International Journal of Differential Equations

Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate

Adnane Boukhouima, Khalid Hattaf, and Noura Yousfi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We propose a fractional order model in this paper to describe the dynamics of human immunodeficiency virus (HIV) infection. In the model, the infection transmission process is modeled by a specific functional response. First, we show that the model is mathematically and biologically well posed. Second, the local and global stabilities of the equilibria are investigated. Finally, some numerical simulations are presented in order to illustrate our theoretical results.

Article information

Int. J. Differ. Equ., Volume 2017, Special Issue (2017), Article ID 8372140, 8 pages.

Received: 11 June 2017
Accepted: 31 July 2017
First available in Project Euclid: 19 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Boukhouima, Adnane; Hattaf, Khalid; Yousfi, Noura. Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate. Int. J. Differ. Equ. 2017, Special Issue (2017), Article ID 8372140, 8 pages. doi:10.1155/2017/8372140.

Export citation


  • Y. A. Rossikhin and M. V. Shitikova, “Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids,” Applied Mechanics Reviews, vol. 50, no. 1, pp. 15–67, 1997.
  • R. J. Marks and M. W. Hall, “Differintegral Interpolation from a Bandlimited Signal's Samples,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 4, pp. 872–877, 1981.
  • G. L. Jia and Y. X. Ming, “Study on the viscoelasticity of cancellous bone based on higher-order fractional models,” in Proceedings of the 2nd International Conference on Bioinformatics and Biomedical Engineering, iCBBE 2008, pp. 1733–1736, chn, May 2008.
  • R. Magin, Fractional Calculus in Bioengineering, Cretical Reviews in Biomedical Engineering 32, vol. 32, 2004.
  • E. Scalas, R. Gorenflo, and F. Mainardi, “Fractional calculus and continuous-time finance,” Physica A. Statistical Mechanics and its Applications, vol. 284, no. 1-4, pp. 376–384, 2000.MR1773804
  • L. Song, S. Xu, and J. Yang, “Dynamical models of happiness with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 616–628, 2010.MR2572195
  • R. Capponetto, G. Dongola, L. Fortuna, and I. Petras, “Fractional order systems: Modelling and control applications,” in World Scientific Series in Nonlinear Science, vol. 72, Series A, Singapore, 2010.
  • K. S. Cole, “Electric conductance of biological systems,” Cold Spring Harbor Symposia on Quantitative Biology, vol. 1, pp. 107–116, 1933.
  • A. A. Arafa, S. Z. Rida, and M. Khalil, “A fractional-order model of HIV infection: numerical solution and comparisons with data of patients,” International Journal of Biomathematics, vol. 7, no. 4, Article ID 1450036, 1450036, 11 pages, 2014.MR3225564
  • A. A. M. Arafa, S. Z. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4 + T-cells during primary infection,” Nonlinear Biomedical Physics, vol. 6, no. 1, article 1, 2012.
  • Y. Liu, J. Xiong, C. Hu, and C. Wu, “Stability analysis for fractional differential equations of an HIV infection model with cure rate,” in Proceedings of the 2016 IEEE International Conference on Information and Automation, IEEE ICIA 2016, pp. 707–711, August 2016.
  • X. Zhou and J. Cui, “Global stability of the viral dynamics with Crowley-Martin functional response,” Bulletin of the Korean Mathematical Society, vol. 48, no. 3, pp. 555–574, 2011.MR2827765
  • G. Huang, W. Ma, and Y. Takeuchi, “Global properties for virus dynamics model with Beddington-DeAngelis functional response,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 22, no. 11, pp. 1690–1693, 2009.MR2569065
  • M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science, vol. 272, no. 5258, pp. 74–79, 1996.
  • P. K. Srivastava and P. Chandra, “Modeling the dynamics of HIV and CD4$^{+}$ T cells during primary infection,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 11, no. 2, pp. 612–618, 2010.MR2571236
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.MR1658022
  • W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007.MR2319693
  • Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007.MR2316514
  • E. Ahmed, A. M. El-Sayed, and H. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rossler, Chua and CHEn systems,” Physics Letters. A, vol. 358, no. 1, pp. 1–4, 2006.MR2244918
  • C. V. De-Leon, “Volterra-type Lyapunov functions for fractional-order epidemic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 24, no. 1-3, pp. 75–85, 2015.MR3313546
  • J. Huo, H. Zhao, and L. Zhu, “The effect of vaccines on backward bifurcation in a fractional order HIV model,” Nonlinear Analysis. Real World Applications, vol. 26, pp. 289–305, 2015.MR3384337
  • Z. Odibat and S. Momani, “An algorithm for the numerical solution of differential equations of fractional order,” Applied Mathematics & Information, vol. 26, no. 1, pp. 15–27, 2008. \endinput