Uniform boundedness and asymptotic behavior of solutions in a chemotaxis model for alopecia areata

Alopecia areata (AA) is an autoimmune disease whose clinical phenotype is characterized by the formation of distinct hairless patterns on the scalp or other parts of the body. In this paper, we study a three-component chemotaxis model for AA, which describes the complex interactions among CD \(4^{+}\) $4^{+}$ T cells, CD \(8^{+}\) $8^{+}$ T cells and interferon-gamma (IFN- \(\gamma \) $\gamma$ ). Our first purpose is to establish the uniform boundedness of classical solutions for the model by self-map method, which extends the corresponding results of Lou and Tao (J Differ Equ 305:401–427, 2021, JDE) and Zhang et al. (Math Biosci Eng 20(5):7922–7942, 2023, MBE) to the case of arbitrary spatial dimensions and non-equidiffusive coefficients. Another purpose is to consider the globally asymptotic stability and convergence rate of the positive equilibrium under either (i) small proliferation rate and large degradation parameters or (ii) weak chemoattractive effect or strong random motions. It is shown under the above two cases that sparse patches occur in or around diseased hair follicles, gradually develop into diffuse or total hair loss and ultimately induce AA.