Abstract
Simple predator-prey type models have brought much insight into the dynamics of both nonspecific and antigen-specific immune responses. However, until now most attention has been focused on examining how the dynamics of interactions between the parasite and the immune system depends on the nature of the function describing the rate of activation or proliferation of immune cells in response to the parasite. In this paper we focus on the term describing the killing of the parasite by cell-mediated immune responses. This term has previously been assumed to be a simple mass-action term dependent solely on the product of the densities of the parasite and the immune cells and does not take into account a handling time (which we define as the time of interaction between an immune cell and its target, during which the immune cell cannot interact with and/or destroy additional targets). We show how the handling time (i) can be incorporated into simple models of nonspecific and specific immunity and (ii) how it affects the dynamics of both nonspecific and antigen-specific immune responses, and in particular the ability of the immune response to control the infection.
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References
Adams, D. O. and T. A. Hamilton (1984). The cell biology of macrophage activation. Annu. Rev. Immunol. 2, 283–318.
Aderem, A. and D.-M. Underhill (1999). Mechanisms of phagocytosis in macrophages. Ann. Rev. Immunol. 17, 593–623.
Agur, Z., D. Abiri and H. T. Van der Ploeg (1989). Ordered appearance of antigenic variants of africain trypanosomes in a mathematical model based on a stochastic process and immune-selection against putative switch intermediates. Proc. Natl Acad. Sci. USA 86, 9626–9630.
Antia, R. and J. Koella (1994). A model of nonspecific immunity. J. Theor. Biol. 168, 141–150.
Antia, R., J. Koella and V. Perrot (1996). Models of the within-host dynamics of persistent mycobacterial infections. Proc. R. Soc. Lond. B 263, 257–263.
Bancroft, G. J., M. J. Bosma, G. C. Bosma and E. R. Unanue (1986). Regulation of macrophage IA expression in mice with severe combined immunodeficiency—induction of IA expression by a T-cell-independent mechanism. J. Immunol. 137, 4–9.
Bancroft, G. J., R. D. Schreiber and E. R. Unanue (1991). Natural immunity: a T-cell independent pathway of macrophage activation defined in the SCID mouse. Immunol. Rev. 124, 5–24.
Bazykin, A. D. (1975). Structural and dynamic stability of model predator-prey systems, IIASA Workshop on Computation of Stability Regions and Equilibria.
Bell, G. I. (1970). Mathematical model of clonal selection and antibody production. Nature 228, 739–744.
Bell, G. I. (1973). Predator-prey equations simulating an immune response. Math. Biosci. 16, 291–314.
Bennett, S. R. M., F. R. Carbone, F. Karamalis, R. A. Flavell, J. F. A. P. Miller and W. R. Heath (1998). Help for cytotoxic-T-cell responses is mediated by CD40 signaling. Nature 393, 478–480.
Berke, G. (1994). The binding and lysis of target cells by cytotoxic lymphocytes: molecular and cellular aspects. Annu. Rev. Immunol. 12, 735–773.
Borghans, J. A. M., R. J. De Boer and L. A. Segel (1996). Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol. 58, 43–63.
Burnett, F. M. (1959). The Clonal Selection Theory of Immunity, Nashville: Vanderbilt University Press, and Cambridge: Cambridge University Press.
Burroughs, N. J. and D. A. Rand (1998). Dynamics of T-cell antagonism enhanced viral diversity and survival. Proc. R. Soc. Lond. B 265, 529–535.
Coddington, E. A. and N. Levinson (1955). Theory of Ordinary Differential Equations, New York: McGraw-Hill.
Coppel, W. A. (1965). Stability and Asymptotic Behavior of Differential Equations, Boston: D. C. Heath.
Crawley, M. J. (1992). Natural Enemies: The Population Biology of Predators, Parasites and Diseases, M. J. Crawley (Ed.), Oxford, Boston: Blackwell Scientific Publications.
De Boer, R. J. and A. S. Perelson (1995). Towards the general function describing T-cell proliferation. J. Theor. Biol. 175, 567–576.
