A Type of Hopf Bifurcation in Models with Holling-II Type Multiple Sclerosis

Multiple autoimmune diseases often exhibit a cyclic pattern of relapse and remission, with significant periods of loss of self-tolerance being interrupted by recurrent autoimmune events. In this article, we explore a specific type of terminally differentiated regulatory T cell $HLA？D{R}^{+}{T}_{Reg}$ cells, and their application in existing autoimmune disease models. We also conduct an in-depth study on a multiple sclerosis model. This model incorporates a Holling-II type functional response mechanism. The focus of the study is to analyze whether the equilibrium points of the system have local asymptotic stability and determine the conditions for the existence of Hopf bifurcation. Furthermore, the direction of Hopf bifurcations and the stability of its periodic solutions can be analyzed through normal form theory and center manifold theorem.