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:

[1]	S. Al-Muhsen, S. Letuve, A. Vazquez-Tello, M. A. Pureza, H. Al-Jahdali, A. S. Bahammam, Q. Hamid and R. Halwani, Th17 cytokines induce pro-fibrotic cytokines release from human eosinophils, Respir. Res., 14 (2013), p34. doi: 10.1186/1465-9921-14-34. Google Scholar
[2]	T. Alarcón, H. M. Byrne and P. K. Maini, Towards whole-organ modelling of tumour growth, Prog. Biophys. Mol. Biol., 85 (2004), 451-472. Google Scholar
[3]	J. F. Alcorn, C. R. Crowe and J. K. Kolls, $T_H$17 cells in asthma and COPD, Annu. Rev. Physiol., 72 (2010), 495-516. Google Scholar
[4]	O. Arino, M. L. Hbid and E. Ait Dads, Delay Differential Equations and Applications, Springer Netherlands, 2006. doi: 10.1007/1-4020-3647-7. Google Scholar
[5]	K. J. Baek, J. Y. Cho, P. Rosenthal, L. E. C. Alexander, V. Nizet and D. H. Broide, Hypoxia potentiates allergen induction of HIF-1$\alpha$, chemokines, airway inflammation, TGF-$\beta$1, and airway remodeling in a mouse model, Clin. Immunol., 147 (2013), 27-37. Google Scholar
[6]	R. L. Bar-Or and L. A. Segel, On the role of a possible dialogue between cytokine and TCR-presentation mechanisms in the regulation of autoimmune disease, J. Theor. Biol., 190 (1998), 161-178. doi: 10.1006/jtbi.1997.0545. Google Scholar
[7]	U. Behn, H. Dambeck and G. Metzner, Modeling Th1-Th2 regulation, allergy, and hyposensitization, in Dynamical Modeling in Biotechnology, World Scientific, 2001, chapter 11, 227-243. doi: 10.1142/9789812813053_0011. Google Scholar
[8]	B. S. Bochner, B. J. Undem and L. M. Lichtenstein, Immunological aspects of allergic asthma, Annu. Rev. Immunol., 12 (1994), 295-335. doi: 10.1146/annurev.iy.12.040194.001455. Google Scholar
[9]	R. E. Callard and A. J. Yates, Immunology and mathematics: Crossing the divide, Immunology, 115 (2005), 21-33. doi: 10.1111/j.1365-2567.2005.02142.x. Google Scholar
[10]	J. Carneiro, J. Stewart, A. Coutinho and G. Coutinho, The ontogeny of class-regulation of CD4$^+$ T lymphocyte populations, Int. Immunol., 7 (1995), 1265-1277. doi: 10.1093/intimm/7.8.1265. Google Scholar
[11]	C. Clemedson and A. Nelson, General biology: The adult organism, in Mechanisms in Radiobiology: Multicellular Organisms (eds. M. Errera and A. Forssberg), Elsevier, 1960, 95-205. doi: 10.1016/B978-1-4832-2829-7.50010-1. Google Scholar
[12]	L. Cosmi, F. Liotta, E. Maggi, S. Romagnani and F. Annunziato, Th17 cells: New players in asthma pathogenesis, Allergy, 66 (2011), 989-998. doi: 10.1111/j.1398-9995.2011.02576.x. Google Scholar
[13]	E. Cutz, H. Levison and D. M. Cooper, Ultrastructure of airways in children with asthma, Histopathology, 2 (1978), 407-421. doi: 10.1111/j.1365-2559.1978.tb01735.x. Google Scholar
[14]	C. Dong, Diversification of T-helper-cell lineages: Finding the family root of IL-17-producing cells, Nat. Rev. Immunol., 6 (2006), 329-334. doi: 10.1038/nri1807. Google Scholar
[15]	C. Dong, $T_H$17 cells in development: An updated view of their molecular identity and genetic programming, Nat. Rev. Immunol., 8 (2008), 337-348. Google Scholar
[16]	S. C. Eisenbarth, D. A. Piggott, J. W. Huleatt, I. Visintin, C. A. Herrick and K. Bottomly, Lipopolysaccharide-enhanced, toll-like receptor 4-dependent T helper cell type 2 responses to inhaled antigen, J. Exp. Med., 196 (2002), 1645-1651. doi: 10.1084/jem.20021340. Google Scholar
[17]	R. L. Elliott and G. C. Blobe, Role of transforming growth factor beta in human cancer, J. Clin. Oncol., 23 (2005), 2078-2093. Google Scholar
[18]	M. A. Fishman and A. S. Perelson, Th1/Th2 differentiation and cross-regulation, Bull. Math. Biol., 61 (1999), 403-436. doi: 10.1006/bulm.1998.0074. Google Scholar
[19]	J. E. Gereda, D. Y. M. Leung, A. Thatayatikom, J. E. Streib, M. R. Price, M. D. Klinnert and A. H. Liu, Relation between house-dust endotoxin exposure, type 1 T-cell development, and allergen sensitisation in infants at high risk of asthma, Lancet, 355 (2000), 1680-1683. doi: 10.1016/S0140-6736(00)02239-X. Google Scholar
[20]	L. Gorelik, S. Constant and R. A. Flavell, Mechanism of transforming growth factor $\beta$-induced inhibition of T helper type 1 differentiation, J. Exp. Med., 195 (2002), 1499-1505. Google Scholar
