Analysis of the dynamics of a mathematical model for HIV infection

Volume 23, Issue 3, pp 181--195 http://dx.doi.org/10.22436/jmcs.023.03.02

Publication Date: November 01, 2020 Submission Date: July 18, 2020 Revision Date: September 15, 2020 Accteptance Date: September 17, 2020

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1091 Downloads
• 2209 Views

Authors

Bhagya Jyoti Nath - Department of Mathematics, Barnagar College, Sorbhog, 781317, Barpeta, Assam, India. Kaushik Dehingia - Department of Mathematics, Gauhati University, Guwahati, 781014, Assam, India. Hemanta Kumar Sarmah - Department of Mathematics, Gauhati University, Guwahati, 781014, Assam, India.

Abstract

Mathematical models are essential tools in the study of different infectious diseases. Researchers have developed other in-host models to investigate HIV dynamics in the human body. In this paper, a mathematical model for the HIV infection of \(CD4^+\) T cells is analyzed. We consider the proliferation of T cells in this study. It is found that there exist two equilibrium states for this model: Infection-free equilibrium state and infected equilibrium state. Local stability is discussed for both infection-free and infected equilibrium states using Routh--Hurwitz criteria. Also, we calculate the basic reproduction number \((R_0)\) for the model with the help of next generation matrix method. The global stability of the infection-free equilibrium point is discussed using Lyapunov's second method. From the stability analysis, it is found that if basic reproduction number \(R_0 \leq 1\), infection of HIV is cleared out, and if \(R_0 >1\), infection of HIV persists. The conditions for global stability of the infected equilibrium point are derived using a geometric approach. We find a parameter region where the infected equilibrium point is globally stable. We carry out numerical simulations to verify the results. Also, the effects of the proliferation rate of uninfected \(CD4^+\) T cells and recovery rate of infected \(CD4^+\) T cells in dynamics of the T cells and free virus are studied using numerical simulations. It is found that small variations of these parameters can change the model's whole dynamics, and infection can be controlled by controlling the proliferation rate and improving the recovery rate.

Share and Cite

Keywords

• HIV infection
• global stability
• \(CD4^+\) T cells
• basic reproduction number

MSC

•  34A34
•  37N25

References

• [1] J. Alvarez-Ramirez, M. Meraz, J. X. Velasco-Hernandez, Feedback control of the chemotherapy of HIV, Int. J. Bifur. Chaos, 10 (2000), 2207--2219
  • View Article
  • Google Scholar


• [2] P. Chandra, Mathematical modeling of HIV dynamics: In vivo, Math. Student, 78 (2009), 7--27
  • View Article
  • Google Scholar


• [3] W. A. Copppel, Stability and Asymptotic Behaviour of Differential Equations, Health, Boston (1965)
  • Google Scholar


• [4] P. V. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibra for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29--48
  • View Article
  • Google Scholar


• [5] P. Essunger, A. S. Perelson, Modelling HIV infection $CD4^+$ T-cell subpopulations, J. Theor. Biol., 170 (1994), 367--391
  • View Article
  • Google Scholar


• [6] T. Gao, W. Wang, X. Liu, Mathematical analysis of an HIV model with impulsive antiretroviral drug doses, Math. Comput. Simulation, 82 (2011), 653--665
  • View Article
  • Google Scholar


• [7] J. P. LaSalle, The Stability of Dynamical Systems,in: Regional Conference Series in Applied Mathematics, SIAM, Philadelaphia (1976)
  • View Article
  • Google Scholar


• [8] M. Y. Li, J. S. Muldowney, A geometric approach to the global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070--1083
  • View Article
  • Google Scholar


• [9] R. H. Martin, Logarithimic norms and projections applied to linear differenetial systems, J. Math. Anal. Appl., 45 (1974), 432--454
  • View Article
  • Google Scholar


• [10] M. A. Nowak, S. Bonhoeffer, G. M. Shaw, R. M. May, Antiviral drug treatment: dynamics of resistance in free virus and infected cell populations, J. Theor. Biol., 184 (1997), 203--217
  • View Article
  • Google Scholar


• [11] M. A. Nowak, R. M. May, Virus dynamics: mathematical principles of immunology and virology: mathematical principles of immunology and virology, Oxford University Press, Oxford (2000)
  • View Article
  • Google Scholar


• [12] A. S. Perelson, D. E. Kirschner, R. J. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114 (1993), 81--125
  • View Article
  • Google Scholar


• [13] A. S. Perelson, P. W. Nelson, Mathematical Analysis of HIV-1 Dynamics in Vivo, SIAM Rev., 41 (1999), 3--44
  • View Article
  • Google Scholar


• [14] L. Rong, M. A. Gilchrist, Z. Feng, A. S. Perelson, Modeling within host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility, J. Theor. Biol., 247 (2007), 804--818
  • View Article
  • Google Scholar


• [15] H. L. Smith, P. D. Leenheer, Virus dynamics: a global analysis, SIAM J. Appl. Math., 63 (2003), 1313--1327
  • View Article
  • Google Scholar


• [16] X. Song, A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281--297
  • View Article
  • Google Scholar


• [17] P. K. Srivastava, M. Banerjee, P. Chandra, Dynamical Model of in-host HIV infection: With Drug therapy and Multi viral strains, J. Biol. Syst., 20 (2012), 303--325
  • View Article
  • Google Scholar


• [18] P. K. Srivastava, P. Chandra, Modeling the dynamics of HIV and $CD4^+$ T cells during primary infection, Nonlinear Anal. Real World Appl., 11 (2010), 612--618
  • View Article
  • Google Scholar


• [19] , World Health Organisation HIV/AIDS Key facts, WHO, ()
  • View Article