SpringerLink
Log in

Existence, stability, and numerical simulations of a fractal-fractional hepatitis B virus model

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper uses a new fractal-fractional operator with a power law-type kernel in the Riemann-Liouville sense to formulate the new fractal-fractional model of hepatitis B virus (HBV) transmission with asymptomatic carriers. The existence of the model’s solutions is demonstrated using Schuder’s fixed point theorem. The Banach fixed point theorem is utilized to prove the uniqueness of the solutions. Solutions’ stability behaviors in the Ulam concept are also discussed. Further, using the newly created numerical scheme based on Newton’s polynomial, the new numerical scheme for HBV is created. Numerical simulations show the accuracy of the approximate solutions of the new numerical method, along with the clear effect of the fractal dimension and fractional order on the spread of the HBV disease.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Atangana, Abdon. Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, solitons & fractals, 2017, vol. 102, p. 396-406.

    Article  MathSciNet  Google Scholar 

  2. Martcheva, M. (2015). An introduction to mathematical epidemiology (Vol. 61, pp. 9–31). New York: Springer.

  3. Cole, K. S. (1933, January). Electric conductance of biological systems. In Cold Spring Harbor symposia on quantitative biology (Vol. 1, pp. 107-116). Cold Spring Harbor Laboratory Press.

  4. Ahmed, E., El-Sayed, A. M. A., & El-Saka, H. A. (2007). Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. Journal of Mathematical Analysis and Applications, 325(1), 542-553.

    Article  MathSciNet  Google Scholar 

  5. Rihan, F. A. (2013, August). Numerical modeling of fractional-order biological systems. In Abstract and Applied Analysis (Vol. 2013). Hindawi.

  6. El-Sayed, A. M. A., Rida, S. Z., & Arafa, A. A. M. (2009). On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber. International Journal of Nonlinear Science, 7(4), 485-492.

    MathSciNet  Google Scholar 

  7. Ding, Y., Wang, Z., & Ye, H. (2011). Optimal control of a fractional-order HIV-immune system with memory. IEEE Transactions on Control Systems Technology, 20(3), 763-769.

    Article  Google Scholar 

  8. Khan, A., Gómez-Aguilar, J. F., Khan, T. S., and Khan, H. (2019). Stability analysis & numerical solutions of fractional order HIV/AIDS model. Chaos, Solitons & Fractals, 122, 119-128.

    Article  MathSciNet  Google Scholar 

  9. Naik, P. A., Yavuz, M., Qureshi, S., Zu, J., & Townley, S. (2020). Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42.

    Article  Google Scholar 

  10. Rezapour, S., Etemad, S., and Mohammadi, H. (2020). A mathematical analysis of a system of Caputo-Fabrizio fractional differential equations for the anthrax disease model in animals. Advances in Difference Equations, 2020(1), 1-30.

    Article  MathSciNet  Google Scholar 

  11. Ahmad, M., Imran, M. A., & Nazar, M. (2020). Mathematical modeling of (Cu A l 2 O 3) water based Maxwell hybrid nanofluids with Caputo-Fabrizio fractional derivative. Advances in Mechanical Engineering, 12(9), 1687814020958841.

    Article  Google Scholar 

  12. Abouelregal, A. E., Akgöz, B., & Civalek, Ö. (2022). Nonlocal thermoelastic vibration of a solid medium subjected to a pulsed heat flux via Caputo-Fabrizio fractional derivative heat conduction. Applied Physics A, 128(8), 660.

    Article  Google Scholar 

  13. Abouelregal, A. E., Nofal, T. A., & Alsharari, F. (2022). A thermodynamic two-temperature model with distinct fractional derivative operators for an infinite body with a cylindrical cavity and varying properties. Journal of Ocean Engineering and Science.

  14. Raslan, W. E. (2021). Fractional mathematical modeling for epidemic prediction of COVID-19 in Egypt. Ain Shams Engineering Journal, 12(3), 3057-3062.

    Article  Google Scholar 

  15. Nortey, S. N., Juga, M., and Bonyah, E. (2022). Fractional order modelling of Anthrax-Listeriosis coinfection with nonsingular Mittag leffler law. Scientific African, e01221.