[21]	L. Gorelik and R. A. Flavell, Abrogation of TGF$\beta$ signaling in T cells leads to spontaneous T cell differentiation and autoimmune disease, Immunity, 12 (2000), 171-181. Google Scholar
[22]	F. Gross, G. Metznerb and U. Behn, Mathematical modelling of allergy and specific immunotherapy: Th1-Th2-Treg interactions, J. Theor. Biol., 269 (2011), 70-78. doi: 10.1016/j.jtbi.2010.10.013. Google Scholar
[23]	G. Grünig, M. Warnock, A. E. Wakil, R. Venkayya, F. Brombacher, D. M. Rennick, D. Sheppard, M. Mohrs, D. D. Donaldson, R. M. Locksley and D. B. Corry, Requirement for IL-13 independently of IL-4 in experimental asthma, Science, 282 (1998), 2261-2263. Google Scholar
[24]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, 1st edition, Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar
[25]	I. Gutcher and B. Becher, APC-derived cytokines and T cell polarization in autoimmune inflammation, J. Clin. Invest., 117 (2007), 1119-1127. doi: 10.1172/JCI31720. Google Scholar
[26]	J. K. Hale, Theory of Functional Differential Equations, vol. 3 of Applied Mathematical Sciences, Springer-Verlag New York, 1977. Google Scholar
[27]	Q. Hamid and M. Tulic, Immunobiology of asthma, Annu. Rev. Physiol., 71 (2009), 489-507. doi: 10.1146/annurev.physiol.010908.163200. Google Scholar
[28]	L. E. Harrington, R. D. Hatton, P. R. Mangan, H. Turner, T. L. Murphy, K. M. Murphy and C. T. Weaver, Interleukin 17-producing $CD4^+$ effector T cells develop via a lineage distinct from the T helper type 1 and 2 lineages, Nat. Immunol., 6 (2005), 1123-1132. doi: 10.1038/ni1254. Google Scholar
[29]	L. E. Harrington, P. R. Mangan and C. T. Weaver, Expanding the effector CD4 T-cell repertoire: The Th17 lineage, Curr. Opin. Immunol., 18 (2006), 349-356. doi: 10.1016/j.coi.2006.03.017. Google Scholar
[30]	B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1981. Google Scholar
[31]	N. A. Hosken, K. Shibuya, A. W. Heath, K. M. Murphy and A. O'Garra, The effect of antigen dose on CD4$^+$ T helper cell phenotype development in a T cell receptor-$\alpha\beta$-transgenic model, J. Exp. Med., 182 (1995), 1579-1584. doi: 10.1084/jem.182.5.1579. Google Scholar
[32]	H. Jiang and L. Chess, An integrated view of suppressor T cell subsets in immunoregulation, J. Clin. Invest., 114 (2004), 1198-1208. doi: 10.1172/JCI23411. Google Scholar
[33]	Y. Kim, H. Lee, N. Dmitrieva, J. Kim, B. Kaur and A. Friedman, Choindroitinase ABC I-mediated enhancement of oncolytic virus spread and anti-tumor efficacy: A mathematical model, PLoS One, 9 (2014), e102499. doi: 10.1371/journal.pone.0102499. Google Scholar
[34]	Y. Kim, S. Lee, Y. Kim, Y. Kim, Y. Gho, H. Hwang and S. Lawler, Regulation of Th1/Th2 cells in asthma development: A mathematical model, Math. Bios. Eng, 10 (2013), 1095-1133. doi: 10.3934/mbe.2013.10.1095. Google Scholar
[35]	Y. Kim and H. Othmer, A hybrid model of tumor-stromal interactions in breast cancer, Bull Math Biol, 75 (2013), 1304-1350. doi: 10.1007/s11538-012-9787-0. Google Scholar
[36]	Y. Kim and S. Roh, A hybrid model for cell proliferation and migration in glioblastoma, Discrete and Continuous Dynamical Systems-B, 18 (2013), 969-1015. doi: 10.3934/dcdsb.2013.18.969. Google Scholar
[37]	Y. Kim, M. Stolarska and H. G. Othmer, A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results, Math. Models Methods Appl. Sci., 17 (2007), 1773-1798. doi: 10.1142/S0218202507002479. Google Scholar
[38]	Y. Kim, M. Stolarska and H. Othmer, The role of the microenvironment in tumor growth and invasion, Prog Biophys Mol Biol, 106 (2011), 353-379. doi: 10.1016/j.pbiomolbio.2011.06.006. Google Scholar
[39]	Y.-K. Kim, S.-Y. Oh, S. G. Jeon, H.-W. Park, S.-Y. Lee, E.-Y. Chun, B. Bang, H.-S. Lee, M.-H. Oh, Y.-S. Kim, J.-H. Kim, Y. S. Gho, S.-H. Cho, K.-U. Min, Y.-Y. Kim and Z. Zhu, Airway exposure levels of lipopolysaccharide determine type 1 versus type 2 experimental asthma, J. Immunol., 178 (2007), 5375-5382. doi: 10.4049/jimmunol.178.8.5375. Google Scholar
