Asian Journal of Pharmaceutical Research and Health Care

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A Study on an HIV Pathogenesis Model with Different Growth rates of Uninfected and Infected CD4+T cells


Affiliations
1 Department of Mathematics, Barnagar College, Sorbhog – 781317, Barpeta, Assam, India
2 Department of Mathematics, Gauhati University, Guwahati – 781014, Assam, India
 

The objective of this paper is to discuss the dynamics of an HIV pathogenesis model with full logistic target cell growth of uninfected T cells and cure rate of infected T cells. Local and global dynamics of both infection-free and infected equilibrium points are rigorously established. It is found that if basic reproduction number R0≤1, the infection is cleared from T cells and if R0>1, the HIV infection persists. Also, we have carried out numerical simulations to verify the results. The existence of non-trivial periodic solution is also studied by means of numerical simulation. Therefore, we find a parameter region where infected equilibrium point is globally stable to make the model biologically significant. From the overall study, it is found that proliferation of T cells cannot be ignored during the study of HIV dynamics for better results and we can focus on a treatment policy which can control the parameters of the model in such a way that the basic reproduction number remains less than or equal to one.


Keywords

HIV, Local and Global Stability, Periodic Solution, Treatment

2010 AMS classification: 34A34, 34D23, 37C25

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  • World Health Organisation HIV/AIDS Key facts. 2020 Jul https://www.who.int/newsroom/fact-sheets/detail/hiv

  • Douek DC, Brenchley JM, Betts MR, Ambrozak DR, Hill BJ, Okamoto Y, Casazza JP, Kuruppu J, Kunstman K, Wolinsky S, Grossman Z. HIV preferentially infects HIV-specific CD4+ T cells. Nature. 2002 May; 417(6884):95–8. PMid: 11986671. https://doi.org/10.1038/417095a

  • Perelson AS, Kirschner DE, Boer R De. Dynamics of HIV infection of CD4+ T cells. Math Biosci. 1993; 114:81–125. https://doi.org/10.1016/0025-5564(93)90043-A

  • Essunger P, Perelson AS. Modelling HIV infection CD4+ T - subpopulations. J Theoret Biol. 1994; 170:367–91. PMid: 7996863. https://doi.org/10.1006/jtbi.1994.1199

  • Wang L, Li MY. Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells. Math Biosci. 2006; 200:44–57. PMid: 16466751. https://doi.org/10.1016/j.mbs.2005.12.026

  • Rong L, Gilchrist MA, Feng Z, Perelson AS. Modeling within host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility. J Theoret Biol. 2007; 247(4):804–18. PMid: 17532343 PMCid: PMC2265667. https://doi.org/10.1016/j.jtbi.2007.04.014

  • Perelson AS, Nelson Patrick W. Mathematical Analysis of HIV-1 dynamics in Vivo. SIAM Review. 1999; 41(1):3–44. https://doi.org/10.1137/S0036144598335107

  • Leenheer P. De, Smith HL. Virus dynamics: A global analysis. SIAM J Appl Math. 2003; 63:1313–27. https://doi.org/10.1137/S0036139902406905

  • Gao T, Wang W, Liu X. Mathematical analysis of an HIV model with impulsive antiretroviral drug doses. Math Comput Simul. 2011 Dec; 82(4):653–65. https://doi.org/10.1016/j.matcom.2011.10.007

  • Srivastava PK, Chandra P. Modeling the dynamics of HIV and CD4+ T cells during primary infection.

  • Nonlinear Anal Real World Appl. 2010 Apr; 11(2):612–8. https://doi.org/10.1016/j.nonrwa.2008.10.037

  • Chandra P. Mathematical modeling of HIV dynamics: In Vivo. Mathematics Student-India. 2009; 78(1):7.

  • Nowak MA, May RM. Virus dynamics. UK: Oxford University Press; 2000.

