hiv/aids Epidemic Mathematical Model, Eur. Phys. J. Plus 136 (2021), 36. https://doi.org/10.1140/epjp/s13360-020-00994-5.
  18. J. Kongson, C. Thaiprayoon, A. Neamvonk, J. Alzabut, W. Sudsutad, Investigation of Fractal-Fractional Hiv Infection by Evaluating the Drug Therapy Effect in the Atangana-Baleanu Sense, Math. Biosci. Eng. 19 (2022), 10762–10808. https://doi.org/10.3934/mbe.2022504.
  19. M. Farman, M. Amin, A. Akgül, A. Ahmad, M.B. Riaz, S. Ahmad, Fractal–fractional Operator for Covid-19 (omicron) Variant Outbreak with Analysis and Modeling, Results Phys. 39 (2022), 105630. https://doi.org/10.1016/j.rinp.2022.105630.
  20. A. Boukhouima, K. Hattaf, E.M. Lotfi, M. Mahrouf, D.F. Torres, N. Yousfi, Lyapunov Functions for Fractional-Order Systems in Biology: Methods and Applications, Chaos, Solitons Fractals 140 (2020), 110224. https://doi.org/10.1016/j.chaos.2020.110224.
  21. M. Amin, M. Farman, A. Akgül, R.T. Alqahtani, Effect of Vaccination to Control Covid-19 with Fractal Fractional Operator, Alex. Eng. J. 61 (2022), 3551–3557. https://doi.org/10.1016/j.aej.2021.09.006.
  22. H. Khan, F. Ahmad, O. Tunç, M. Idrees, On Fractal-Fractional Covid-19 Mathematical Model, Chaos, Solitons Fractals 157 (2022), 111937. https://doi.org/10.1016/j.chaos.2022.111937.
  23. T. Gunasekar, S. Manikandan, S. Haque, M. Suba, N. Mlaiki, Fractal-fractional Mathematical Modeling of Monkeypox Disease and Analysis of Its Ulam–hyers Stability, Bound. Value Probl. 2025 (2025), 20. https://doi.org/10.1186/s13661-025-02013-x.
  24. T. Gunasekar, S. Manikandan, M. Suba, A. Akgül, A Fractal-Fractional Mathematical Model for Covid-19 and Tuberculosis Using Atangana–baleanu Derivative, Math. Comput. Model. Dyn. Syst. 30 (2024), 857–881. https://doi.org/10.1080/13873954.2024.2426608.
  25. S. Manikandan, T. Gunasekar, A. Kouidere, K.A. Venkatesan, K. Shah, T. Abdeljawad, Mathematical Modelling of Hiv/aids Treatment Using Caputo–fabrizio Fractional Differential Systems, Qual. Theory Dyn. Syst. 23 (2024), 149. https://doi.org/10.1007/s12346-024-01005-z.
  26. H. Khan, K. Alam, H. Gulzar, S. Etemad, S. Rezapour, A Case Study of Fractal-Fractional Tuberculosis Model in China: Existence and Stability Theories Along with Numerical Simulations, Math. Comput. Simul. 198 (2022), 455–473. https://doi.org/10.1016/j.matcom.2022.03.009.
  27. T. Gunasekar, S. Manikandan, V. Govindan, P. D, J. Ahmad, W. Emam, I. Al-Shbeil, Symmetry Analyses of Epidemiological Model for Monkeypox Virus with Atangana–baleanu Fractional Derivative, Symmetry 15 (2023), 1605. https://doi.org/10.3390/sym15081605.