[40]	Y.-S. Kim, S.-W. Hong, J.-P. Choi, T.-S. Shin, H.-G. Moon, E.-J. Choi, S. G. Jeon, S.-Y. Oh, Y. S. Gho, Z. Zhu and Y.-K. Kim, Vascular endothelial growth factor is a key mediator in the development of T cell priming and its polarization to type 1 and type 17 T helper cells in the airways, J. Immunol., 183 (2009), 5113-5120. doi: 10.4049/jimmunol.0901566. Google Scholar
[41]	T. A. Krouskop, T. M. Wheeler, F. Kallel, B. S. Garra and T. Hall, Elastic moduli of breast and prostate tissues under compression, Ultrason. Imaging, 20 (1998), 260-274. doi: 10.1177/016173469802000403. Google Scholar
[42]	Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993. Google Scholar
[43]	C. L. Langrish, Y. Chen, W. M. Blumenschein, J. Mattson, B. Basham, J. D. Sedgwick, T. McClanahan, R. A. Kastelein and D. J. Cua, IL-23 drives a pathogenic T cell population that induces autoimmune inflammation, J. Exp. Med., 201 (2005), 233-240. doi: 10.1084/jem.20041257. Google Scholar
[44]	S. Lee, H. 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 (2014), 1045-1080. doi: 10.1007/s11538-014-9946-6. Google Scholar
[45]	Y. K. Lee, H. Turner, C. L. Maynard, J. R. Oliver, D. Chen, C. O. Elson and C. T. Weaver, Late developmental plasticity in the T helper 17 lineage, Immunity, 30 (2009), 92-107. doi: 10.1016/j.immuni.2008.11.005. Google Scholar
[46]	C. M. Lloyd and C. M. Hawrylowicz, Regulatory T cells in asthma, Immunity, 31 (2009), 438-449. doi: 10.1016/j.immuni.2009.08.007. Google Scholar
[47]	M. S. Maddur, P. Miossec, S. V. Kaveri and J. Bayry, Th17 cells: Biology, pathogenesis of autoimmune and inflammatory diseases, and therapeutic strategies, Am. J. Pathol., 181 (2012), 8-18. doi: 10.1016/j.ajpath.2012.03.044. Google Scholar
[48]	A. O. Magnan, L. G. Mély, C. A. Camilla, M. M. Badier, F. A. Montero-Julian, C. M. Guillot, B. B. Casano, S. J. Prato, V. Fert, P. Bongrand and D. Vervloet, Assessment of the Th1/Th2 paradigm in whole blood in atopy and asthma: Increased IFN-$\gamma$-producing CD8(+) T cells in asthma, Am. J. Respir. Crit. Care Med., 161 (2000), 1790-1796. doi: 10.1164/ajrccm.161.6.9906130. Google Scholar
[49]	S. Marino, I. Hogue, C. Ray and D. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar
[50]	O. Michel, R. Ginanni, J. Duchateau, F. Vertongen, B. Bon and R. Sergysels, Domestic endotoxin exposure and clinical severity of asthma, Clin. Exp. Allergy, 21 (1991), 441-448. doi: 10.1111/j.1365-2222.1991.tb01684.x. Google Scholar
[51]	H.-G. Moon, Y.-M. Tae, Y.-S. Kim, S. G. Jeon, S.-Y. Oh, Y. S. Gho, Z. Zhu and Y.-K. Kim, Conversion of Th17-type into Th2-type inflammation by acetyl salicylic acid via the adenosine and uric acid pathway in the lung, Allergy, 65 (2010), 1093-1103. doi: 10.1111/j.1398-9995.2010.02352.x. Google Scholar
[52]	B. F. Morel, J. Kalagnanam and P. A. Morel, Mathematical modeling of Th1-Th2 dynamics, in Theoretical and Experimental Insights into Immunology (eds. A. S. Perelson and G. Weisbuch), vol. 66 of Nato ASI Subseries H:, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 1992, 171-190. doi: 10.1007/978-3-642-76977-1_11. Google Scholar
[53]	T. R. Mosmann and S. Sad, The expanding universe of T-cell subsets: Th1, Th2 and more, Immunol. Today, 17 (1996), 138-146. doi: 10.1016/0167-5699(96)80606-2. Google Scholar
[54]	T. R. Mosmann, H. Cherwinski, M. W. Bond, M. A. Giedlin and R. L. Coffman, Two types of murine helper T cell clone. I. definition according to profiles of lymphokine activities and secreted proteins, J. Immunol., 136 (1986), 2348-2357. Google Scholar
[55]	T. R. Mosmann and R. L. Coffman, TH1 and TH2 cells: Different patterns of lymphokine secretion lead to different functional properties, Annu. Rev. Immunol., 7 (1989), 145-173. doi: 10.1146/annurev.iy.07.040189.001045. Google Scholar
[56]	E. Muraille, O. Leo and M. Kaufman, The role of antigen presentation in the regulation of class-specific (Th1/Th2) immune responses, J. Biol. Syst., 3 (1995), 397-408. doi: 10.1142/S021833909500037X. Google Scholar
[57]	K. M. Murphy, P. Travers and M. Walport, Janeway's Immunobiology, 7th edition, Garland Science, 2007. Google Scholar
