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

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  • Aliyu A. Yusuf

    (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)

  • Isabel P. Figueiredo

    (Departamento de Matemática, Instituto Superior de Engenharia do Porto, Instituto Politécnico do Porto, 4200-072 Porto, Portugal)

  • Atefeh Afsar

    (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)

  • Nigel J. Burroughs

    (Systems Biology Centre, University of Warwick, Coventry CV4 7AL, UK)

  • Alberto A. Pinto

    (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)

  • Bruno M. P. M. Oliveira

    (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
    Faculdade de Ciências da Nutrição e Alimentação, Universidade do Porto, 4200-465 Porto, Portugal)

Abstract

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.

Suggested Citation

  • Aliyu A. Yusuf & Isabel P. Figueiredo & Atefeh Afsar & Nigel J. Burroughs & Alberto A. Pinto & Bruno M. P. M. Oliveira, 2020. "The Effect of a Linear Tuning between the Antigenic Stimulations of CD4 + T Cells and CD4 + Tregs," Mathematics, MDPI, vol. 8(2), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:293-:d:323388
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    Cited by:

    1. Ricardo López-Ruiz, 2022. "Mathematical Biology: Modeling, Analysis, and Simulations," Mathematics, MDPI, vol. 10(20), pages 1-2, October.

    More about this item

    Keywords

    CD4 + T cells; CD4 + Tregs; bifurcation; eigenvalues; ODE model;
    All these keywords.

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