Approximate Equilibria for a T Cell and Treg Model

We analyse a model of immune response by T cells (CD4), where regulatory T cells (Tregs) act by inhibiting IL-2 secretion. We introduced an asymmetry reflecting that the difference between the growth and death rates can be higher for the active T cells and the active Tregs than for the inactive T cells and inactive Tregs. This asymmetry mimics the presence of memory T cells. In this paper we start by analysing the model in the absence of Tregs. We obtain an explicit formula that gives approximately the antigenic stimulation of T cells from the concentration of Tregs. Afterwards, we present an explicit formula that describes approximately the balance between the concentration of T cells and the concentration of Tregs; and an explicit formula that relates approximately the antigenic stimulation of T cells, the concentration of T cells and the concentration of Tregs. For our parameter values, the relation between the antigenic stimulation of T cells and the concentration of T cells is an hysteresis that is unfold when some of the parameters are changed. We also consider a linear tuning between the antigenic stimulation of T cells and the antigenic stimulation of Tregs. Again, we have obtained an explicit formula relating approximately the antigenic stimulation of T cells, the concentration of T cells and the concentration of Tregs. With it, we can explain the appearance of an isola and a transcritical bifurcation