[58]	T. Nakagiri, M. Inoue, M. Minami, Y. Shintani and M. Okumura, Immunology mini-review: the basics of $T_H$17 and interleukin-6 in transplantation, Transplant. Proc., 44 (2012), 1035-1040. Google Scholar
[59]	M. F. Neurath, S. Finotto and L. H. Glimcher, The role of Th1/Th2 polarization in mucosal immunity, Nat. Med., 8 (2002), 567-573. doi: 10.1038/nm0602-567. Google Scholar
[60]	K. Oh, M. W. Seo, G. Y. Lee, O.-J. Byoun, H.-R. Kang, S.-H. Cho and D.-S. Lee, Airway epithelial cells initiate the allergen response through transglutaminase 2 by inducing IL-33 expression and a subsequent Th2 response, Respir. Res., 14 (2013), 35-43. doi: 10.1186/1465-9921-14-35. Google Scholar
[61]	M. J. Paszek and V. M. Weaver, The tension mounts: mechanics meets morphogenesis and malignancy, J. Mammary Gland Biol. Neoplasia, 9 (2004), 325-342. doi: 10.1007/s10911-004-1404-x. Google Scholar
[62]	A. Ray, A. Khare, N. Krishnamoorthy, Z. Qi and P. Ray, Regulatory T cells in many flavors control asthma, Mucosal Immunol., 3 (2010), 216-229. doi: 10.1038/mi.2010.4. Google Scholar
[63]	J. Richter, G. Metzner and U. Behn, Mathematical modelling of venom immunotherapy, J. Theor. Med., 4 (2002), 119-132. doi: 10.1080/10273660290022172. Google Scholar
[64]	D. S. Robinson, Regulatory T cells and asthma, Clin. Exp. Allergy, 39 (2009), 1314-1323. doi: 10.1111/j.1365-2222.2009.03301.x. Google Scholar
[65]	S. Romagnani, Atopic allergy and other hypersensitivities interactions between genetic susceptibility, innocuous and/or microbial antigens and the immune system, Curr. Opin. Immunol., 9 (1997), 773-775. doi: 10.1016/S0952-7915(97)80176-8. Google Scholar
[66]	S. Sakaguchi, Regulatory T cells: Key controllers of immunologic self-tolerance, Cell, 101 (2000), 455-458. Google Scholar
[67]	R. A. Seder and W. E. Paul, Acquisition of lymphokine-producing phenotype by CD4$^+$ T cells, Annu. Rev. Immunol., 12 (1994), 635-673. Google Scholar
[68]	R. Vogel and U. Behn, Th1-Th2 regulation and allergy: Bifurcation analysis of the non-autonomous system, in Mathematical Modeling of Biological Systems, Volume II (eds. A. Deutsch, R. Parra, R. J. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky and H. Metz), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2008, 145-155. Google Scholar
[69]	Y. Y. Wan, Multi-tasking of helper T cells, Immunology, 130 (2010), 166-171. doi: 10.1111/j.1365-2567.2010.03289.x. Google Scholar
[70]	M. Wills-Karp, J. Luyimbazi, X. Xu, B. Schofield, T. Y. Neben, C. L. Karp and D. D. Donaldson, Interleukin-13: Central mediator of allergic asthma, Science, 282 (1998), 2258-2261. doi: 10.1126/science.282.5397.2258. Google Scholar
[71]	M. Wills-Karp, J. Santeliz and C. L. Karp, The germless theory of allergic disease: Revisiting the hygiene hypothesis, Nat. Rev. Immunol., 1 (2001), 69-75. doi: 10.1038/35095579. Google Scholar
[72]	Y. Yang, H.-L. Zhang and J. Wu, Role of T regulatory cells in the pathogenesis of asthma, Chest, 138 (2010), 1282-1283. doi: 10.1378/chest.10-1440. Google Scholar
[73]	A. Yates, C. Bergmann, J. L. Van Hemmen, J. Stark and R. Callard, Cytokine-modulated regulation of helper T cell populations, J. Theor. Biol., 206 (2000), 539-560. doi: 10.1006/jtbi.2000.2147. Google Scholar
[74]	A. Yates, R. Callard and J. Stark, Combining cytokine signalling with T-bet and GATA-3 regulation in Th1 and Th2 differentiation: A model for cellular decision-making, J. Theor. Biol., 231 (2004), 181-196. doi: 10.1016/j.jtbi.2004.06.013. Google Scholar
[75]	M. Yazdanbakhsh, P. G. Kremsner and R. van Ree, Allergy, parasites, and the hygiene hypothesis, Science, 296 (2002), 490-494. doi: 10.1126/science.296.5567.490. Google Scholar
[76]	Y. Zhao, J. Yang, Y. dong Gao and W. Guo, Th17 immunity in patients with allergic asthma, Int. Arch. Allergy Immunol., 151 (2010), 297-307. doi: 10.1159/000250438. Google Scholar
[77]	L. Zhou, I. I. Ivanov, R. Spolski, R. Min, K. Shenderov, T. Egawa, D. E. Levy, W. J. Leonard and D. R. Littman, IL-6 programs $T_H$-17 cell differentiation by promoting sequential engagement of the IL-21 and IL-23 pathways, Nat. Immunol., 8 (2007), 967-974. Google Scholar

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References:

