Differential Equations and Dynamical Systems

, Volume 22, Issue 2, pp 181–190 | Cite as

A Delay Virus Dynamics Model with General Incidence Rate

  • Khalid HattafEmail author
  • Noura Yousfi
  • Abdessamad Tridane
Original Research


In this paper, the dynamical behavior of a virus dynamics model with general incidence rate and intracellular delay is studied. Lyapunov functionals are constructed and LaSalle invariance principle for delay differential equation is used to establish the global asymptotic stability of the disease-free equilibrium and the chronic infection equilibrium. The results obtained show that the global dynamics are completely determined by the value of a certain threshold parameter called the basic reproduction number \(R_0\) and under some assumptions on the general incidence function. Our results extend the known results on delay virus dynamics considered in other papers and suggest useful methods to control virus infection. These results can be applied to a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models.


Virus dynamics General incidence rate Intracellular delay Global stability Lyapunov functional 



The authors would like to thank the anonymous referee for his/her valuable comments on the first version of the manuscript which have led to an improvement in this revised version.


  1. 1.
    Li, M.Y., Shu, H.: Global dynamics of an in-host viral model with intracellular delay. Bull. Math. Biol. 72, 1492–1505 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Hattaf, K., Yousfi, N.: A Delay differential equation model of HIV with therapy and cure Rate. Int. J. Nonlinear Sci. 12, 503–512 (2011)MathSciNetGoogle Scholar
  3. 3.
    Gourley, S.A., Kuang, Y., Nagy, J.D.: Dynamics of a delay differential model of hepatitis B virus. J. Biol. Dyn. 2, 140–153 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Eikenberry, S., Hews, S., Nagy, J.D., Kuang, Y.: The dynamics of a delay model of HBV infection with logistic hepatocyte growth. Math. Biosci. Eng. 6, 283–299 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Tian, X., Xu, R.: Asymptotic properties of a hepatitis B virus infection model with time delay. Disc. Dyn. Nat. Soc. 21 pp. (2010). doi: 10.1155/2010/182340
  6. 6.
    Li, D., Ma, W.: Asymptotic properties of an HIV-1 infection model with time delay. J. Math. Anal. Appl. 335, 683–691 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Shi, X., Zhou, X., Song, X.: Dynamical behavior of a delay virus dynamics model with CTL immune response. Nonlinear Anal. RWA 11, 1795–1809 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Huang, G., Ma, W., Takeuchi, Y.: Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response. Appl. Math. Lett. 24(7), 1199–1203 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Nakata, Y.: Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response. Nonlinear Anal. TMA 74, 2929–2940 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Hattaf, K., Yousfi, N., Tridane, A.: Mathematical analysis of a virus dynamics model with general incidence rate and cure rate. Nonlinear Anal. RWA 13, 1866–1872 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Hale, J., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    LaSalle, J.P.: The Stability of Dnamical Systems. Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1976)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2013

Authors and Affiliations

  • Khalid Hattaf
    • 1
    Email author
  • Noura Yousfi
    • 1
  • Abdessamad Tridane
    • 2
  1. 1.Department of Mathematics and Computer Science, Faculty of Sciences Ben M’sikHassan II UniversityCasablancaMorocco
  2. 2.Department of Applied Sciences and MathematicsArizona State UniversityMesaUSA

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