Revealing the role of the effector-regulatory t cell loop on autoimmune disease symptoms via nonlinear analysis.

• This paper studies the chronic, multiphases, and relapse-remitting autoimmune disease outcomes. • Our results show that stable states, recurrent oscillations, and three co-existing limit cycles determine the rich disease outcomes. • The mathematical theory is related to the well-known Hilbert's 16th problem. • Numerical simulation demonstrates that the solution (effector T cell population) travels between two stable periodic solutions in a random manner. • Our approach presented in this paper is not only theoretically significant, but also very useful in predicting dynamical behaviors in real systems. In this paper, we investigate the influence of the effector-regulatory (Teff-Treg) T cell interaction on the T-cell-mediated autoimmune disease dynamics. The simple 3-dimensional Teff-Treg model is derived from the two-step model reduction of an established 5-dimensional model. The reduced 4- and 3-dimensional models preserve the dynamical behaviors in the original 5-dimensional model, which represents the chronic and relapse-remitting autoimmune symptoms. Moreover, we find three co-existing limit cycles in the reduced 3-dimensional model, in which two stable periodic solutions enclose an unstable one. The existence of multiple limit cycles provides a new mechanism to explain varying oscillating amplitudes of lesion grade in multiple sclerosis. The complex multiphase symptom could be caused by a noise-driven Teff population traveling between two coexisting stable periodic solutions. The simulated phase portrait and time history of coexisting limit cycles are given correspondingly. [ABSTRACT FROM AUTHOR]