MAP-PDMA

Bruno M. P. M. Oliveira

FCNAUP and LIAAD – INESC TEC – University of Porto

Abstract ::  We use a set of ordinary differential equations (ODE) to study mathematically the effect of regulatory T cells (Tregs) in the control of immune responses by CD4+ T cells. T cells trigger an immune response in the presence of their specific antigen, while regulatory T cells (Tregs) play a role in limiting auto-immune diseases due to their immune-suppressive ability, see Pinto et al. [5], Yusuf et al. [6] and references within. We fitted this model to quantitative data regarding the CD4+ T cell numbers from the 28 days following the infection of mice with lymphocytic choriomeningitis virus LCMV. We observed the proliferation of T cells and, to a lower extent, Tregs during the immune activation phase following infection and subsequently, during the contraction phase, a smooth transition from faster to slower death rates, see Afsar et al. [1]. 

Furthermore, we have obtained 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 region of bistability, where 2 stable equilibria exist. Making a cross section along the antigenic stimulation of T cells parameter, we observe an hysteresis bounded by two thresholds of antigenic stimulation of T cells. Moreover, there are values of the slope parameter of the tuning, between the antigenic stimulation of T cells and the antigenic stimulation of Tregs, for which an isolacenter bifurcation appear and, for some other values, there is a transcritical bifurcation, see Yusuf et al. [6] and references within. 

Time evolutions of this model were also used to simulate the appearance of autoimmunity both due to cross-reactivity or due to bystander proliferation, and to simulate the suppression of the autoimmune line of T cells after a different line of T cells responds to a pathogen infection, see Burroughs et al. [2, 3] and Oliveira et al. [4].

[1] A. Afsar, F. Martins, B. M. P. M. Oliveira, and A. A. Pinto. A fit of CD4 + T cell immune response to an infection by lymphocytic choriomeningitis virus. Mathematical Biosciences and Engineering, 16(6):70097021, 2019.

[2] N. J. Burroughs, B. M. P. M. Oliveira, and A. A. Pinto. Regulatory T cell adjustment of quorum growth thresholds and the control of local immune responses. Journal of Theoretical Biology, 241:134141, 2006.

[3] N. J. Burroughs, M. Ferreira, B. M. P. M. Oliveira, and A. A. Pinto. Autoimmunity arising from bystander proliferation of T cells in an immune response model. Mathematical and Computer Modelling, 53:13891393, 2011.

[4] B. M. P. M. Oliveira, R. Trinchet, M. V. Otero-Espinar, A. A. Pinto, and N. J. Burroughs. Modelling the suppression of autoimmunity after pathogen infection. Mathematical Methods in the Applied Sciences, 41(18):85658570, 2018.

[5] A. A. Pinto, N. J. Burroughs, F. Ferreira, and B. M. P. M. Oliveira. Dynamics of immunological models. Acta Biotheoretica, 58:391404, 2010. [6] A. A. Yusuf, Isabel P. Figueiredo, A. Afsar, N. J. Burroughs, B. M. P. M. Oliveira, and A. A. Pinto. The effect of a linear tuning between the antigenic stimulations of CD4+T cells and CD4+ Tregs. Mathematics, 58:391404, 2010.

 Seminar Room of DMat-UMinho (3.08), and via zoom at 

https://videoconf-colibri.zoom.us/j/94769148463?pwd=YWRVYjEzMkk2bmtMY2Q2OEhFM0hUdz09

Seminar for the Doctoral Program in Applied Mathematics (MAP-PDMA Seminar)