Many autoimmune diseases exhibit an alternating pattern of relapses and remissions in which the apparent self-tolerance phase is interrupted by periodic autoimmune episodes. In this paper, we introduce a class of terminally differentiated effector T cells to an existing model of autoimmune disease and investigate the stability and Hopf branching phenomenon in a model of multiple sclerosis with a saturable functional response. First, we explore the local asymptotic stability of the equilibrium point and propose conditions for the existence of Hopf branching. Finally, with the help of canonical type theory and the central manifold theorem, we analyze the direction of Hopf branching and the stability of branching periodic solutions.
References
[1]
Kumar, B.V., Connors, T.J. and Farber, D.L. (2018) Human T Cell Development, Localization, and Function Throughout Life. Immunity, 48, 202-213. https://doi.org/10.1016/j.immuni.2018.01.007
[2]
Murphy, K. and Weaver, C. (2016) Janeway’s Immunobiology. Garland Science.
[3]
Janeway, C.A., Travers, P., Walport, M. and Shlomchik, M.J. (2005) Immunobiology: The Immune System in Health and Disease. 6th Edition, Garland.
[4]
Grigoriadis, N. and van Pesch, V. (2015) A Basic Overview of Multiple Sclerosis Immunopathology. EuropeanJournalofNeurology, 22, 3-13. https://doi.org/10.1111/ene.12798
[5]
Weatherley, G., Araujo, R.P., Dando, S.J. and Jenner, A.L. (2023) Could Mathematics Be the Key to Unlocking the Mysteries of Multiple Sclerosis? BulletinofMathematicalBiology, 85, Article No. 75. https://doi.org/10.1007/s11538-023-01181-0
[6]
Dendrou, C.A., Fugger, L. and Friese, M.A. (2015) Immunopathology of Multiple Sclerosis. NatureReviewsImmunology, 15, 545-558. https://doi.org/10.1038/nri3871
[7]
Polman, C.H., Reingold, S.C., Banwell, B., Clanet, M., Cohen, J.A., Filippi, M., et al. (2011) Diagnostic Criteria for Multiple Sclerosis: 2010 Revisions to the McDonald Criteria. AnnalsofNeurology, 69, 292-302. https://doi.org/10.1002/ana.22366
[8]
Alexander, H.K. and Wahl, L.M. (2010) Self-Tolerance and Autoimmunity in a Regulatory T Cell Model. BulletinofMathematicalBiology, 73, 33-71. https://doi.org/10.1007/s11538-010-9519-2
[9]
Zhang, W., Wahl, L.M. and Yu, P. (2014) Modeling and Analysis of Recurrent Autoimmune Disease. SIAMJournalonAppliedMathematics, 74, 1998-2025. https://doi.org/10.1137/140955823
[10]
Zhang, W., Wahl, L.M. and Yu, P. (2014) Viral Blips May Not Need a Trigger: How Transient Viremia Can Arise in Deterministic In-Host Models. SIAM Review, 56, 127-155. https://doi.org/10.1137/130937421
[11]
Jeschke, J.M., Kopp, M. and Tollrian, R. (2002) Predator Functional Responses: Discriminating between Handling and Digesting Prey. Ecological Monographs, 72, 95-112. https://doi.org/10.1890/0012-9615(2002)072[0095:pfrdbh]2.0.co;2
[12]
Dawes, J.H.P. and Souza, M.O. (2013) A Derivation of Holling’s Type I, II and III Functional Responses in Predator-Prey Systems. JournalofTheoreticalBiology, 327, 11-22. https://doi.org/10.1016/j.jtbi.2013.02.017
[13]
Yu, P. (2005) Closed-Form Conditions of Bifurcation Points for General Differential Equations. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 15, 1467-1483. https://doi.org/10.1142/s0218127405012582
[14]
Hinrichsen, D. and Pritchard, A.J. (2004) Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness. Springer.
[15]
Xie, J.H., Le, Y. and Li, D. (2018) Nonlinear Dynamics. Science Press.
[16]
Lucchinetti, C., Brück, W., Parisi, J., Scheithauer, B., Rodriguez, M. and Lassmann, H. (2000) Heterogeneity of Multiple Sclerosis Lesions: Implications for the Pathogenesis of Demyelination. AnnalsofNeurology, 47, 707-717. https://doi.org/10.1002/1531-8249(200006)47:6<707::aid-ana3>3.0.co;2-q
[17]
Roncador, G., Brown, P.J., Maestre, L., Hue, S., Martínez-Torrecuadrada, J.L., Ling, K., et al. (2005) Analysis of FOXP3 Protein Expression in Human CD4+ CD25+ Regulatory T Cells at the Single-Cell Level. European Journal of Immunology, 35, 1681-1691. https://doi.org/10.1002/eji.200526189
[18]
Vélez de Mendizábal, N., Carneiro, J., Solé, R.V., Goñi, J., Bragard, J., Martinez-Forero, I., et al. (2011) Modeling the Effector-Regulatory T Cell Cross-Regulation Reveals the Intrinsic Character of Relapses in Multiple Sclerosis. BMCSystemsBiology, 5, Article No. 114. https://doi.org/10.1186/1752-0509-5-114
[19]
Baecher‐Allan, C. and Hafler, D.A. (2006) Human Regulatory T Cells and Their Role in Autoimmune Disease. ImmunologicalReviews, 212, 203-216. https://doi.org/10.1111/j.0105-2896.2006.00417.x
[20]
Putnam, A.L., Brusko, T.M., Lee, M.R., Liu, W., Szot, G.L., Ghosh, T., et al. (2009) Expansion of Human Regulatory T-Cells from Patients with Type 1 Diabetes. Diabetes, 58, 652-662. https://doi.org/10.2337/db08-1168
