ABSTRACT

We study a mathematical model of immune responses by T cells. T cells are part of the leukocytes (white blood cells) and function by targeting “invader” antigens, see [3,6] and references within. This model consists of a set of ordinary differential equations for the two states of T cells, the two states of Regulatory T cells (Tregs) and for the Interleukine 2 (IL- 2, a signalling protein) [1-7]. We study the time evolution [1,5], equilibria and bifurcations [1,2,4,7]. The relation between the concentration of T cells and its antigenic stimulation is a hysteresis, characterized by a region of bistability bounded by two thresholds of antigenic stimulation of T cells. We study the effect of an asymmetry in the death rates of the cells [4,5]. With this asymmetry, the antigenic stimulation controls the local population of the regulatory T cells [4,5]. When we consider a positive correlation between the antigenic stimulation of T cells and the antigenic stimulation of the regulatory T cells we observe, for some parameter values, that the rate of variation of the levels of antigenic stimulation determines if an immune response arises or if the regulatory T cells are able to maintain control [4]. This behaviour is due to the presence of a transcritical bifurcation [4].

References

- [1] N. J. Burroughs, B. M. P. M. Oliveira, A. A. Pinto, Regulatory T cell adjustment of quorum growth thresholds and the control of local immune responses, Journal of Theoretical Biology 241 (2006) 134–141.
- [2] N. J. Burroughs, B. M. P. M. Oliveira, A. A. Pinto, H. J. T. Sequeira, Sensibility of the quorum growth thresholds controlling local immune responses, Mathematical and Computer Modelling 47 (7-8) (2008) 714–725.
- [3] A. A. Pinto, N. J. Burroughs, F. Ferreira, B. M. P. M. Oliveira, Dynamics of immuno- logical models, Acta Biotheoretica 58 (2010) 391–404.
- [4] N. J. Burroughs, M. Ferreira, B. M. P. M. Oliveira, A. A. Pinto, A transcritical bifur- cation in an immune response model, Journal of Difference Equations and Applications 17 (7) (2011) 1101–1106.
- [5] N. J. Burroughs, M. Ferreira, B. M. P. M. Oliveira, A. A. Pinto, Autoimmunity arising from bystander proliferation of T cells in an immune response model, Mathematical and Computer Modeling 53 (2011) 1389–1393.
- [6] N. J. Burroughs, B. M. P. M. Oliveira, A. A. Pinto, M. Ferreira, Immune response dynamics, Mathematical and Computer Modelling 53 (2011) 1410–1419.
- [7] B. M. P. M. Oliveira, I. P. Figueiredo, N. J. Burroughs, A. A. Pinto, Approximate equilibria for a T cell and Treg model, Applied Mathematics and Information Sciences 9 (5) (2015) 2221–2231.