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Open AccessArticle

The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs

Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal
Laboratório de Inteligência Artificial e Apoio à Decisão (LIAAD)–Instituto de Engenharia de Sistemas e Computadores, Tecnologia e Ciência (INESC TEC), 4200-465 Porto, Portugal
Department of Mathematics, Kano University of Science and Technology, Wudil, 3244 Kano, Nigeria
Departamento de Matemática, Instituto Superior de Engenharia do Porto, Instituto Politécnico do Porto, 4200-072 Porto, Portugal
Systems Biology Centre, University of Warwick, Coventry CV4 7AL, UK
Faculdade de Ciências da Nutrição e Alimentação, Universidade do Porto, 4200-465 Porto, Portugal
Authors to whom correspondence should be addressed.
Mathematics 2020, 8(2), 293;
Received: 21 November 2019 / Revised: 14 February 2020 / Accepted: 14 February 2020 / Published: 21 February 2020
(This article belongs to the Special Issue Mathematical Biology: Modeling, Analysis, and Simulations)
We study the equilibria of an Ordinary Differencial Equation (ODE) system where CD4 + effector or helper T cells and Regulatory T cells (Tregs) are present. T cells trigger an immune response in the presence of their specific antigen. Regulatory T cells (Tregs) play a role in limiting auto-immune diseases due to their immune-suppressive ability. Here, we present explicit exact formulas that give the relationship between the concentration of T cells, the concentration of Tregs, and the antigenic stimulation of T cells, when the system is at equilibria, stable or unstable. We found a parameter region of bistability, limited by two thresholds of antigenic stimulation of T cells (hysteresis). Moreover, there are values of the slope parameter of the tuning for which an isola-center bifurcation appears, and, for some other values, there is a transcritical bifurcation. We also present time evolutions of the ODE system. View Full-Text
Keywords: CD4+ T cells; CD4+ Tregs; bifurcation; eigenvalues; ODE model CD4+ T cells; CD4+ Tregs; bifurcation; eigenvalues; ODE model
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Yusuf, A.A.; Figueiredo, I.P.; Afsar, A.; Burroughs, N.J.; Pinto, A.A.; Oliveira, B.M.P.M. The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs. Mathematics 2020, 8, 293.

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