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

Article Contents

2016, Volume 21, Issue 10: 3575-3602. Doi: 10.3934/dcdsb.2016111

This issue Previous Article Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes Next Article Optimal contraception control for a nonlinear population model with size structure and a separable mortality

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

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

Received: September 2015

Revised: September 2016

Published: December 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 92C45, 92C50; Secondary: 92B05.

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