[1]	S. Al-Muhsen, S. Letuve, A. Vazquez-Tello, M. A. Pureza, H. Al-Jahdali, A. S. Bahammam, Q. Hamid and R. Halwani, Th17 cytokines induce pro-fibrotic cytokines release from human eosinophils, Respir. Res., 14 (2013), p34. doi: 10.1186/1465-9921-14-34. Google Scholar
[2]	T. Alarcón, H. M. Byrne and P. K. Maini, Towards whole-organ modelling of tumour growth, Prog. Biophys. Mol. Biol., 85 (2004), 451-472. Google Scholar
[3]	J. F. Alcorn, C. R. Crowe and J. K. Kolls, $T_H$17 cells in asthma and COPD, Annu. Rev. Physiol., 72 (2010), 495-516. Google Scholar
[4]	O. Arino, M. L. Hbid and E. Ait Dads, Delay Differential Equations and Applications, Springer Netherlands, 2006. doi: 10.1007/1-4020-3647-7. Google Scholar
[5]	K. J. Baek, J. Y. Cho, P. Rosenthal, L. E. C. Alexander, V. Nizet and D. H. Broide, Hypoxia potentiates allergen induction of HIF-1$\alpha$, chemokines, airway inflammation, TGF-$\beta$1, and airway remodeling in a mouse model, Clin. Immunol., 147 (2013), 27-37. Google Scholar
[6]	R. L. Bar-Or and L. A. Segel, On the role of a possible dialogue between cytokine and TCR-presentation mechanisms in the regulation of autoimmune disease, J. Theor. Biol., 190 (1998), 161-178. doi: 10.1006/jtbi.1997.0545. Google Scholar
[7]	U. Behn, H. Dambeck and G. Metzner, Modeling Th1-Th2 regulation, allergy, and hyposensitization, in Dynamical Modeling in Biotechnology, World Scientific, 2001, chapter 11, 227-243. doi: 10.1142/9789812813053_0011. Google Scholar
[8]	B. S. Bochner, B. J. Undem and L. M. Lichtenstein, Immunological aspects of allergic asthma, Annu. Rev. Immunol., 12 (1994), 295-335. doi: 10.1146/annurev.iy.12.040194.001455. Google Scholar
[9]	R. E. Callard and A. J. Yates, Immunology and mathematics: Crossing the divide, Immunology, 115 (2005), 21-33. doi: 10.1111/j.1365-2567.2005.02142.x. Google Scholar
[10]	J. Carneiro, J. Stewart, A. Coutinho and G. Coutinho, The ontogeny of class-regulation of CD4$^+$ T lymphocyte populations, Int. Immunol., 7 (1995), 1265-1277. doi: 10.1093/intimm/7.8.1265. Google Scholar
[11]	C. Clemedson and A. Nelson, General biology: The adult organism, in Mechanisms in Radiobiology: Multicellular Organisms (eds. M. Errera and A. Forssberg), Elsevier, 1960, 95-205. doi: 10.1016/B978-1-4832-2829-7.50010-1. Google Scholar
[12]	L. Cosmi, F. Liotta, E. Maggi, S. Romagnani and F. Annunziato, Th17 cells: New players in asthma pathogenesis, Allergy, 66 (2011), 989-998. doi: 10.1111/j.1398-9995.2011.02576.x. Google Scholar
[13]	E. Cutz, H. Levison and D. M. Cooper, Ultrastructure of airways in children with asthma, Histopathology, 2 (1978), 407-421. doi: 10.1111/j.1365-2559.1978.tb01735.x. Google Scholar
[14]	C. Dong, Diversification of T-helper-cell lineages: Finding the family root of IL-17-producing cells, Nat. Rev. Immunol., 6 (2006), 329-334. doi: 10.1038/nri1807. Google Scholar
[15]	C. Dong, $T_H$17 cells in development: An updated view of their molecular identity and genetic programming, Nat. Rev. Immunol., 8 (2008), 337-348. Google Scholar
[16]	S. C. Eisenbarth, D. A. Piggott, J. W. Huleatt, I. Visintin, C. A. Herrick and K. Bottomly, Lipopolysaccharide-enhanced, toll-like receptor 4-dependent T helper cell type 2 responses to inhaled antigen, J. Exp. Med., 196 (2002), 1645-1651. doi: 10.1084/jem.20021340. Google Scholar
[17]	R. L. Elliott and G. C. Blobe, Role of transforming growth factor beta in human cancer, J. Clin. Oncol., 23 (2005), 2078-2093. Google Scholar
[18]	M. A. Fishman and A. S. Perelson, Th1/Th2 differentiation and cross-regulation, Bull. Math. Biol., 61 (1999), 403-436. doi: 10.1006/bulm.1998.0074. Google Scholar
[19]	J. E. Gereda, D. Y. M. Leung, A. Thatayatikom, J. E. Streib, M. R. Price, M. D. Klinnert and A. H. Liu, Relation between house-dust endotoxin exposure, type 1 T-cell development, and allergen sensitisation in infants at high risk of asthma, Lancet, 355 (2000), 1680-1683. doi: 10.1016/S0140-6736(00)02239-X. Google Scholar
[20]	L. Gorelik, S. Constant and R. A. Flavell, Mechanism of transforming growth factor $\beta$-induced inhibition of T helper type 1 differentiation, J. Exp. Med., 195 (2002), 1499-1505. Google Scholar
