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Stability Analysis of a Mathematical Model of the Immune Response with Delays

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References

  1. Akira, S., K. Takeda and T. Kaisho (2001), Toll-like receptors: critical proteins linking innate and acquired immunity, Nat. Immunol., 2, 675.

    Article  Google Scholar 

  2. Antia, R., C. T. Bergstrom, S. S. Pilyugin, S. M. Kaech and R. Ahmed (2003), Models of CD8+ responses: 1. What is the antigen-independent proliferation program, J. Theor. Biol., 221, 585.

    Article  MathSciNet  Google Scholar 

  3. Baker, C. T. H. and N. Ford (1988), Convergence of linear multistep methods for a class of delay-integro-differential equations. In:International Series of Numerical Mathematics 86, Birkhäuser Verlag Basel.

    Google Scholar 

  4. Beretta, E. and Y. Kuang (2002), Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33, 1144.

    Article  MathSciNet  Google Scholar 

  5. Capuano, S. V., D. A. Croix, S. Pawar, A. Zinovik, A. Myers, P. L. Lin, S. Bissel, C. Fuhrman, E. Klein and J. L. Flynn (2003), Experimental Mycobacterium tuberculosis infection of cynomolgus macaques closely resembles the various manifestations of human M. tuberculosis infection. Infect. Immun., 71, 5831.

    Article  Google Scholar 

  6. Fearon, D.T. and R. M. Locksley (1996), The instructive role of innate immunity in the acquired immune response Science, 272, 50.

    Article  Google Scholar 

  7. Flesch, I. E., S. H. Kaufmann (1990), Activation of tuberculostatic macrophage functions by gamma interferon, interleukin-4, and tumor necrosis factor, Infect. Immun., 58, 2675.

    Google Scholar 

  8. Guermonprez, P., J. Valladeau, L. Zitvogel, C. Thery, and S. Amigorena (2002), Antigen presentation and T cell stimulation by dendritic cells, Annu. Rev. Immunol.., 20, 621.

    Article  Google Scholar 

  9. Hale, J. K. and S. M. Verduyn Lunel (1993), Introduction to functional differential equations, Springer Verlag, New York.

    MATH  Google Scholar 

  10. Harty, J. T., A. R. Tvinnereim and D. W. White (2000), CD8+ T cell effector mechanisms in resistance to infection Annu. Rev. Immunol., 18, 275.

    Article  Google Scholar 

  11. Holt, P. G. and M. A. Schon-Hegrad (1987), Localization of T cells, macrophages and dendritic cells in rat respiratory tract tissue: implications for immune function studies, Immunology, 62, 349.

    Google Scholar 

  12. Holt, P.G. (2000), Antigen presentation in the lung, Am. J. Respir. Crit. Care Med., 162, 151.

    Google Scholar 

  13. Janeway, Jr. C. A. and R. Medzhitov (2002), Innate immune recognition, Annu. Rev. Immunol., 20, 197.

    Article  Google Scholar 

  14. Janeway, Jr. C. A. (1989), Approaching the asymptote? Evolution and revolution in immunology, Cold Spring Harb. Symp. Quant. Biol., 54 Pt 1, 1.

    Google Scholar 

  15. Janeway, Jr. C. A. (2001), Immunobiology 5: the immune system in health and disease, Garland Pub., New York.

    Google Scholar 

  16. Janeway, Jr. C. A. (2002), A trip through my life with an immunological theme Annu. Rev. Immunol., 20, 1.

    Article  Google Scholar 

  17. Jenkins, M. K., A. Khoruts, E. Ingulli, D. L. Mueller, S. J. McSorley, R. L. Reinhardt, A. Itano and K.A. Pape (2001), In vivo activation of antigen-specific CD4 T cells, Annu. Rev. Immunol., 19, 23.

    Article  Google Scholar 

  18. Kuang, Y. (1993), Delay differential equations: with applications in population dynamics, Academic Press, Cambridge, Mass.

    MATH  Google Scholar 

  19. Lewinsohn, D. M., T. T. Bement, J. Xu, D. H. Lynch, K. H. Grabstein, S. G. Reed and Alderson, M. R. (1998), Human purified protein derivativespecific CD4+ T cells use both CD95-dependent and CD95-independent cytolytic mechanisms, J. Immunol., 160, 2374.

    Google Scholar 

  20. Lurie, M. B. (1964), Resistance to tuberculosis: experimental studies in native and acquired defensive mechanisms, Harvard Univ. Press, Cambridge, Mass.

    Google Scholar 

  21. Marino, S. and D. E. Kirschner (2004), The human immune response to Mycobacterium tuberculosis in lung and lymph node, J. Theor. Biol. 2274, 463.

    Article  Google Scholar 

  22. Marino, S., S. Pawar, C. L. Fuller, T. A. Reinhart, J. L. Flynn and D. E. Kirschner, (2004), Dendritic cell trafficking and antigen presentation in the human immune response to Mycobacterium tuberculosis, J. Immunol. 1731, 494.

    Google Scholar 

  23. Medzhitov R. and C. A. Janeway, Jr. (1997), Innate immunity: the virtues of a nonclonal system of recognition, Cell, 91, 295.

