International Journal of Differential Equations

A Fractional Order Model for Viral Infection with Cure of Infected Cells and Humoral Immunity

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


In this paper, we study the dynamics of a viral infection model formulated by five fractional differential equations (FDEs) to describe the interactions between host cells, virus, and humoral immunity presented by antibodies. The infection transmission process is modeled by Hattaf-Yousfi functional response which covers several forms of incidence rate existing in the literature. We first show that the model is mathematically and biologically well-posed. By constructing suitable Lyapunov functionals, the global stability of equilibria is established and characterized by two threshold parameters. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Article information

Int. J. Differ. Equ., Volume 2018, Special Issue (2018), Article ID 1019242, 12 pages.

Received: 9 September 2018
Accepted: 7 November 2018
First available in Project Euclid: 10 January 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Boukhouima, Adnane; Hattaf, Khalid; Yousfi, Noura. A Fractional Order Model for Viral Infection with Cure of Infected Cells and Humoral Immunity. Int. J. Differ. Equ. 2018, Special Issue (2018), Article ID 1019242, 12 pages. doi:10.1155/2018/1019242.

Export citation


  • K. Hattaf, N. Yousfi, and A. Tridane, “Global stability analysis of a generalized virus dynamics model with the immune response,” Canadian Applied Mathematics Quarterly, vol. 20, no. 4, pp. 499–518, 2012.MR3134709
  • M. Maziane, K. Hattaf, and N. Yousfi, “Global stability for a class of HIV infection models with cure of infected cells in eclipse stage and CTL immune response,” International Journal of Dynamics and Control, 2016.
  • X. Wang, Y. Tao, and X. Song, “Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response,” Nonlinear Dynamics, vol. 66, no. 4, pp. 825–830, 2011.MR2859606
  • C. Lv, L. Huang, and Z. Yuan, “Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 1, pp. 121–127, 2014.MR3142453
  • 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.
  • A. Murase, T. Sasaki, and T. Kajiwara, “Stability analysis of pathogen-immune interaction dynamics,” Journal of Mathematical Biology, vol. 51, no. 3, pp. 247–267, 2005.MR2206233
  • M. A. Obaid and A. M. Elaiw, “Stability of Virus Infection Models with Antibodies and Chronically Infected Cells,” Abstract and Applied Analysis, vol. 2014, Article ID 650371, 12 pages, 2014.MR3193532
  • M. A. Obaid, “Dynamical behaviors of a nonlinear virus infection model with latently infected cells and immune response,” Journal of Computational Analysis and Applications, vol. 21, no. 1, pp. 182–193, 2016.MR3469734
  • H. F. Huo, Y. L. Tang, and L. X. Feng, “A Virus Dynamics Model with Saturation Infection and Humoral Immunity,” Int. Journal of Math. Analysis, vol. 6, no. 40, 2012.
  • A. M. Elaiw, “Global stability analysis of humoral immunity virus dynamics model including latently infected cells,” Journal of Biological Dynamics, vol. 9, no. 1, pp. 215–228, 2015.MR3420650
  • A. M. Elaiw and N. H. AlShamrani, “Global properties of nonlinear humoral immunity viral infection models,” International Journal of Biomathematics, vol. 8, no. 5, 1550058, 53 pages, 2015.MR3383483
  • J. X. Velasco-Herna'ndez, J. A. Garci'a, and D. E. Kirschner, “Remarks on modeling host-viral dynamics and treatment, Mathematical Approaches for Emerging and Reemerging Infectious Diseases,” An Introduction to Models, Methods, and Theory, vol. 125, pp. 287–308, 2002.
  • A. S. Perelson, “Modelling viral and immune system dynamics,” Nature Reviews Immunology, vol. 2, no. 1, pp. 28–36, 2002.
  • R. L. Magin, “Fractional calculus models of complex dynamics in biological tissues,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1586–1593, 2010.
  • A. A. Stanislavsky, “Memory effects and macroscopic manifestation of randomness,” Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 61, no. 5, pp. 4752–4759, 2000.
  • M. Saeedian, M. Khalighi, N. Azimi-Tafreshi, G. R. Jafari, and M. Ausloos, “Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 95, no. 2, Article ID 022409, 2017.
  • 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 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 '08), pp. 1733–1736, Shanghai, China, May 2008.
  • R. Magin, “Fractional calculus in bioengineering,” Cretical reviews in biomedical engineering, vol. 32, pp. 13–77, 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
  • R. Capponetto, G. Dongola, L. Fortuna, and I. Petras, “Fractional order systems: Modelling and control applications,” World Scientific Series in Nonlinear Science, Series A, vol. 72, 2010.
  • V. D. Djordjević, J. Jarić, B. Fabry, J. J. Fredberg, and D. Stamenović, “Fractional derivatives embody essential features of cell rheological behavior,” Annals of Biomedical Engineering, vol. 31, no. 6, pp. 692–699, 2003.
  • K. S. Cole, “Electric conductance of biological systems,” Cold Spring Harbor Symposium on Quantitative Biology, vol. 1, pp. 107–116, 1933.