  16. Chu, Y. M., Yassen, M. F., Ahmad, I., Sunthrayuth, P., & Khan, M. A. (2022). A FRACTIONAL SARS-COV-2 MODEL WITH ATANGANA-BALEANU DERIVATIVE: APPLICATION TO FOURTH WAVE. Fractals, 30(08), 2240210.

  17. El hadj Moussa, Y., Boudaoui, A., Ullah, S., Bozkurt, F., Abdeljawad, T., & Alqudah, M. A. (2021). Stability analysis and simulation of the novel Corornavirus mathematical model via the Caputo fractional-order derivative: A case study of Algeria. Results in physics, 26, 104324.

  18. Shen, W. Y., Chu, Y. M., ur Rahman, M., Mahariq, I., and Zeb, A. (2021). Mathematical analysis of HBV and HCV co-infection model under nonsingular fractional order derivative. Results in Physics, 28, 104582.

  19. Chen, S. B., Rajaee, F., Yousefpour, A., Alcaraz, R., Chu, Y. M., Gómez-Aguilar, J. F., ... & Jahanshahi, H. (2021). Antiretroviral therapy of HIV infection using a novel optimal type-2 fuzzy control strategy. Alexandria engineering journal, 60(1), 1545-1555.

  20. Chu, Y. M., Ali, A., Khan, M. A., Islam, S., and Ullah, S. (2021). Dynamics of fractional order COVID-19 model with a case study of Saudi Arabia. Results in Physics, 21, 103787.

    Article  Google Scholar 

  21. Ullah, S., Altaf Khan, M., & Farooq, M. (2018). Modeling and analysis of the fractional HBV model with Atangana-Baleanu derivative. The European Physical Journal Plus, 133(8), 1-18.

    Article  Google Scholar 

  22. Din, A., Li, Y., Khan, F. M., Khan, Z. U., & Liu, P. (2022). On Analysis of fractional order mathematical model of Hepatitis B using Atangana-Baleanu Caputo (ABC) derivative. Fractals, 30(01), 2240017.

    Article  Google Scholar 

  23. Danane, J., Allali, K., & Hammouch, Z. (2020). Mathematical analysis of a fractional differential model of HBV infection with antibody immune response. Chaos, Solitons & Fractals, 136, 109787.

    Article  MathSciNet  Google Scholar 

  24. Shah, S. A. A., Khan, M. A., Farooq, M., Ullah, S., & Alzahrani, E. O. (2020). A fractional order model for Hepatitis B virus with treatment via Atangana-Baleanu derivative. Physica A: Statistical Mechanics and its Applications, 538, 122636.

    Article  MathSciNet  Google Scholar 

  25. Gul, N., Bilal, R., Algehyne, E. A., Alshehri, M. G., Khan, M. A., Chu, Y. M., & Islam, S. (2021). The dynamics of fractional order Hepatitis B virus model with asymptomatic carriers. Alexandria Engineering Journal, 60(4), 3945-3955.

    Article  Google Scholar 

  26. Asamoah, J. K. K. (2022). Fractal-fractional model and numerical scheme based on Newton polynomial for Q fever disease under Atangana-Baleanu derivative. Results in Physics, 34, 105189.

    Article  Google Scholar 

  27. Najafi, H., Etemad, S., Patanarapeelert, N., Asamoah, J. K. K., Rezapour, S., & Sitthiwirattham, T. (2022). A study on dynamics of CD4+ T-cells under the effect of HIV-1 infection based on a mathematical fractal-fractional model via the Adams-Bashforth scheme and Newton polynomials. Mathematics, 10(9), 1366.

    Article  Google Scholar 

  28. Li, X. P., Al Bayatti, H., Din, A., & Zeb, A. (2021). A vigorous study of fractional order COVID-19 model via ABC derivatives. Results in Physics, 29, 104737.

    Article  Google Scholar 

  29. Priya, P., & Sabarmathi, A. (2022). Caputo Fractal Fractional Order Derivative of Soil Pollution Model Due to Industrial and Agrochemical. International Journal of Applied and Computational Mathematics, 8(5), 1-22.