    Article  Google Scholar 

  24. Medzhitov, R. and C. A. Janeway, Jr. (2000), Innate immunity, N. Engl. J. Med., 343, 338.

    Article  Google Scholar 

  25. Medzhitov, R. and C. A. Janeway, Jr. (2002), Decoding the patterns of self and nonself by the innate immune system. Science, 296, 298.

    Article  Google Scholar 

  26. Mercer R. R., M. L. Russell, V. L. Roggli and J. D. Crapo (1994), Cell number and distribution in human and rat airways, Am. J. Respir. Cell. Mol. Biol., 10, 613.

    Google Scholar 

  27. Murali-Krishna, K., L. L. Lau, S. Sambhara, F. Lemonnier, J. Altman and R. Ahmed (1999), Persistence of memory CD8 T cells in MHC class I-deficient mice, Science, 286, 1377.

    Article  Google Scholar 

  28. Murray, J. D. (2002), Mathematical biology, 3rd edn., Springer, New York.

    MATH  Google Scholar 

  29. North R. J. and A. A. Izzo (1993), Mycobacterial virulence. Virulent strains of Mycobacteria tuberculosis have faster in vivo doubling times and are better equipped to resist growth-inhibiting functions of macrophages in the presence and absence of specific immunity, J. Exp. Med., 177, 1723.

    Article  Google Scholar 

  30. Shampine, L. F. and S. Thompson, Solving DDEs with Matlab, manuscript, URL: http://www.radford.edu/thompson/webddes

    Google Scholar 

  31. Silver, R. F., Q. Li, J. J. Ellner (1998b), Expression of virulence of Mycobacterium tuberculosis within human monocytes: virulence correlates with intracellular growth and induction of tumor necrosis factor alpha but not with evasion of lymphocyte-dependent monocyte effector functions, Infect. Immun., 66, 1190.

    Google Scholar 

  32. Silver, R. F., Q. Li, W. H. Boom and J. J. Ellner (1998a), Lymphocyte-dependent inhibition of growth of virulent Mycobacterium tuberculosis H37Rv within human monocytes: requirement for CD4+ T cells in purified protein derivative-positive, but not in purified protein derivative-negative subjects, J. Immunol., 160, 2408.

    Google Scholar 

  33. Sprent, J. and A. Basten (1973), Circulating T and B lymphocytes of the mouse. II. Lifespan, Cell. Immunol., 7, 40.

    Article  Google Scholar 

  34. Sprent, J., C. D. Surh (2002), T cell memory Annu. Rev. Immunol., 20, 551.

    Article  Google Scholar 

  35. Stone, K. C., R. R. Mercer, P. Gehr, B. Stockstill and J. D. Crapo (1992), Allometric relationships of cell numbers and size in the mammalian lung, Am. J. Respir. Cell. Mol. Biol., 6, 235.

    Google Scholar 

  36. Surh, C. D. and J. Sprent (2002), Regulation of naive and memory T-cell homeostasis, Microbes Infect., 4, 51.

    Article  Google Scholar 

  37. Swain, S. L., H. Hu and G. Huston (1999), Class II-independent generation of CD4 memory T cells from effectors Science, 286, 1381.

    Article  Google Scholar 

  38. Takeda, K., T. Kaisho and S. Akira (2003), Toll-like receptors, Annu. Rev. Immunol., 21, 335.

    Article  Google Scholar 

  39. Tan, J. S., D. H. Canaday, W. H. Boom, K. N. Balaji, S. K. Schwander and E. A. Rich (1997), Human alveolar T lymphocyte responses to Mycobacterium tuberculosis antigens: role for CD4+ and CD8+ cytotoxic T cells and relative resistance of alveolar macrophages to lysis, J. Immunol., 159, 290.

    Google Scholar 

  40. Tsukaguchi, K., K. N. Balaji and W. H. Boom (1995), CD4+ alpha beta T cell and gamma delta T cell responses to Mycobacterium tuberculosis. Similarities and differences in Ag recognition, cytotoxic effector function, and cytokine production, J. Immunol., 154, 1786.

    Google Scholar 

  41. Van Furth, R., M. C. Diesselhoff-den Dulk and H. Mattie (1973), Quantitative study on the production and kinetics of mononuclear phagocytes during an acute inflammatory reaction, J. Exp. Med., 138, 1314.

    Article  Google Scholar 

  42. Wigginton, J. E. and D. E. Kirschner (2001), A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. Immunol., 166, 1951.

    Google Scholar 

  43. Wong, P. and E. G. Pamer (2003), CD8 T cell responses to infectious pathogens, Annu. Rev. Immunol., 21, 29.

    Article  Google Scholar 

  44. Zinkernagel, R. M. (2003), On natural and artificial vaccinations, Annu. Rev. Immunol., 21, 515

    Article  Google Scholar 

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Beretta, E., Carletti, M., Kirschner, D.E., Marino, S. (2007). Stability Analysis of a Mathematical Model of the Immune Response with Delays. In: Takeuchi, Y., Iwasa, Y., Sato, K. (eds) Mathematics for Life Science and Medicine. Biological and Medical Physics, Biomedical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34426-1_8

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