Modeling and analysis of recurrent autoimmune disease

Many autoimmune diseases are characterized by a pattern of recurrence and remission, in which periods of apparent self-tolerance are punctuated by intervals of recurring autoimmunity. We introduce a newly discovered class of terminally differentiated regulatory T cells, HLA-DR+ T_Reg cells, into an existing autoimmune disease model. Our newly developed 4-dimensional model exhibits recurrent dynamics, which are preserved in a reduced and rescaled 3-dimensional model as well. Applying dynamical systems theory, we analyze the dynamics underlying this behavior in both the 4-dimensional and 3-dimensional models and further prove that the recurrent behavior (or oscillation) arises due to a Hopf bifurcation or a persistent oscillation rather than from homoclinic orbits. Numerical simulations are conducted to verify the analytical results and identify the recurrent parameter region.