Global and exponential stabilization of a chemotactic alopecia areata model with signal-dependent diffusion and sensitivity

This paper focuses on the large time behavior of a three-component chemotaxis model with signal-dependent diffusion and sensitivity for alopecia areata (AA) which is a noncontagious autoimmune disorder. The model describes the complex interactions among the CD$4^{+}$ T cells, CD$8^{+}$ T cells and interferon-gamma (IFN-$\gamma$). Firstly, we use a method of weighted energy estimates to establish the uniform-in-time boundedness of classical solutions for the system when nonlinear proliferation rate is small compared with density-dependent death rates. Then, the globally exponentially asymptotic stability of positive equilibrium is proved if nonlinear proliferation rate and density-dependent death rates are within a particular range, and signal-dependent sensitivities are small relative to signal-dependent diffusions. Finally, some numerical simulations are performed to reveal the spatio-temporal dynamics of system in two space dimensions. It is shown that sparse and homogeneous hairless patches occur in or around diseased hair follicles (HFs) and eventually induce a stable AA if signal-dependent chemotactic effect is weak.