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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Dynamics of Two Delays Differential Equations Model of HIV Pathogenesis with Absorption and Saturation Incidence

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 435--444 | DOI:10.5890/DNC.2021.09.007

Vinoth Sivakumar , Jayakumar Thippan, Prasantha Bharathi Dhandapani

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, India

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Abstract

In this paper, we proposed and analyzed time delays HIV pathogenesis model with absorption and saturation incidence. We derived the basic reproductive number $R_{0}$ which is used to show the stability of the disease-free and infected steady states. Further, we studied the effect of the time delay of the infected steady state. In addition, we examined the existence of Hopf bifurcation on infected steady-state and the model exhibits Hopf bifurcation by using delay as a bifurcation parameter. Numerical simulations are provided to illustrate the corresponding theoretical results.

References

  1. [1]  Culshaw, R.V. and Ruan, S. (2000), A delay-differential equation model of HIV infection of CD4$^{+}$ T-cells, Mathematical Biosciences, 165(1), 27-39.
  2. [2]  Miao, H., Teng, Z., Kang, C., and Muhammadhaji, A. (2016), Stability analysis of a virus infection model with humoral immunity response and two time delays, Mathematical Methods in the Applied Sciences, 39(12), 3434-3449.
  3. [3]  Perelson, A.S., Kirschner, D.E., and De Boer, R. (1993), Dynamics of HIV Infection of CD4$^+$ T-cells, Math. Biosci, 114, 81-125.
  4. [4]  Wang, X., Tao, Y., and Song, X. (2010), A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dyn, 62, 67-72.
  5. [5]  Herz, V. Bonhoeffer, S., and Anderson, R. (1996), Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93, 7247-7251.
  6. [6]  Li, M.Y. and Shu, H. (2010), Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol, 72, 1492-1505.
  7. [7]  Song, H., Jiang, W., and Liu, S. (2015), Virus dynamics model with intracellular delays and immune response, Math. Biosci. Eng., 12, 185-208.
  8. [8]  Wang, Z. and Xu, R. (2012), Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response, Commun. Nonlinear Sci., 17, 964-978.
  9. [9]  Song, X. Neumann, A.U. (2007), Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329, 281-297.
  10. [10]  Xu, R. (2011), Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl. 375, 75-81.
  11. [11]  Hale, J.K. and Verduyn Lunel, S.M. (1993), Introduction to Functional-Differential Equations, Applied Mathematical Sciences, Springer: New York.
  12. [12]  Gautam, R. (2012), Reproduction numbers for infections with free-living pathogens growing in the environment, J. Biol. Dyn 6(2), 923-940.
  13. [13]  Van den Driessche, P. and Watmough, J. (2002), Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission, Mathematical Biosciences, 180, 29-48.
  14. [14]  Hassard, B.D. Kazarinoff, N.D. and Wan, Y.H. (1981), Theory and Applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge:UK.
  15. [15]  Song, X., Zhou, X., and Zhao, X. (2010), Properties of stability and Hopf bifurcation for a HIV infection model with time delay, Appl. Math. Model, 34, 1511-1523.