[21]	L. Gorelik and R. A. Flavell, Abrogation of TGF$\beta$ signaling in T cells leads to spontaneous T cell differentiation and autoimmune disease, Immunity, 12 (2000), 171-181. Google Scholar
[22]	F. Gross, G. Metznerb and U. Behn, Mathematical modelling of allergy and specific immunotherapy: Th1-Th2-Treg interactions, J. Theor. Biol., 269 (2011), 70-78. doi: 10.1016/j.jtbi.2010.10.013. Google Scholar
[23]	G. Grünig, M. Warnock, A. E. Wakil, R. Venkayya, F. Brombacher, D. M. Rennick, D. Sheppard, M. Mohrs, D. D. Donaldson, R. M. Locksley and D. B. Corry, Requirement for IL-13 independently of IL-4 in experimental asthma, Science, 282 (1998), 2261-2263. Google Scholar
[24]	J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, 1st edition, Springer-Verlag New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar
[25]	I. Gutcher and B. Becher, APC-derived cytokines and T cell polarization in autoimmune inflammation, J. Clin. Invest., 117 (2007), 1119-1127. doi: 10.1172/JCI31720. Google Scholar
[26]	J. K. Hale, Theory of Functional Differential Equations, vol. 3 of Applied Mathematical Sciences, Springer-Verlag New York, 1977. Google Scholar
[27]	Q. Hamid and M. Tulic, Immunobiology of asthma, Annu. Rev. Physiol., 71 (2009), 489-507. doi: 10.1146/annurev.physiol.010908.163200. Google Scholar
[28]	L. E. Harrington, R. D. Hatton, P. R. Mangan, H. Turner, T. L. Murphy, K. M. Murphy and C. T. Weaver, Interleukin 17-producing $CD4^+$ effector T cells develop via a lineage distinct from the T helper type 1 and 2 lineages, Nat. Immunol., 6 (2005), 1123-1132. doi: 10.1038/ni1254. Google Scholar
[29]	L. E. Harrington, P. R. Mangan and C. T. Weaver, Expanding the effector CD4 T-cell repertoire: The Th17 lineage, Curr. Opin. Immunol., 18 (2006), 349-356. doi: 10.1016/j.coi.2006.03.017. Google Scholar
[30]	B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1981. Google Scholar
[31]	N. A. Hosken, K. Shibuya, A. W. Heath, K. M. Murphy and A. O'Garra, The effect of antigen dose on CD4$^+$ T helper cell phenotype development in a T cell receptor-$\alpha\beta$-transgenic model, J. Exp. Med., 182 (1995), 1579-1584. doi: 10.1084/jem.182.5.1579. Google Scholar
[32]	H. Jiang and L. Chess, An integrated view of suppressor T cell subsets in immunoregulation, J. Clin. Invest., 114 (2004), 1198-1208. doi: 10.1172/JCI23411. Google Scholar
[33]	Y. Kim, H. Lee, N. Dmitrieva, J. Kim, B. Kaur and A. Friedman, Choindroitinase ABC I-mediated enhancement of oncolytic virus spread and anti-tumor efficacy: A mathematical model, PLoS One, 9 (2014), e102499. doi: 10.1371/journal.pone.0102499. Google Scholar
[34]	Y. Kim, S. Lee, Y. Kim, Y. Kim, Y. Gho, H. Hwang and S. Lawler, Regulation of Th1/Th2 cells in asthma development: A mathematical model, Math. Bios. Eng, 10 (2013), 1095-1133. doi: 10.3934/mbe.2013.10.1095. Google Scholar
[35]	Y. Kim and H. Othmer, A hybrid model of tumor-stromal interactions in breast cancer, Bull Math Biol, 75 (2013), 1304-1350. doi: 10.1007/s11538-012-9787-0. Google Scholar
[36]	Y. Kim and S. Roh, A hybrid model for cell proliferation and migration in glioblastoma, Discrete and Continuous Dynamical Systems-B, 18 (2013), 969-1015. doi: 10.3934/dcdsb.2013.18.969. Google Scholar
[37]	Y. Kim, M. Stolarska and H. G. Othmer, A hybrid model for tumor spheroid growth in vitro I: Theoretical development and early results, Math. Models Methods Appl. Sci., 17 (2007), 1773-1798. doi: 10.1142/S0218202507002479. Google Scholar
[38]	Y. Kim, M. Stolarska and H. Othmer, The role of the microenvironment in tumor growth and invasion, Prog Biophys Mol Biol, 106 (2011), 353-379. doi: 10.1016/j.pbiomolbio.2011.06.006. Google Scholar
[39]	Y.-K. Kim, S.-Y. Oh, S. G. Jeon, H.-W. Park, S.-Y. Lee, E.-Y. Chun, B. Bang, H.-S. Lee, M.-H. Oh, Y.-S. Kim, J.-H. Kim, Y. S. Gho, S.-H. Cho, K.-U. Min, Y.-Y. Kim and Z. Zhu, Airway exposure levels of lipopolysaccharide determine type 1 versus type 2 experimental asthma, J. Immunol., 178 (2007), 5375-5382. doi: 10.4049/jimmunol.178.8.5375. Google Scholar
