Bulletin of Mathematical Biology

, Volume 67, Issue 1, pp 101–114 | Cite as

Backward bifurcation in a model for HTLV-I infection of CD4+ T cells

Article

Abstract

Human T-cell Lymphotropic Virus Type I (HTLV-I) primarily infects CD4+ helper T cells. HTLV-I infection is clinically linked to the development of Adult T-cell Leukemia/Lymphoma and of HTLV-I Associated Myelopathy/Tropical Spastic Paraparesis, among other illnesses. HTLV-I transmission can be either horizontal through cell-to-cell contact, or vertical through mitotic division of infected CD4+ T cells. It has been observed that HTLV-I infection has a high proviral load but a low rate of proviral genetic variation. This suggests that vertical transmission through mitotic division of infected cells may play an important role.We consider and analyze a mathematical model for HTLV-I infection of CD4+ T cells that incorporates both horizontal and vertical transmission. Among interesting dynamical behaviors of the model is a backward bifurcation which raises many new challenges to effective infection control.

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References

  1. Bangham, C. R.M., 2000. The immune response to HTLV-I. Curr. Opin. Immunol. 12, 397–402.CrossRefGoogle Scholar
  2. Bangham, C. R.M., Hall, S.E., Jeffery, K.J.M., Vine, A.M., Witkover, A., Nowak, M.A., Usuku, K., Osame, M., 1999. Genetic control and dynamics of the cellular immune response to the human T-cell leukemia virus, HTLV-I. Philos. Trans. R. Soc. Lond. B Biol. Sci. 354, 691–700.CrossRefGoogle Scholar
  3. Brauer, F., Noel, J.A., 1968. Qualitative Theory of Ordinary Differential Equations. Benjamin, New York.Google Scholar
  4. Cann, A.J., Chen, I.S.Y., 1996. Human T-cell leukemia virus types I and II. In: Fieds, B.N., Knipe, D.M., Howley, P.M. et al. (Eds.), Fields Virology, 3rd edn. Lippincott-Raven Publishers, Philadelphia, pp. 1849–1880.Google Scholar
  5. Dushoff, J., 1996. Incorporating immunological ideas in epidemiological models. J. Theor. Biol. 180, 181–187.CrossRefGoogle Scholar
  6. Dushoff, J., Huang, W., Castillo-Chavez, C., 1998. Backwards bifurcations and catastrophe in simple models of fatal diseases. J. Math. Biol. 36, 227–248.MathSciNetCrossRefGoogle Scholar
  7. Gómez-Acevedo, H., Li, M.Y., 2002. Global dynamics of a mathematical model for HTLV-I infection of T cells. Canad. Appl. Math. Quart. 10, 71–86.Google Scholar
  8. Grant, C., Barmak, K., Alefantis, T., Yao, J., Jacobson, S., Wigdahl, B., 2002. Human T cell leukemia virus type I and neurologic disease: events in bone marrow, peripheral blood, and central nervous system during normal immune surveillance and neuroinflammation. J. Cell. Phys. 190, 133–159.CrossRefGoogle Scholar
  9. Greenhalgh, D., Diekmann, O., de Jong, M.C.M., 2000. Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity. Math. Biosci. 165, 1–25.MathSciNetCrossRefGoogle Scholar
  10. Hadeler, K.P., van den Driessche, P., 1997. Backward bifurcation in epidemic control. Math. Biosci. 146, 15–35.MathSciNetCrossRefGoogle Scholar
  11. Hale, J.K., 1969. Ordinary Differential Equations. John Wiley&Sons, New York.Google Scholar
  12. Kribs-Zaleta, C.M., Martcheva, M., 2002. Vaccination strategies and backward bifurcation in an age-sinceinfection structured model. Math. Biosci. 177-178, 317–332.MathSciNetCrossRefGoogle Scholar
  13. Kribs-Zaleta, C.M., Velasco-Hernandez, J., 2000. A simple vaccination model with multiple endemic states. Math. Biosci. 164, 183–201.CrossRefGoogle Scholar
  14. Martcheva, M., Thieme, H.R., 2003. Progression age enhanced backward bifurcation in an epidemic model with super-infection. J. Math. Biol. 46, 385–424.MathSciNetCrossRefGoogle Scholar
  15. Medley, G.F., Lindop, N.A., Edmunds, W.J., Nokes, D.J., 2001. Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control. Nat. Med. 7, 619–624.CrossRefGoogle Scholar
  16. Mortreux, F., Gabet, A.-S., Wattel, E., 2003. Molecular and cellular aspects of HTLV-I associated leukemogenesis in vivo. Leukemia 17, 26–38.CrossRefGoogle Scholar
  17. Mortreux, F., Kazanji, M., Gabet, A.-S., de Thoisy, B., Wattel, E., 2001. Two-step nature of human T-cell leukemia virus type 1 replication in experimentally infected squirrel monkeys (Saimiri sciureus). J. Virol. 75, 1083–1089.CrossRefGoogle Scholar
  18. Nelson, P.W., Murray, J.D., Perelson, A.S., 2000. A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163, 201–215.MathSciNetCrossRefGoogle Scholar
  19. Nowak, M.A., May, R.M., 2000. Virus Dynamics. Oxford University Press, Oxford.Google Scholar
  20. Perelson, A.S., 1989. Modeling the interaction of the immune system with HIV. In: Castillo-Chavez, C. (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology. Lect. Notes Biomath, vol. 83. Springer, Berlin, pp. 350–370.Google Scholar
  21. Pugliese, A., 1990. Population models for diseases with no recovery. J. Math. Biol. 28, 65–82.MATHMathSciNetCrossRefGoogle Scholar
  22. van den Driessche, P., Watmough, J., 2000. A simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40, 525–540.MathSciNetCrossRefGoogle Scholar
  23. Wang, L., Li, M.Y., Kirschner, D., 2002. Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression. Math. Biosci. 179, 207–217.MathSciNetCrossRefGoogle Scholar
  24. Wattel, E., Vartanian, J.-P., Pannetier, C., Wain-Hobson, S., 1995. Clonal expansion of human T-cell leukemia virus type I-infected cells in asymptomatic and symptomatic carriers without malignancy. J. Virol. 69, 2863–2868.Google Scholar
  25. Wodarz, D., Nowak, M.A., Bangham, C.R.M., 1999. The dynamics of HTLV-I and the CTL response. Immun. Today 20, 220–227.CrossRefGoogle Scholar
  26. Wucherpfennig, K.W., Höllsberg, P., Richardson, J.H., Benjamin, D., Hafler, D.A., 1992. T-cell activation by autologous human T-cell leukemia virus type I-infected T-cell clones. Proc. Natl. Acad. Sci. USA 89, 2110–2114.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2005

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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