Grakoui, A., S. K. Bromley, C. Sumen, M. M. Davis, A. S. Shaw, P. M. Allen and M. L. Dustin (1999). The immunological synapse: a molecular machine controlling T-cell activation. Science 285, 221–227.
Greenberg, S. and S. G. Silverstein (1993). Phagocytosis, in Fundamental Immunology, 3rd edn, New York: Raven Press, pp. 941–965.
Hale, J. K. and H. Kocak (1991). Dynamics and Bifurcations, New York-Berlin: Springer-Verlag.
Hofbauer, J. and J. W.-H. So (1990). Multiple limit cycles for predator-prey models. Math. Biosci. 99, 71–75.
Hsu, S.-B. (1978). On global stability of a predator-prey system. Math. Biosci. 39, 1–10.
Kevrikidis, I.-G., A.-D. Zecha and A.-S. Perelson (1988). Modeling dynamical aspects of the immune response; T cell proliferation and the effect of IL-Z in Theoretical Immunology, A.-S. Perelson (Ed.), Reading, MA: Addison-Wesley, pp. 167–197.
Kuby, J. (1997). Immunology, 3rd edn, New York: H. Freeman.
McLean, A. R. and T. B. L. Kirkwood (1990). A model of human immunodeficiency virus infection in T helper cell clones. J. Theor. Biol. 147, 177–203.
Murali-Krishna, K., J. D. Altman, M. Suresh, D. J. D. Sourdive, A. J. Zajac, J. D. Miller, J. Slansky and R. Ahmed (1998). Counting antigen-specific CD8+ T cells: A reevaluation of bystander activation during viral infection. Immunity 8, 177–187.
Perelson, A. S. and G. I. Bell (1982). Delivery of lethal hits by cytotoxic T lymphocytes in multicellular conjugates occurs sequentially but at random times. J. Immunol. 129, 2796–2801.
Pilyugin, S., J. Mittler and R. Antia (1997). Modeling T-cell proliferation: an investigation of the consequences of the Hayflick limit. J. Theor. Biol. 186, 117–129.
Razvi, E. S., R. M. Welsh and H. I. McFarland (1995). In vivo state of antiviral CTL precursors. J. Immunol. 154, 620–632.
Ridge, J. P., F. Di Rosa and P. Matzinger (1998). A conditioned dendritic cell can be a temporal bridge between a CD4+ T-helper and a T-killer cell. Nature 393, 474–478.
Schweitzer, A. N., J. Swinton and R. M. Anderson (1992). Complex outcomes in mouse leishmaniasis: a model for the dynamics of the Th1 response, in Theoretical and Experimental Insights into Immunology, A. S. Perelson and G. Weisbuch (Eds), NATO ASI Series H66, pp. 191–202, Berlin: Springer-Verlag.
Valitutti, J. R., J. A. Sullivan, G. L. Mandell and V. H. Engelhard (1996). Different responses are elicited in cytotoxic T lymphocytes by different levels of T cell receptor occupancy. J. Exp. Med. 183, 1917–1921.
Weiss, A. (1993). T lymphocyte activation, in Fundamental Immunology, New York: Raven Press, pp. 467–505.
Yanelli, J. R., J. A. Sullivan, G. L. Mandell and V. H. Engelhard (1986). Reorientation and fusion of cytotoxic T lymphocyte granules after interaction with target cells as determined by high resolution cinemicrography. J. Immunol. 136, 377–382.
Zagury, D., J. Bernard, N. Thierness, M. Feldman and G. Berke (1975). Isolation and characterization of individual functionally reactive cytotoxic T lymphocytes: conjugation, killing and recycling at the single cell level. Eur. J. Immunol. 5, 818–822.
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Pilyugin, S.S., Antia, R. Modeling immune responses with handling time. Bull. Math. Biol. 62, 869–890 (2000). https://doi.org/10.1006/bulm.2000.0181
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DOI: https://doi.org/10.1006/bulm.2000.0181