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  4. Global stability of fractional partial differential equations applied to the biological system modeling a viral infection with Hattaf time-fractional derivative

Global stability of fractional partial differential equations applied to the biological system modeling a viral infection with Hattaf time-fractional derivative

MMC.
2024;
Volume 11, Number 2
: pp. 430–437
https://doi.org/10.23939/mmc2024.02.430
Received: December 20, 2023
Revised: May 05, 2024
Accepted: May 10, 2024

Assadiki F., Hattaf K., Yousfi N.  Global stability of fractional partial differential equations applied to the biological system modeling a viral infection with Hattaf time-fractional derivative.  Mathematical Modeling and Computing. Vol. 11, No. 2, pp. 430–437 (2024)

Authors:
  1. F. Assadiki
  2. K. Hattaf
  3. N. Yousfi
1
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca
2
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca; Equipe de Recherche en Modélisation et Enseignement des Mathématiques (ERMEM), Centre Régional des Métiers de l'Education et de la Formation (CRMEF)
3
Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M'Sick, Hassan II University of Casablanca

In this article, we study the global stability of fractional partial differential equations applied to the biological system modeling a viral infection.  The reaction in the proposed biological system is described by the new generalized Hattaf fractional (GHF) derivative.  However, the diffusion is modeled by the Laplacian operator.

