Lesion–immune feedback induces bistability and oscillatory dynamics in a reduced model of multiple sclerosis

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Abstract

Relapsing–remitting multiple sclerosis (RRMS) is characterized by recurrent immune activation, lesion development, and marked variability in disease activity. To investigate how lesion- associated activity may feed back on immune regulation, we extend an established effector T cell– regulatory T cell (Teff–Treg) model by introducing a normalized lesion-activity variable and a feedback factor that amplifies antigen-presentation-related immune activation. The resulting closed-loop system links autoreactive effector T-cell activity, lesion-associated activity, and subsequent immune reactivation through lesion–immune feedback. Positivity, global existence, and uniform boundedness hold in the feasible region. Positive equilibria satisfy α = Φ_κ(S ), with S the combined regulatory strength. The feedback-free branch is monotone, and $\mathcal{F}$ > 1 gives backward bending and at least two positive equilibria for α_{S N} < α < α_c, though their exact number and stability remain analytically undetermined. For the working parameters, branch reconstruction and full-Jacobian spectral checks reveal a nondegenerate fold and coexisting stable tolerant and active equilibria separated by a saddle; time integration shows initial-condition dependence, and a schematic branch-following construction illustrates hysteresis-like switching. Routh–Hurwitz conditions, together with conditional saddle- node and Hopf criteria, are derived. Numerical verification identifies a transverse imaginary-pair crossing and a negative first Lyapunov coefficient, consistent with a supercritical Hopf bifurcation, with simulations showing sustained coupled oscillations. The computed two-parameter boundaries concern equilibrium existence and local stability in the scanned regime, not a global classification of attractors or calibrated clinical relapse intervals.

Keywords

relapsing-remitting multiple sclerosis / lesion feedback / bistability / saddle-node bifurcation / Hopf bifurcation

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References

[1] A. Compston, A. Coles, Multiple sclerosis, Lancet, 372 (2008), 1502-1517. https://doi.org/10.1016/S0140-6736(08)61620-7

[2] H. Lassmann, Multiple sclerosis pathology, Cold Spring Harbor Perspect. Med., 8 (2018), a028936. https://doi.org/10.1101/cshperspect.a028936

[3] F. D. Lublin, S. C. Reingold, J. A. Cohen, G. R. Cutter, P. S. Sørensen, A. J. Thompson, et al., Defining the clinical course of multiple sclerosis: the 2013 revisions, Neurology, 83 (2014), 278-286. https://doi.org/10.1212/WNL.0000000000000560

[4] M. P. McGinley, C. H. Goldschmidt, A. D. Rae-Grant, Diagnosis and treatment of multiple sclerosis: a review, JAMA, 325 (2021), 765-779. https://doi.org/10.1001/jama.2020.26858

[5] T. Kuhlmann, M. Moccia, T. Coetzee, J. A. Cohen, J. Correale, J. Graves, et al., Multiple sclerosis progression: time for a new mechanism-driven framework, Lancet Neurol., 22 (2023), 78-88. https://doi.org/10.1016/S1474-4422(22)00289-7

[6] M. Bisi, M. Groppi, G. Martalò, R. Travaglini, Derivation from kinetic theory and 2-D pattern analysis of chemotaxis models for multiple sclerosis, J. Math. Biol., 91 (2025), 43. https://doi.org/10.1007/s00285-025-02282-1

[7] C. A. Dendrou, L. Fugger, M. A. Friese, Immunopathology of multiple sclerosis, Nat. Rev. Immunol., 15 (2015), 545-558. https://doi.org/10.1038/nri3871

[8] B. Hemmer, M. Kerschensteiner, T. Korn, Role of the innate and adaptive immune responses in the course of multiple sclerosis, Lancet Neurol., 14 (2015), 406-419. https://doi.org/10.1016/S1474-4422(14)70305-9

[9] M. Sospedra, R. Martin, Immunology of multiple sclerosis, Semin. Neurol., 36 (2016), 115-127. https://doi.org/10.1055/s-0036-1579739

[10] J. Goverman, Autoimmune T cell responses in the central nervous system, Nat. Rev. Immunol., 9 (2009), 393-407. https://doi.org/10.1038/nri2550

[11] S. Sakaguchi, T. Yamaguchi, T. Nomura, M. Ono, Regulatory T cells and immune tolerance, Cell, 133 (2008), 775-787. https://doi.org/10.1016/j.cell.2008.05.009

