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2010 | 20 | 3 | 601-612
Tytuł artykułu

A mathematical model of HIV-1 infection including the saturation effect of healthy cell proliferation

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we derive a model describing the dynamics of HIV-1 infection in tissue culture where the infection spreads directly from infected cells to healthy cells trough cell-to-cell contact. We assume that the infection rate between healthy and infected cells is a saturating function of cell concentration. Our analysis shows that if the basic reproduction number does not exceed unity then infected cells are cleared and the disease dies out. Otherwise, the infection is persistent with the existence of an infected equilibrium. Numerical simulations indicate that, depending on the fraction of cells surviving the incubation period, the solutions approach either an infected steady state or a periodic orbit.
Słowa kluczowe
Rocznik
Tom
20
Numer
3
Strony
601-612
Opis fizyczny
Daty
wydano
2010
otrzymano
2009-05-31
poprawiono
2010-03-28
Twórcy
  • Laboratory of Applied Mathematics, Badji-Mokhtar University, BP 12, Annaba 23000, Algeria
  • Institute of Mathematics of Bordeaux, UMR CNRS 52 51, Case 36, Université Victor Segalen Bordeaux 2, 3 ter place de la Victoire, F 33076 Bordeaux Cedex, France
Bibliografia
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  • Culshaw, R., Ruan, S. and Webb, G. (2003). A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, Journal of Mathematical Biology 46(5): 425-444.
  • De Boer, R.J. and Perelson, A.S. (1995). Towards a general function describing T-cell proliferation, Journal of Theoretical Biology 175(4): 567-576.
  • De Leenheer, P. and Smith, H.L. (2003). Virus dynamics: A global analysis, SIAM Journal of Applied Mathematics 63(4): 1313-1327.
  • Diekmann, O., Heesterbeek, J.A.P. and Metz, J. (1990). On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology 29(4): 365-382.
  • Dimitrov, D.S., Willey, R.L., Sato, H., Chang, L.J., Blumenthal, R. and Martin, M.A. (1993). Quantitation of human immunodeficiency virus type 1 infection kinetics, Journal of Virology 67(4): 2182-2190.
  • Esteva-Peralta, L. and Velasco-Hernandez, J.X. (2002). M-matrices and local stability in epidemic models, Mathematical and Computer Modelling 36(4-5): 491-501.
  • Grossman, Z., Feinberg, M.B. and Paul, W. (1998). Multiple modes of cellular activation and virus transmission in HIV infection: A role for chronically and latently infected cells in sustaining viral replication, Proceedings of the National Academy of Sciences of the United States of America 95(11): 6314-6319.
  • Hassard, B.D., Kazarinoff, N.D. and Wan, Y.H. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge.
  • Herz, A. V.M., Bonhoeffer, S., Anderson, R.M., May, R.M. and Nowak, M. A. (1996). Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proceedings of the National Academy of Sciences of the United States of America 93(14): 7247-7251.
  • Kirschner, D. E. (1996). Using mathematics to understand HIV immune system, Notices of the American Mathematical Society 43(2): 191.
  • Kirschner, D.E. and Webb, G. (1996). A model for treatment strategy in the chemotherapy of AIDS, Bulletin of Mathematical Biology 58(2): 167-190.
  • MacDonald, N. (1978). Time Delays in Biological Models, Springer-Verlag, New York, NY.
  • Mittler, J.E., Sulzer, B., Neumann, A.U. and Perelson, A.S. (1998). Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Mathematical Biosciences 152(2): 143-163.
  • Nelson, P.W., Murray, J.D. and Perelson, A.S. (2000). A model of HIV-1 pathogenesis that includes an intracellular delay, Mathematical Biosciences 163(2): 201-215.
  • Nelson, P.W. and Perelson, A.S. (2002). Mathematical analysis of delay differential equation models of HIV-1 infection, Mathematical Biosciences 179(1): 73-94.
  • Nowak, M.A. and Bangham, R.M. (1996). Population dynamics of immune responses to persistent viruses, Science 272(5258): 74-79.
  • Nowak, M.A. and May, R.M. (2000). Virus Dynamics, Oxford University Press, New York, NY.
  • Perelson, A.S. (1989). Modelling the interaction of the immune system with HIV, in C. Castillo-Chavez (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology, Springer-Verlag, New York, NY, pp. 350-370.
  • Perelson, A.S., Kirschner, D.E. and Boer, R.D. (1993). Dynamics of HIV infection of CD4⁺ T-cells, Mathematical Biosciences 114(1): 81-125.
  • Perelson, A.S. and Nelson, P.W. (1999). Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review 41(1): 3-44.
  • Philips, D. M. (1994). The role of cell-to-cell transmission in HIV infection, AIDS 8(6): 719-731.
  • Smith, H.L. (1995). Monotone Dynamical Systems, American Mathematical Society, Providence, RI.
  • Spouge, J.L., Shrager, R.I. and Dimitrov, D.S. (1996). HIV-1 infection kinetics in tissue cultures, Mathematical Biosciences 138(1): 1-22.
  • Tam, J. (1999). Delay effect in a model for virus replication, IMA Journal of Mathematics Applied to Medicine and Biology 16(1): 29-37.
  • Thieme, H.R. (1993). Persistence under relaxed pointdissipativity (with application to an endemic model), SIAM Journal on Mathematical Analysis 24(2): 407.
  • van den Driessche, P. and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180(1-2): 29-48.
  • Wang, L. and Li, M.Y. (2006). Mathematical analysis of the global dynamics of a model for HIV infection of CD4⁺ T-cells, Mathematical Biosciences 200(1): 4-57.
  • Zhu, H. and Smith, H.L. (1994). Stable periodic orbits for a class of three dimensional competitive systems, Journal of Differential Equations 110(1): 143-156.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv20i3p601bwm
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