[40]	Y.-S. Kim, S.-W. Hong, J.-P. Choi, T.-S. Shin, H.-G. Moon, E.-J. Choi, S. G. Jeon, S.-Y. Oh, Y. S. Gho, Z. Zhu and Y.-K. Kim, Vascular endothelial growth factor is a key mediator in the development of T cell priming and its polarization to type 1 and type 17 T helper cells in the airways, J. Immunol., 183 (2009), 5113-5120. doi: 10.4049/jimmunol.0901566. Google Scholar
[41]	T. A. Krouskop, T. M. Wheeler, F. Kallel, B. S. Garra and T. Hall, Elastic moduli of breast and prostate tissues under compression, Ultrason. Imaging, 20 (1998), 260-274. doi: 10.1177/016173469802000403. Google Scholar
[42]	Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, 1993. Google Scholar
[43]	C. L. Langrish, Y. Chen, W. M. Blumenschein, J. Mattson, B. Basham, J. D. Sedgwick, T. McClanahan, R. A. Kastelein and D. J. Cua, IL-23 drives a pathogenic T cell population that induces autoimmune inflammation, J. Exp. Med., 201 (2005), 233-240. doi: 10.1084/jem.20041257. Google Scholar
[44]	S. Lee, H. 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 (2014), 1045-1080. doi: 10.1007/s11538-014-9946-6. Google Scholar
[45]	Y. K. Lee, H. Turner, C. L. Maynard, J. R. Oliver, D. Chen, C. O. Elson and C. T. Weaver, Late developmental plasticity in the T helper 17 lineage, Immunity, 30 (2009), 92-107. doi: 10.1016/j.immuni.2008.11.005. Google Scholar
[46]	C. M. Lloyd and C. M. Hawrylowicz, Regulatory T cells in asthma, Immunity, 31 (2009), 438-449. doi: 10.1016/j.immuni.2009.08.007. Google Scholar
[47]	M. S. Maddur, P. Miossec, S. V. Kaveri and J. Bayry, Th17 cells: Biology, pathogenesis of autoimmune and inflammatory diseases, and therapeutic strategies, Am. J. Pathol., 181 (2012), 8-18. doi: 10.1016/j.ajpath.2012.03.044. Google Scholar
[48]	A. O. Magnan, L. G. Mély, C. A. Camilla, M. M. Badier, F. A. Montero-Julian, C. M. Guillot, B. B. Casano, S. J. Prato, V. Fert, P. Bongrand and D. Vervloet, Assessment of the Th1/Th2 paradigm in whole blood in atopy and asthma: Increased IFN-$\gamma$-producing CD8(+) T cells in asthma, Am. J. Respir. Crit. Care Med., 161 (2000), 1790-1796. doi: 10.1164/ajrccm.161.6.9906130. Google Scholar
[49]	S. Marino, I. Hogue, C. Ray and D. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 254 (2008), 178-196. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar
[50]	O. Michel, R. Ginanni, J. Duchateau, F. Vertongen, B. Bon and R. Sergysels, Domestic endotoxin exposure and clinical severity of asthma, Clin. Exp. Allergy, 21 (1991), 441-448. doi: 10.1111/j.1365-2222.1991.tb01684.x. Google Scholar
[51]	H.-G. Moon, Y.-M. Tae, Y.-S. Kim, S. G. Jeon, S.-Y. Oh, Y. S. Gho, Z. Zhu and Y.-K. Kim, Conversion of Th17-type into Th2-type inflammation by acetyl salicylic acid via the adenosine and uric acid pathway in the lung, Allergy, 65 (2010), 1093-1103. doi: 10.1111/j.1398-9995.2010.02352.x. Google Scholar
[52]	B. F. Morel, J. Kalagnanam and P. A. Morel, Mathematical modeling of Th1-Th2 dynamics, in Theoretical and Experimental Insights into Immunology (eds. A. S. Perelson and G. Weisbuch), vol. 66 of Nato ASI Subseries H:, Springer-Verlag Berlin Heidelberg, Berlin, Germany, 1992, 171-190. doi: 10.1007/978-3-642-76977-1_11. Google Scholar
[53]	T. R. Mosmann and S. Sad, The expanding universe of T-cell subsets: Th1, Th2 and more, Immunol. Today, 17 (1996), 138-146. doi: 10.1016/0167-5699(96)80606-2. Google Scholar
[54]	T. R. Mosmann, H. Cherwinski, M. W. Bond, M. A. Giedlin and R. L. Coffman, Two types of murine helper T cell clone. I. definition according to profiles of lymphokine activities and secreted proteins, J. Immunol., 136 (1986), 2348-2357. Google Scholar
[55]	T. R. Mosmann and R. L. Coffman, TH1 and TH2 cells: Different patterns of lymphokine secretion lead to different functional properties, Annu. Rev. Immunol., 7 (1989), 145-173. doi: 10.1146/annurev.iy.07.040189.001045. Google Scholar
[56]	E. Muraille, O. Leo and M. Kaufman, The role of antigen presentation in the regulation of class-specific (Th1/Th2) immune responses, J. Biol. Syst., 3 (1995), 397-408. doi: 10.1142/S021833909500037X. Google Scholar
[57]	K. M. Murphy, P. Travers and M. Walport, Janeway's Immunobiology, 7th edition, Garland Science, 2007. Google Scholar
