Multiple steady states in a mathematical model for interactions between T cells and macrophages

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May 2013, 18(3): 769-782. doi: 10.3934/dcdsb.2013.18.769

Multiple steady states in a mathematical model for interactions between T cells and macrophages

Alan D. Rendall ^1,

1.	Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Potsdam, Germany

Received August 2011 Revised November 2012 Published December 2012

Abstract
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The aim of this paper is to prove results about the existence and stability of multiple steady states in a system of ordinary differential equations introduced by R. Lev Bar-Or [5] to model the interactions between T cells and macrophages. Previous results showed that for certain values of the parameters these equations have three stationary solutions, two of which are stable. Here it is shown that there are values of the parameters for which the number of stationary solutions is at least seven and the number of stable stationary solutions at least four. This requires approaches different to those used in existing work on this subject. In addition, a rather explicit characterization is obtained of regions of parameter space for which the system has a given number of stationary solutions.

Keywords: Dynamical systems, qualitative behaviour, immunology..

Mathematics Subject Classification: Primary: 92C37; Secondary: 37N2.

Citation: Alan D. Rendall. Multiple steady states in a mathematical model for interactions between T cells and macrophages. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 769-782. doi: 10.3934/dcdsb.2013.18.769

References:

[1]	R. Callard, Decision-making by the immune response, Immun. and Cell Biol., 85 (2007), 300-305.
[2]	R. de Waal Malefyt, J. Haanen, H. Spits, M.-G. Roncarolo, A. te Velde, C. Figdor, K. Johnson, R. Kastelein, H. Yssel and J. E. de Vries, Interleukin 10 (IL-10) and viral IL-10 strongly reduce antigen-specific human T cell proliferation by diminishing the antigen-presenting capacity of monocytes via downregulation of Class II major histocompatibility complex expression, J. Exp. Med., 174 (1991), 915-924.
[3]	M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I: Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013.
[4]	E. Klipp, R. Herwig, A. Kowald, C. Wierling and H. Lehrach, "Systems Biology in Practice," Wiley-VCH, Weinheim, 2005.
[5]	R. Lev Bar-Or, Feedback mechanisms between T helper cells and macrophages in the determination of the immune response, Math. Biosci., 163 (2000), 35-58.
[6]	R. M. May, Uses and abuses of mathematics in biology, Science, 303 (2004), 790-793.
[7]	K. M. Murphy, P. Travers and M. Walport, "Janeway's Immunobiology," Garland Science, New York, 2007.
[8]	A. D. Rendall, Analysis of a mathematical model for interactions between T cells and macrophages, Electr. J. Diff. Eq., 2010 (2010), 115.
[9]	I. Roitt, J. Brostoff, D. Male and D. Roth, "Immunology," Mosby, New York, 2006.
[10]	L. Steinman, A brief history of Th17, the first major revision of the Th1/Th2 hypothesis of T cell-mediated tissue damage, Nature Med., 13 (2007), 139-145.
[11]	H.-J. van den Hamm and R. J. de Boer, From the two-dimensional Th1 and Th2 phenotypes to high-dimensional models for gene regulation, Int. Immunol., 20 (2008), 1269-1277.
[12]	A. Yates, R. Callard and J. Stark, Combining cytokine signalling with T-bet and GATA-3 regulation in Th1 and Th2 differentiation: A model for cellular decision-making, J. Theor. Biol., 231 (2004), 181-196. doi: 10.1016/j.jtbi.2004.06.013.

show all references

References:

[1]	R. Callard, Decision-making by the immune response, Immun. and Cell Biol., 85 (2007), 300-305.
[2]	R. de Waal Malefyt, J. Haanen, H. Spits, M.-G. Roncarolo, A. te Velde, C. Figdor, K. Johnson, R. Kastelein, H. Yssel and J. E. de Vries, Interleukin 10 (IL-10) and viral IL-10 strongly reduce antigen-specific human T cell proliferation by diminishing the antigen-presenting capacity of monocytes via downregulation of Class II major histocompatibility complex expression, J. Exp. Med., 174 (1991), 915-924.
[3]	M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I: Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013.
[4]	E. Klipp, R. Herwig, A. Kowald, C. Wierling and H. Lehrach, "Systems Biology in Practice," Wiley-VCH, Weinheim, 2005.
[5]	R. Lev Bar-Or, Feedback mechanisms between T helper cells and macrophages in the determination of the immune response, Math. Biosci., 163 (2000), 35-58.
[6]	R. M. May, Uses and abuses of mathematics in biology, Science, 303 (2004), 790-793.
[7]	K. M. Murphy, P. Travers and M. Walport, "Janeway's Immunobiology," Garland Science, New York, 2007.
[8]	A. D. Rendall, Analysis of a mathematical model for interactions between T cells and macrophages, Electr. J. Diff. Eq., 2010 (2010), 115.
[9]	I. Roitt, J. Brostoff, D. Male and D. Roth, "Immunology," Mosby, New York, 2006.
[10]	L. Steinman, A brief history of Th17, the first major revision of the Th1/Th2 hypothesis of T cell-mediated tissue damage, Nature Med., 13 (2007), 139-145.
[11]	H.-J. van den Hamm and R. J. de Boer, From the two-dimensional Th1 and Th2 phenotypes to high-dimensional models for gene regulation, Int. Immunol., 20 (2008), 1269-1277.
[12]	A. Yates, R. Callard and J. Stark, Combining cytokine signalling with T-bet and GATA-3 regulation in Th1 and Th2 differentiation: A model for cellular decision-making, J. Theor. Biol., 231 (2004), 181-196. doi: 10.1016/j.jtbi.2004.06.013.

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