Stability issues in a biological model of self and non-self immune regulation

In this paper, we analyze the stability of a model for the Self/Non-self discrimination. The model simulates how the T helper cells (Th), namely lymphocytes developed in the thymus, learn which antigens are self and which ones are non-self. All antigen-responsive cells are born without effector activity and, in the initial stage, have two pathways open to them: inactivation and activation. The choice between these two paths depends on the presence or absence of effector T-helpers (eTh). Thus, the problem is to provide a model for the origin of the first eTh. A kinetic formulation can be modelled with 6 variables. To have a steady state, the system starts at particular levels of populations and has to stay at these levels as long as there is no external action. A stable steady state is reached when the net change of each population equals zero. We analyze the instability of the system in respect to the SI, the proportion of anti-self cells in the total number of repertoire (anti-self and anti-non self cells). In addition, we explore the spectrum of the control parameter SI and we discover a point where the stability changes its behavior.