# American Institute of Mathematical Sciences

2006, 3(1): 237-248. doi: 10.3934/mbe.2006.3.237

## An improved model of t cell development in the thymus and its stability analysis

 1 Department of Mathematics and Mechanics, Applied Science College, University of Science and Technology Beijing, Beijing 100083, China 2 Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083

Received  February 2005 Revised  May 2005 Published  November 2005

Based on some important experimental dates, in this paper we shall introduce time delays into Mehrs's non-linear differential system model which is used to describe proliferation, differentiation and death of T cells in the thymus (see, for example, [3], [6], [7] and [9]) and give a revised nonlinear differential system model with time delays. By using some classical analysis techniques of functional differential equations, we also consider local and global asymptotic stability of the equilibrium and the permanence of the model.
Citation: Hongjing Shi, Wanbiao Ma. An improved model of t cell development in the thymus and its stability analysis. Mathematical Biosciences & Engineering, 2006, 3 (1) : 237-248. doi: 10.3934/mbe.2006.3.237
 [1] Alan D. Rendall. Multiple steady states in a mathematical model for interactions between T cells and macrophages. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 769-782. doi: 10.3934/dcdsb.2013.18.769 [2] Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945 [3] Wenbo Cheng, Wanbiao Ma, Songbai Guo. A class of virus dynamic model with inhibitory effect on the growth of uninfected T cells caused by infected T cells and its stability analysis. Communications on Pure & Applied Analysis, 2016, 15 (3) : 795-806. doi: 10.3934/cpaa.2016.15.795 [4] Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159 [5] Robert Artebrant, Aslak Tveito, Glenn T. Lines. A method for analyzing the stability of the resting state for a model of pacemaker cells surrounded by stable cells. Mathematical Biosciences & Engineering, 2010, 7 (3) : 505-526. doi: 10.3934/mbe.2010.7.505 [6] Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 [7] Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 31-46. doi: 10.3934/dcdss.2020002 [8] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [9] La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 [10] Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147 [11] Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1415-1446. doi: 10.3934/dcds.2010.27.1415 [12] Wei-Ming Ni, Yaping Wu, Qian Xu. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271 [13] Pietro Baldi. Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 883-903. doi: 10.3934/dcds.2006.15.883 [14] Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917 [15] Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95 [16] Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305 [17] Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57 [18] Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39 [19] Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028 [20] Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445

2018 Impact Factor: 1.313