  • LaSalle JP. The stability of dynamical systems. SIAM; 1976. PMCid: PMC1411100. https://doi.org/10.21236/ADA031020

  • Ho DD, Neumann AU, Perelson AS, Chen W, Leonard JM, Markowitz M. Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature. 1995; 373:123–6. PMid: 7816094. https://doi.org/10.1038/373123a0

  • Sachsenberg N, Perelson AS, Yerly S, Schockmel GA, Leduc D, Hirschel B, Perrin L. Turnover of CD4+ and CD8+ T lymphocytes in HIV-1 infection as measured by ki-67 antigen. J Exp Med. 1998; 187:1295–303. PMid: 9547340 PMCid: PMC2212238. https://doi.org/10.1084/jem.187.8.1295

  • Fan X, Brauner CM,Wittkop L. Mathematical analysis of a HIV model with quadratic logistic growth term. Discrete and Continuous Dynamical Systems Series-B. 2012; 17(7):2359–85. https://doi.org/10.3934/dcdsb.2012.17.2359

  • Song X, Neumann AU. Global stability and periodic solution of the viral dynamics. J Math Anal Appl. 2007 May; 329(1):281–97. https://doi.org/10.1016/j.jmaa.2006.06.064

  • Wang X, Song X. Global stability and periodic solution of a model for HIV infection of CD4+ T cells.

  • Appl Math Comput. 2007 Jun; 189(2):1331–40. https://doi.org/10.1016/j.amc.2006.12.044

  • Culshaw RV, Ruan S. A delay-differential equation model of HIV infection of CD4+ T-cells. Math Biosci.

  • May; 165(1):27–39. https://doi.org/10.1016/S00255564(00)00006-7

  • Merdan M, Gokdogan A, Yildirim A. On the numerical solution of the model for HIV infection of CD4+ T cells. Comput Math. Appl. 2011 Jul; 62(1):118–23. https://doi.org/10.1016/j.camwa.2011.04.058

  • Zack JA, Arrigo SJ, Weitsman SR, Go AS, Haislip A, Chen IS. HIV-1 entry into quiescent primary lymphocytes: molecular analysis reveals a labile, latent viral structure. Cell. 1990 Apr; 61(2):213–22. https://doi.org/10.1016/00928674(90)90802-L

  • Zack JA, Haislip AM, Krogstad P, Chen IS. Incompletely reverse-transcribed human immunodeficiency virus type 1 genomes in quiescent cells can function as intermediates in the retroviral life cycle. J Virol. 1992 Mar; 66(3):1717– 25. PMid: 1371173 PMCid: PMC240919. https://doi.org/10.1128/JVI.66.3.1717-1725.1992

  • Gradshteyn IS, Ryzhik IM. Routh-Hurwitz theorem, Tables of Integrals, Series and Products. San Diego: Academic Press; 2000.

  • Li MY, Muldowney JS. A geometric approach to global-stability problems. SIAM J Math Anal. 1996 Jul; 27(4):1070–83. https://doi.org/10.1137/S0036141094266449

  • Coppel WA. Stability and asymptotic behaviour of differential equations. Health, Boston; 1965.

  • Fiedler M. Additive compound matrices and inequalities for eigen values of stochastic matrices. Czech. Math J. 1974; 24(3): 392–402. https://doi.org/10.21136/CMJ.1974.101253

  • Muldowney JS. Compound matrices and ordinary differential equations. Rocky Mount. J Math. 1990 Oct: 857-72. https://doi.org/10.1216/rmjm/1181073047

  • Van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci. 2002 Nov; 180(1-2):29–48. https://doi.org/10.1016/S00255564(02)00108-6

  • Bani-Yaghoub M, Gautam R, Shuai Z, Van Den Driessche P, Ivanek R. Reproduction numbers for infections with free-living pathogens growing in the environment. J Biol Dyn. 2012 Mar; 6(2):923–40. PMid: 22881277. https://doi.org/10.1080/17513758.2012.693206


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  • A Study on an HIV Pathogenesis Model with Different Growth rates of Uninfected and Infected CD4+T cells

Abstract Views: 286  |  PDF Views: 118

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


The objective of this paper is to discuss the dynamics of an HIV pathogenesis model with full logistic target cell growth of uninfected T cells and cure rate of infected T cells. Local and global dynamics of both infection-free and infected equilibrium points are rigorously established. It is found that if basic reproduction number R0≤1, the infection is cleared from T cells and if R0>1, the HIV infection persists. Also, we have carried out numerical simulations to verify the results. The existence of non-trivial periodic solution is also studied by means of numerical simulation. Therefore, we find a parameter region where infected equilibrium point is globally stable to make the model biologically significant. From the overall study, it is found that proliferation of T cells cannot be ignored during the study of HIV dynamics for better results and we can focus on a treatment policy which can control the parameters of the model in such a way that the basic reproduction number remains less than or equal to one.


Keywords


HIV, Local and Global Stability, Periodic Solution, Treatment

2010 AMS classification: 34A34, 34D23, 37C25


References





DOI: https://doi.org/10.18311/ajprhc%2F2020%2F25775