Equilibrium of a kinetic model of oscillating patterns for chronic autoimmune diseases

Abstract

We study the equilibrium states of a non-linear kinetic system of integro-partial differential equations describing the well known relapsingremitting stages observed in many chronic autoimmune diseases. The kinetic system describes the immune system at the cellular level in terms of conservative, proliferative and destructive interactions between population of cells, with the addition of natural death rates and a constant input source for one of the cell populations. The macroscopic equations describing the global behaviour of the populations have been obtained from the kinetic system by averaging with respect to the activity variable. They constitute an autonomous dynamical system possessing very interesting dynamical properties in terms of equilibrium states, their stability and existence of bifurcations. We study the equilibrium solutions of the kinetic system and relate these states to the equilibrium solutions of the corresponding macroscopic equations. We also analyse the stability of the equilibrium solutions of the kinetic system by establishing an H-theorem for certain variants of the model. First, we deal with the case in which only conservative interactions are involved in the dynamics, and then we study the case in which also non conservative interactions are involved under the assumption that proliferative interactions are counterbalanced by destructive interactions. Finally, we determine numerically equilibrium solutions of the conservative system exhibiting a stable behaviour, and then we obtain both unstable and stable solutions for the non conservative kinetic system.