[12] D. A. A. Vignali, L. W. Collison, C. J. Workman, How regulatory T cells work, Nat. Rev. Immunol., 8 (2008), 523-532. https://doi.org/10.1038/nri2343

[13] M. T. Cencioni, M. Mattoscio, R. Magliozzi, A. Bar-Or, P. A. Muraro, B cells in multiple sclerosis-from targeted depletion to immune reconstitution therapies, Nat. Rev. Neurol., 17 (2021), 399-414. https://doi.org/10.1038/s41582-021-00498-5

[14] M. Absinta, D. Maric, M. Gharagozloo, T. Garton, M. D. Smith, J. Jin, et al., A lymphocyte-microglia-astrocyte axis in chronic active multiple sclerosis, Nature, 597 (2021), 709-714. https://doi.org/10.1038/s41586-021-03892-7

[15] P. Ostkamp, M. Deffner, A. Schulte-Mecklenbeck, C. Wünsch, I-N. Lu, G. F. Wu, et al., A single-cell analysis framework allows for characterization of CSF leukocytes and their tissue of origin in multiple sclerosis, Sci. Transl. Med., 14 (2022), eadc9778. https://doi.org/10.1126/scitranslmed.adc9778

[16] T. V. Lanz, R. C. Brewer, P. P. Ho, J. Moon, K. M. Jude, D. Fernandez, et al., Clonally expanded B cells in multiple sclerosis bind EBV EBNA1 and GlialCAM, Nature, 603 (2022), 321-327. https://doi.org/10.1038/s41586-022-04432-7

[17] H. K. Alexander, L. M. Wahl, Self-tolerance and autoimmunity in a regulatory T cell model, Bull. Math. Biol., 73 (2011), 33-71. https://doi.org/10.1007/s11538-010-9519-2

[18] W. Zhang, L. M. Wahl, P. Yu, Modeling and analysis of recurrent autoimmune disease, SIAM J. Appl. Math., 74 (2014), 1998-2025. https://doi.org/10.1137/140955823

[19] W. Zhang, P. Yu, Hopf and generalized Hopf bifurcations in a recurrent autoimmune disease model, Int. J. Bifurcation Chaos, 26 (2016), 1650079. https://doi.org/10.1142/S0218127416500796

[20] W. Zhang, P. Yu, Revealing the role of the effector-regulatory t cell loop on autoimmune disease symptoms via nonlinear analysis, Commun. Nonlinear Sci. Numer. Simul., 93 (2021), 105529. https://doi.org/10.1016/j.cnsns.2020.105529

[21] J. M. Oliveira, R. Travaglini, Reaction-diffusion systems derived from kinetic theory for multiple sclerosis, Math. Models Methods Appl. Sci., 34 (2024), 1279-1308. https://doi.org/10.1142/S0218202524500222

[22] Y. Lebel, T. Milo, A. Bar, A. Mayo, U. Alon, Excitable dynamics of flares and relapses in autoimmune diseases, iScience, 26 (2023), 108084. https://doi.org/10.1016/j.isci.2023.108084

[23] M. Menale, R. Travaglini, A nonconservative kinetic model under the action of an external force field for modeling the medical treatment of autoimmune response, Commun. Nonlinear Sci. Numer. Simul., 137 (2024), 108126. https://doi.org/10.1016/j.cnsns.2024.108126

[24] N. Moise, A. Friedman, A mathematical model of the multiple sclerosis plaque, J. Theor. Biol., 512 (2021), 110532. https://doi.org/10.1016/j.jtbi.2020.110532

[25] V. K. Tuohy, M. Yu, L. Yin, J. A. Kawczak, J. M. Johnson, P. M. Mathisen, et al., The epitope spreading cascade during progression of experimental autoimmune encephalomyelitis and multiple sclerosis, Immunol. Rev., 164 (1998), 93-100. https://doi.org/10.1111/j.1600-065X.1998.tb01211.x

[26] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory , Springer, New York, 2004. https://doi.org/10.1007/978-1-4757-3978-7

[27] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , Springer, New York, 1983. https://doi.org/10.1007/978-1-4612-1140-2

[28] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, et al., SciPy 1.0: fundamental algorithms for scientific computing in Python, Nat. Methods, 17 (2020), 261-272. https://doi.org/10.1038/s41592-019-0686-2