Global Stability Analysis of the Equilibrium State of an Improved Time-Delayed Dynamics Model to Demonstrate the Development of T Cells in the Thymus

Abstract

In this paper, based on some biological meaning, triple-negative  T cells (TN) and the immature singlepositive T cells (CD3^{−}4^{+}8^{−} and CD3^{−}4^{−}8^{+}) have been  introduced into well known Mehr’s non-linear differential  equation which is used to describe proliferation, differentiation  and death of T cells in the thymus (Modeling positive and negative selection and differentiation processes in the thymus,  Journal of Theoretical Biology, 175 (1995) 103-126), and a class of improved nonlinear differential system with seven state  variables and time delays has been proposed. Then, by using  quasi-steady-state approximation and some classical analysis  techniques of functional differential equations, the local and global asymptotic stability of the equilibrium of the system have  been analysed. Finally, some numerical simulations are given to  summarize the applications of the theoretical results.