Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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A DELAY-DIFFERENTIAL EQUATION MODEL OF HIV INFECTION OF CD4+ T-CELLS

  • SONG, XINYU (Department of Mathematics Xinyang Normal University) ;
  • CHENG, SHUHAN (College of Information Science and Engineering Shandong Agricultural University)
  • Published : 2005.09.01
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Abstract

In this paper, we introduce a discrete time to the model to describe the time between infection of a CD4$^{+}$ T-cells, and the emission of viral particles on a cellular level. We study the effect of the time delay on the stability of the endemically infected equilibrium, criteria are given to ensure that the infected equilibrium is asymptotically stable for all delay. We also obtain the condition for existence of an orbitally asymptotically stable periodic solution.

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References

  1. D. Ho, A. Neumann, A. Perelson, W. Chen, J. Leonard, and M. Markowitz, Rapid turnover of plasma virions and $CD4^+$ lymphocytes in HIV-1 infection,Nature 373 (1995), 123-126 https://doi.org/10.1038/373123a0
  2. X. Wei, S. Ghosh, M. Taylor, V. Johnson, E. Emini, P. Deutsch, J. Lifson, S. Bonhoeffer, M. Nowak, B. Hahn, S. Saag, and G. Shaw, Viral dynamics in human immunodeficiency virus type 1 infection, Nature 373 (1995), 117 https://doi.org/10.1038/373117a0
  3. A. Perelson, A. Neumann, M. Markowitz, J. Leonard, and D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science 271 (1996), 1582
  4. A. Perelson, P. Essunger, Y. Cao, M. Vesanen, A. Hurley, K. Saksela, M. Markowitz, and D. Ho, Decay characteristics of HIV-1-infected compartments during combination therapy, Nature 387 (1997), 188 https://doi.org/10.1038/387188a0
  5. A. Neumann, N. Lam, H. Dahari, D. Gretch, T. Wiley, T. Layden, and A. Perel- son, Hepatitis C viral dynamics in vivo and antiviral efficacy of the interferon-ff therapy, Science 282 (1998), 103-107
  6. A. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev. 41 (1999), no. 1, 3-44 https://doi.org/10.1137/S0036144598335107
  7. V. Herz, S. Bonhoeffer, R. Anderson, R. May, and M. Nowak, Viral dynamics in vivo: limitations on estimations on intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996), 7247-7251 https://doi.org/10.1073/pnas.93.14.7247
  8. Z. Grossman, M. Feinberg, V. Kuznetsov, D. Dimitrov, and W. Paul, HIV infection: how effective is drug combination treatment, Immunol. Today 19 (1998), 528 https://doi.org/10.1016/S0167-5699(98)01353-X
  9. Z. Grossman, M. Polis, M. Feinberg, I. Levi, S. Jankelevich, R. Yarchoan, J. Boon, F. de Wolf, J. Lange, J. Goudsmit, D. Dimitrov, and W. Paul, Ongoing HIV dissemination during HAART, Nat. Med. 5 (1999), 1099 https://doi.org/10.1038/13410
  10. J. Mittler, B. Sulzer, A. Neumann, and A. Perelson, Influence of delayed virus production on viral dynamics in HIV-1 infected patients, Math. Biosci. 152 (1998), 143 https://doi.org/10.1016/S0025-5564(98)10027-5
  11. J. Tam, Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol. 16 (1999), 29 https://doi.org/10.1093/imammb/16.1.29
  12. P. Nelson, J. Murray, and A. Perelson,A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci. 163 (2000), 201 https://doi.org/10.1016/S0025-5564(99)00055-3
  13. J. Mittler, M. Markowitz, D. Ho, and A. Perelson,Refined estimates for HIV-1 clearance rate and intracellular delay, AIDS, 13 (1999), 1415 https://doi.org/10.1097/00002030-199907300-00023
  14. P. Nelson and A. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci. 179 (2002), 73-94 https://doi.org/10.1016/S0025-5564(02)00099-8
  15. H. I. Freedman and V. Sree Hari Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol. 45 (1983), 991-1003 https://doi.org/10.1007/BF02458826
  16. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977
  17. J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), 388-396 https://doi.org/10.1137/0520025
  18. H. Smith, Monotone semiflows generated by functional differential equations, J. Differential Equations 66 (1987), 420-442 https://doi.org/10.1016/0022-0396(87)90027-1
  19. K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic, Dordrecht/Norwell, MA
  20. Hsiu-Rong Zhu and H. Smith, Stable periodic orbits for a class three dimentional competitive systems, J. Differential Equations 110 (1994), 143-156 https://doi.org/10.1006/jdeq.1994.1063
  21. E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002), 1144-1165 https://doi.org/10.1137/S0036141000376086

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