Dynamics of Immunological Models
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Dynamics of Immunological Models A. A. Pinto • N. J. Burroughs • M. Ferreira B. M. P. M. Oliveira
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Received: 17 June 2010 / Accepted: 5 July 2010 / Published online: 4 August 2010 Springer Science+Business Media B.V. 2010
Abstract We analyse the effect of the regulatory T cells (Tregs) in the local control of the immune responses by T cells. We obtain an explicit formula for the level of antigenic stimulation of T cells as a function of the concentration of T cells and the parameters of the model. The relation between the concentration of the T cells and the antigenic stimulation of T cells is an hysteresis, that is unfold for some A. A. Pinto (&) Department of Mathematics, Faculty of Sciences, University of Porto; LIAAD-INESC Porto LA, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal e-mail: [email protected] A. A. Pinto Center of Mathematics and Department of Mathematics and Applications, School of Sciences, University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal N. J. Burroughs Mathematics Institute and Warwick Systems Biology Centre, University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected] M. Ferreira ESEIG, Instituto Polite´cnico do Porto, Rua D. Sancho I, number 981, 4480-876 Vila do Conde, Portugal e-mail: [email protected] M. Ferreira LIAAD-INESC Porto LA, University of Porto, Porto, Portugal M. Ferreira University of Minho, Campus de Gualtar, 4710-057 Braga, Portugal B. M. P. M. Oliveira Faculdade de Cieˆncias da Nutric¸a˜o e Alimentac¸a˜o, Universidade do Porto; LIAAD-INESC Porto LA, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal e-mail: [email protected]
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parameter values. We study the appearance of autoimmunity from cross-reactivity between a pathogen and a self antigen or from bystander proliferation. We also study an asymmetry in the death rates. With this asymmetry we show that the antigenic stimulation of the Tregs is able to control locally the population size of Tregs. Other effects of this asymmetry are a faster immune response and an improvement in the simulations of the bystander proliferation. The rate of variation of the levels of antigenic stimulation determines if the outcome is an immune response or if Tregs are able to maintain control due to the presence of a transcritical bifurcation for some tuning between the antigenic stimuli of T cells and Tregs. This behavior is explained by the presence of a transcritical bifurcation. Keywords
Immunology Tregs Secretion inhibition ODE model
1 Introduction This chapter surveys, essentially, the works presented in the papers [2, 3, 4, 5, 6] and [7]. The primary function of the immune system is the protection of the host from pathogen invasion. During such an invasion, T cells specific to the antigen proliferate and under most circumstances successfully remove the pathogen. However, the immune system can also target self antigens (autoimmunity) and cause tissue damage and death. Regulatory T cells, or Tregs, have emerged as a fundamental component of the T cell repertoire,
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