  • F. A. Rihan, M. Sheek-Hussein, A. Tridane, and R. Yafia, “Dynamics of hepatitis C virus infection: mathematical modeling and parameter estimation,” Mathematical Modelling of Natural Phenomena, vol. 12, no. 5, pp. 33–47, 2017.MR3716909
  • N. Khodabakhshi, S. M. Vaezpour, and D. Baleanu, “On Dynamics Of a Fractional-order Model Of HCV Infection,” Journal of Mathematical Analysis, vol. 8, no. 1, pp. 16–27, 2017.MR3636733
  • X. Zhou and Q. Sun, “Stability analysis of a fractional-order HBV infection model,” International Journal of Advances in Applied Mathematics and Mechanics, vol. 2, no. 2, pp. 1–6, 2014.MR3339946
  • S. S. M. and Y. A. M., “On a fractional-order model for HBV infection with cure of infected cells,” Journal of the Egyptian Mathematical Society, vol. 25, no. 4, pp. 445–451, 2017.MR3732356
  • Y. Ding and H. Ye, “A fractional-order differential equation model of HIV infection of CD4+ T-cells,” Mathematical and Computer Modelling, vol. 50, no. 3-4, pp. 386–392, 2009.
  • C. M. Pinto and A. R. Carvalho, “The role of synaptic transmission in a HIV model with memory,” Applied Mathematics and Computation, vol. 292, pp. 76–95, 2017.MR3542541
  • A. Boukhouima, K. Hattaf, and N. Yousfi, “Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate,” International Journal of Differential Equations, vol. 2017, Article ID 8372140, 8 pages, 2017.MR3696023
  • J. A. Deans and S. Cohen, “Immunology of malaria,” Annual Review of Microbiology, vol. 37, pp. 25–49, 1983.
  • I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, vol. 198, Academic press, 1998.
  • K. Hattaf and N. Yousfi, “A class of delayed viral infection models with general incidence rate and adaptive immune response,” International Journal of Dynamics and Control, vol. 4, no. 3, pp. 254–265, 2016.MR3536136
  • D. Riad, K. Hattaf, and N. Yousfi, “Dynamics of Capital-labour Model with Hattaf-Yousfi Functional Response,” British Journal of Mathematics & Computer Science, vol. 18, no. 5, pp. 1–7, 2016.
  • M. Mahrouf, K. Hattaf, and N. Yousfi, “Dynamics of a stochastic viral infection model with immune response,” Mathematical Modelling of Natural Phenomena, vol. 12, no. 5, pp. 15–32, 2017.MR3716908
  • K. Hattaf, N. Yousfi, and A. Tridane, “Stability analysis of a virus dynamics model with general incidence rate and two delays,” Applied Mathematics and Computation, vol. 221, pp. 514–521, 2013.MR3091948
  • P. Crowley and E. Martin, “Functional responses and interference within and between year classes of a dragonfly population,” Journal of the North American Benthological Society, vol. 8, pp. 211–221, 1989.
  • S. Xu, “Global stability of the virus dynamics model with Crowley-Martin functional response,” Electronic Journal of Qualitative Theory of Differential Equations, No. 9, 10 pages, 2012.MR2878794
  • J. R. Beddington, “Mutual interference between parasites or predat ors and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331–340, 1975.
  • D. L. DeAngelis, R. A. Goldstein, and R. V. ONeill, “A model for trophic interaction,” Ecology, pp. 881–892, 1975.
  • L. Rong, M. A. Gilchrist, Z. Feng, and A. S. Perelson, “Modeling within-host HIV-1 dynamics and the evolution of drug resistance: trade-offs between viral enzyme function and drug susceptibility,” Journal of Theoretical Biology, vol. 247, no. 4, pp. 804–818, 2007.MR2479625
  • Z. Hu, W. Pang, F. Liao, and W. Ma, “Analysis of a cd4+ t cell viral infection model with a class of saturated infection rate,” Discrete and Continuous Dynamical Systems - Series B, vol. 19, no. 3, pp. 735–745, 2014.
  • M. Maziane, E. M. Lotfi, K. Hattaf, and N. Yousfi, “Dynamics of a Class of HIV Infection Models with Cure of Infected Cells in Eclipse Stage,” Acta Biotheoretica, vol. 63, no. 4, pp. 363–380, 2015.
  • J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Springer, Berlin, Germany, 1993.MR1243878
  • J. K. Hale, “Sufficient conditions for stability and instability of autonomous functional-differential equations,” Journal of Differential Equations, vol. 1, pp. 452–482, 1965.MR0183938
  • T. A. Burton, Stability and Periodic sSolutions of Ordinary and Functional Differential Equations, Academic Press, Orlando, Fla, USA, 1985.MR837654
  • B. S. Ogundare, “Stability and boundedness properties of solutions to certain fifth order nonlinear differential equations,” Matematicki Vesnik, vol. 61, no. 4, pp. 257–268, 2009.MR2578506
  • C. Tunç, “A study of the stability and boundedness of the solutions of nonlinear differential equations of the fifth order,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 4, pp. 519–529, 2002.MR1902691
  • C. Tunç, “New results on the stability and boundedness of nonlinear differential equations of fifth order with multiple deviating arguments,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 3, pp. 671–682, 2013.MR3071757
  • 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
  • C. Vargas-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
  • D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Computational Eng. in Sys. Appl, vol. 2, p. 963, Lille, France, 1996.
  • E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1, pp. 1–4, 2006.
  • E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 542–553, 2007.MR2273544 \endinput