    Article  MathSciNet  Google Scholar 

  30. Atangana, A., & Araz, S. I. (2021). New numerical scheme with Newton polynomial: theory, methods, and applications. Academic Press.

    Google Scholar 

  31. Alkahtani, B. S. T. (2020). A new numerical scheme based on Newton polynomial with application to Fractional nonlinear differential equations. Alexandria Engineering Journal, 59(4), 1893-1907.

    Article  Google Scholar 

  32. Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation & Applications, 1(2), 73-85.

    Google Scholar 

  33. Saad, K. M., Atangana, A., & Baleanu, D. (2018). New fractional derivatives with non-singular kernel applied to the Burgers equation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(6).

  34. Pandey, P., Chu, Y. M., Gómez-Aguilar, J. F., Jahanshahi, H., & Aly, A. A. (2021). A novel fractional mathematical model of COVID-19 epidemic considering quarantine and latent time. Results in physics, 26, 104286.

    Article  Google Scholar 

  35. Ali, Z., Rabiei, F., Shah, K., & Khodadadi, T. (2021). Fractal-fractional order dynamical behavior of an HIV/AIDS epidemic mathematical model. The European Physical Journal Plus, 136(1), 36.

    Article  Google Scholar 

  36. Chu, Y. M., Zarin, R., Khan, A., & Murtaza, S. (2023). A vigorous study of fractional order mathematical model for SARS-CoV-2 epidemic with Mittag-Leffler kernel. Alexandria Engineering Journal, 71, 565-579.

    Article  Google Scholar 

  37. Li, Z., Liu, Z., & Khan, M. A. (2020). Fractional investigation of bank data with fractal-fractional Caputo derivative. Chaos, Solitons & Fractals, 131, 109528.

    Article  MathSciNet  Google Scholar 

  38. Guran, L., Akgül, E. K., Akgül, A., & Bota, M. F. (2022). Remarks on fractal-fractional Malkus Waterwheel model with computational analysis. Symmetry, 14(10), 2220.

    Article  Google Scholar 

  39. Zhong, J. F., Gul, N., Bilal, R., **a, W. F., Khan, M. A., Muhammad, T., & Islam, S. (2021). A fractal-fractional order Atangana-Baleanu model for Hepatitis B virus with asymptomatic class. Physica Scripta, 96(7), 074001.

    Article  Google Scholar 

  40. Zhao, S., Xu, Z., & Lu, Y. (2000). A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China. International journal of epidemiology, 29(4), 744-752.

    Article  Google Scholar 

  41. Simelane, S. M., & Dlamini, P. G. (2021). A fractional order differential equation model for hepatitis B virus with saturated incidence. Results in Physics, 24, 104114.

    Article  Google Scholar 

  42. Ullah, S., Altaf Khan, M., & Farooq, M. (2018). A new fractional model for the dynamics of the hepatitis B virus using the Caputo-Fabrizio derivative. The European Physical Journal Plus, 133, 1-14.

    Article  Google Scholar 

  43. Habenom, H., Suthar, D. L., Baleanu, D., & Purohit, S. D. (2021). A numerical simulation on the effect of vaccination and treatments for the fractional hepatitis b model. Journal of Computational and Nonlinear Dynamics, 16(1), 011004.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Applied Mathematics, Kasdi Merbah University, 30000, Ouargla, Algeria

    Meroua Medjoudja & Mohammed El hadi Mezabia

  2. Department of Mathematics, College of Arts & Sciences, Najran University, Najran, Kingdom of Saudi Arabia

    Fawaz K. Alalhareth

  3. Laboratory of Mathematics Modeling and Applications, University of Adrar, Adrar, Algeria

    Ahmed Boudaoui

Authors
  1. Meroua Medjoudja
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammed El hadi Mezabia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fawaz K. Alalhareth
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ahmed Boudaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmed Boudaoui.

Additional information

Communicated by NM Bujurke.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Medjoudja, M., El hadi Mezabia, M., Alalhareth, F.K. et al. Existence, stability, and numerical simulations of a fractal-fractional hepatitis B virus model. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00612-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13226-024-00612-5

Keywords

Search

Navigation