[58]	T. Nakagiri, M. Inoue, M. Minami, Y. Shintani and M. Okumura, Immunology mini-review: the basics of $T_H$17 and interleukin-6 in transplantation, Transplant. Proc., 44 (2012), 1035-1040. Google Scholar
[59]	M. F. Neurath, S. Finotto and L. H. Glimcher, The role of Th1/Th2 polarization in mucosal immunity, Nat. Med., 8 (2002), 567-573. doi: 10.1038/nm0602-567. Google Scholar
[60]	K. Oh, M. W. Seo, G. Y. Lee, O.-J. Byoun, H.-R. Kang, S.-H. Cho and D.-S. Lee, Airway epithelial cells initiate the allergen response through transglutaminase 2 by inducing IL-33 expression and a subsequent Th2 response, Respir. Res., 14 (2013), 35-43. doi: 10.1186/1465-9921-14-35. Google Scholar
[61]	M. J. Paszek and V. M. Weaver, The tension mounts: mechanics meets morphogenesis and malignancy, J. Mammary Gland Biol. Neoplasia, 9 (2004), 325-342. doi: 10.1007/s10911-004-1404-x. Google Scholar
[62]	A. Ray, A. Khare, N. Krishnamoorthy, Z. Qi and P. Ray, Regulatory T cells in many flavors control asthma, Mucosal Immunol., 3 (2010), 216-229. doi: 10.1038/mi.2010.4. Google Scholar
[63]	J. Richter, G. Metzner and U. Behn, Mathematical modelling of venom immunotherapy, J. Theor. Med., 4 (2002), 119-132. doi: 10.1080/10273660290022172. Google Scholar
[64]	D. S. Robinson, Regulatory T cells and asthma, Clin. Exp. Allergy, 39 (2009), 1314-1323. doi: 10.1111/j.1365-2222.2009.03301.x. Google Scholar
[65]	S. Romagnani, Atopic allergy and other hypersensitivities interactions between genetic susceptibility, innocuous and/or microbial antigens and the immune system, Curr. Opin. Immunol., 9 (1997), 773-775. doi: 10.1016/S0952-7915(97)80176-8. Google Scholar
[66]	S. Sakaguchi, Regulatory T cells: Key controllers of immunologic self-tolerance, Cell, 101 (2000), 455-458. Google Scholar
[67]	R. A. Seder and W. E. Paul, Acquisition of lymphokine-producing phenotype by CD4$^+$ T cells, Annu. Rev. Immunol., 12 (1994), 635-673. Google Scholar
[68]	R. Vogel and U. Behn, Th1-Th2 regulation and allergy: Bifurcation analysis of the non-autonomous system, in Mathematical Modeling of Biological Systems, Volume II (eds. A. Deutsch, R. Parra, R. J. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky and H. Metz), Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2008, 145-155. Google Scholar
[69]	Y. Y. Wan, Multi-tasking of helper T cells, Immunology, 130 (2010), 166-171. doi: 10.1111/j.1365-2567.2010.03289.x. Google Scholar
[70]	M. Wills-Karp, J. Luyimbazi, X. Xu, B. Schofield, T. Y. Neben, C. L. Karp and D. D. Donaldson, Interleukin-13: Central mediator of allergic asthma, Science, 282 (1998), 2258-2261. doi: 10.1126/science.282.5397.2258. Google Scholar
[71]	M. Wills-Karp, J. Santeliz and C. L. Karp, The germless theory of allergic disease: Revisiting the hygiene hypothesis, Nat. Rev. Immunol., 1 (2001), 69-75. doi: 10.1038/35095579. Google Scholar
[72]	Y. Yang, H.-L. Zhang and J. Wu, Role of T regulatory cells in the pathogenesis of asthma, Chest, 138 (2010), 1282-1283. doi: 10.1378/chest.10-1440. Google Scholar
[73]	A. Yates, C. Bergmann, J. L. Van Hemmen, J. Stark and R. Callard, Cytokine-modulated regulation of helper T cell populations, J. Theor. Biol., 206 (2000), 539-560. doi: 10.1006/jtbi.2000.2147. Google Scholar
[74]	A. Yates, R. Callard and J. Stark, Combining cytokine signalling with T-bet and GATA-3 regulation in Th1 and Th2 differentiation: A model for cellular decision-making, J. Theor. Biol., 231 (2004), 181-196. doi: 10.1016/j.jtbi.2004.06.013. Google Scholar
[75]	M. Yazdanbakhsh, P. G. Kremsner and R. van Ree, Allergy, parasites, and the hygiene hypothesis, Science, 296 (2002), 490-494. doi: 10.1126/science.296.5567.490. Google Scholar
[76]	Y. Zhao, J. Yang, Y. dong Gao and W. Guo, Th17 immunity in patients with allergic asthma, Int. Arch. Allergy Immunol., 151 (2010), 297-307. doi: 10.1159/000250438. Google Scholar
[77]	L. Zhou, I. I. Ivanov, R. Spolski, R. Min, K. Shenderov, T. Egawa, D. E. Levy, W. J. Leonard and D. R. Littman, IL-6 programs $T_H$-17 cell differentiation by promoting sequential engagement of the IL-21 and IL-23 pathways, Nat. Immunol., 8 (2007), 967-974. Google Scholar

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