The role of local kinetics in a three-component chemotaxis model for Alopecia Areata

Abstract

This manuscript considers the homogeneous Neumann initial-boundary value problem for a three-component chemotaxis system describing the spatio-temporal Alopecia Areata dynamics. The model addresses the complex interactions between CD4+ T cells, CD8+ T cells and interferon-gamma (IFN-γ): Both types of immune cells secrete the chemical that diffuses and degrades; conversely, the T cells are activated by the chemical but decay due to density-dependent death. In addition to random motion, the T cells bias their movement toward the concentration gradient of IFN-γ. To compare with previous chemotaxis models, a distinctive feature of this system is that CD8+ T cells additionally proliferate in a nonlinear manner with the help of CD4+ T cells. Given suitably regular initial data, it is shown that either small logistic damping in the two-dimensional case or strong logistic damping in the three-dimensional setting can prevent any blow-up of classical solutions to the problem. Moreover, we find a specific parameter regime for which the unique coexistence equilibrium is globally convergent.