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

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 di...

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Bibliographic Details
Published in:Mathematics (Basel) 2020-02, Vol.8 (2), p.293
Main Authors: Yusuf, Aliyu, Figueiredo, Isabel, Afsar, Atefeh, Burroughs, Nigel, Pinto, Alberto, Oliveira, Bruno
Format: Article
Language:English
Subjects:
bifurcation
cd4+ t cells
cd4+ tregs
eigenvalues
ode model
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math8020293