fractional partial differential equation
Hattaf time-fractional derivative
diffusion
global stability
  1. Xu R., Ma Z. E.  An HBV model with diffusion and time delay.  Journal of Theoretical Biology.  257 (3), 499–509 (2009).
  2. Fang Z., Gao C.  Lyapunov function partial differential equations for chemical reaction networks: Some special cases.  SIAM Journal on Applied Dynamical Systems.  18 (2), 1163–1199 (2019).
  3. Mohd M H.  Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology.  Dynamical Systems, Bifurcation Analysis and Applications.  117–132 (2019).
  4. Holmes E. E., Lewis M. A., Banks J. E., Veit R. R.  Partial differential equations in ecology: spatial interactions and population dynamics.  Ecology.  75 (1), 17–29 (1994).
  5. Thabet H., Kendre S.  A fractional mathematical model with nonlinear partial differential equations for transmission dynamics of severe acute respiratory syndrome coronavirus 2 infection.  Healthcare Analytics.  4, 100209 (2023).
  6. Wang H., Yamamoto N.  Using a partial differential equation with Google Mobility data to predict COVID-19 in Arizona.  Mathematical Biosciences and Engineering.  17 (5), 4891–4904 (2020).
  7. Ghaffari A., Moura S., Krsti M.  Modeling, control, and stability analysis of heterogeneous thermostatically controlled load populations using partial differential equations.  Journal of Dynamic Systems, Measurement, and Control.  137 (10), 101009 (2015).
  8. Dorofeenko V., Shorish J.  Partial differential equation modelling for stochastic fixed strategy distributed systems.  Journal of Economic Dynamics and Control.  29 (1–2), 335–367 (2005).
  9. Boucekkine R., Camacho C., Fabbri G.  On the optimal control of some parabolic partial differential equations arising in economics.  Serdica Mathematical Journal.  39 (3), 331–354 (2013).
  10. Wang C., Zhao X., Zhang Y., Lv Z.  Global existence and fixed–time synchronization of a hyperchaotic financial system governed by semi-linear parabolic partial differential equations equipped with the homogeneous Neumann boundary condition.  Entropy.  25 (2), 359 (2023).
  11. Wang K., Wang W.  Propagation of HBV with spatial dependence.  Mathematical Biosciences.  210 (1), 78–95 (2007).
  12. Wang K., Wang W., Song S.  Dynamics of an HBV model with diffusion and delay.  Journal of Theoretical Biology.  253 (1), 36–44 (2008).
  13. Alagha A. D., Elaiw A. M.  Stability of a general reaction-diffusion HIV-1 dynamics model with humoral immunity.  The European Physical Journal Plus.  134, 390 (2019).
  14. Hattaf K., Yousfi N.  Global properties of a diffusive HBV infection model with cell-to-cell transmission and three distributed delays.  Disease Prevention and Health Promotion in Developing Countries.  117–131 (2020).
  15. Hattaf K., Yousfi N.  A generalized HBV model with diffusion and two delays.  Computers & Mathematics with Applications.  69 (1), 31–40 (2015).
  16. Manna K., Chakrabarty S. P.  Chronic hepatitis B infection and HBV DNA–containing capsids: Modeling and analysis.  Communications in Nonlinear Science and Numerical Simulation.  22 (1–3), 383–395 (2015).
  17. Manna K.  Dynamics of a diffusion–driven HBV infection model with capsids and time delay.  International Journal of Biomathematics.  10 (05), 1750062 (2017).
  18. Guo T., Liu H., Liu H., Xu C., Yan F.  Global stability of a diffusive and delayed HBV infection model with HBV DNA containing capsids and general incidence rate.  Discrete and Continuous Dynamical Systems–B.  23 (10), 4223–4242 (2018).
  19. Nawaz Y., Arif M. S., Shatanawi W.  A new numerical scheme for time fractional diffusive seair model with non–linear incidence rate: An application to computational biology.  Fractal and Fractional.  6 (2), 78 (2022).
  20. Hattaf K., Yousfi N.  Global stability for fractional diffusion equations in biological systems.  Complexity.  2020, 5476842 (2020).
  21. Hamadneh T., Hioual A., Alsayyed O., Al-Khassawneh Y. A., Al-Husban A., Ouannas A.  Local Stability, Global Stability, and Simulations in a Fractional Discrete Glycolysis Reaction–Diffusion Model.  Fractal and Fractional.  7 (8), 587 (2023).
  22. Ahmed N., Tahira S. S., Imran M., Rafiq M., Rehman M. A., Younis M.  Numerical analysis of auto-catalytic glycolysis model.  AIP Advances.  9 (8), 085213 (2019).
  23. Noufaey K. S.  Stability analysis for Selkov–Schnakenberg reaction–diffusion system.  Open Mathematics.  19 (1), 46–62 (2021).
  24. Wu G. C., Baleanu D., Zeng S. D., Deng Z. G.  Discrete fractional diffusion equation.  Nonlinear Dynamics.  80, 281–286 (2015).
  25. Hattaf K.  A new generalized definition of fractional derivative with non-singular kernel.  Computation.  8 (2), 49 (2020).
  26. Atangana A., Baleanu D.  New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model.  Thermal Science.  20 (2), 763–769 (2016).
  27. Al-Refai M.  On weighted Atangana–Baleanu fractional operators.  Advances in Difference Equations.  2020, 3 (2020).
  28. Caputo M., Fabrizio M.  A new definition of fractional derivative without singular kernel.  Progress in Fractional Differentiation and Applications.  1 (2), 73–85 (2015).
  29. Hattaf K.  On some properties of the new generalized fractional derivative with non-singular kernel.  Mathematical Problems in Engineering.  2021, 1580396 (2021).
  30. Hattaf K., Yousfi N., Tridane A.  Stability analysis of a virus dynamics model with general incidence rate and two delays.  Applied Mathematics and Computation.  221, 514–521 (2013).
  31. Hattaf K.  On the stability and numerical scheme of fractional differential equations with application to biology.  Computation.  10 (6), 97 (2022).
  32. Beddington J. R.  Mutual interference between parasites or predators and its effect on searching efficiency.  Journal of Animal Ecology.  44 (1), 331–340 (1975).
  33. DeAngelis D. L., Goldstein R. A., O'Neill R. V.  A model for trophic interaction.  Ecology.  56 (4), 881–892 (1975).
  34. Crowley P. H., Martin E. K.  Functional responses and interference within and between year classes of a dragonfly population.  Journal of the North American Benthological Society.  8 (3), 211–221 (1989).
  35. Hattaf K., Yousfi N.  A class of delayed viral infection models with general incidence rate and adaptive immune response.  International Journal of Dynamics and Control.  4, 254–265 (2015).
  36. Nangue A., Tchiffo B. N.  Global analysis of a spatiotemporal cellular model for the transmission of hepatitis C virus with Hattaf–Yousfi functional response.  Preprint arXiv:2111.03312 (2021).
  37. Phukan J., Dutta H.  Dynamic analysis of a fractional order SIR model with specific functional response and Holling type II treatment rate.  Chaos, Solitons & Fractals.  175 (1), 114005 (2023).
  38. Silva C. J., Torres D. F.  On SICA models for HIV transmission.  Mathematical Modelling and Analysis of Infectious Diseases. 155–179 (2020).
  39. Lotfi E. M., Mahrouf M., Maziane M., Silva C J., Torres D. F., Yousfi N.  A minimal HIV–AIDS infection model with general incidence rate and application to Morocco data.  Statistics, Optimization and Information Computing.  7 (3), 588–603 (2019).