Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype

Jisun  Lim; Seongwon  Lee; Yangjin  Kim

doi:10.3934/dcdsb.2016111

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December 2016, 21(10): 3575-3602. doi: 10.3934/dcdsb.2016111

Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype

Jisun Lim ^1, , Seongwon Lee ^2, and Yangjin Kim ^3,

1.	School of Biological Sciences, Seoul National University, Seoul 08826, South Korea
2.	Division of Mathematical Models, National Institute for Mathematical Sciences, Daejeon 34047, South Korea
3.	Department of Mathematics, Konkuk University, Seoul, 05029, South Korea

Received September 2015 Revised September 2016 Published November 2016

Abstract
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Airway exposure of lipopolysaccharide (LPS) is shown to regulate type I and type II helper T cell induced asthma. While high doses of LPS derive Th1- or Th17-immune responses, low LPS levels lead to Th2 responses. In this paper, we analyze a mathematical model of Th1/Th2/Th17 asthma regulation suggested by Lee (S. Lee, H.J. Hwang, and Y. Kim, Modeling the role of TGF-$\beta$ in regulation of the Th17 phenotype in the LPS-driven immune system, Bull Math Biol., 76 (5), 1045-1080, 2014) and show that the system can undergo a Hopf bifurcation at a steady state of the Th17 phenotype for high LPS levels in the presence of time delays in inhibition pathways of two key regulators: IL-4/Th2 activities ($H$) and TGF-$\beta$ levels ($G$). The time delays affect the phenotypic switches among the Th1, Th2, and Th17 phenotypes in response to time-dependent LPS doses via nonlinear crosstalk between $H$ and $G$. An extended reaction-diffusion model also predicts coexistence of these phenotypes under various biochemical and bio-mechanical conditions in the heterogeneous microenvironment.

Keywords: mathematical model, asthma, Delay differential equation, TGF$\beta$., Th1/Th2/Th17, Hopf bifurcation.

Mathematics Subject Classification: Primary: 92C45, 92C50; Secondary: 92B0.

Citation: Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111

References:

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