Research article Special Issues

Global boundedness and stability for a chemotaxis model of Boló’s concentric sclerosis

  • Received: 27 June 2020 Accepted: 20 July 2020 Published: 31 July 2020
  • Baló's concentric sclerosis (BCS) is considered a variant of inflammatory demyelinating disease closely related to multiple sclerosis characterized by a discrete concentrically layered lesion in the cerebal white matter. Khonsari and Calvez (Plos ONE. 2(2007)) proposed a parabolic-elliptic-ODE chemotaxis model for BCS which describes the evolution of the densities of activated macrophages, cytokine and apoptotic oligodendrocytes. Because "classically activated" M1 microglia can produce cytotoxicity, we introduce a linear production term from the activated microglia in the ODE for pro-inflammatory cytotoxic. For the new BCS chemotaxis model, we first investigate the uniform boundedness and global existence of classical solutions, and then get a range of the chemosensitive rate χ where the unique positive equilibrium point is exponentially asymptotically stable.

    Citation: Xiaoli Hu, Shengmao Fu. Global boundedness and stability for a chemotaxis model of Boló’s concentric sclerosis[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5134-5146. doi: 10.3934/mbe.2020277

    Related Papers:

  • Baló's concentric sclerosis (BCS) is considered a variant of inflammatory demyelinating disease closely related to multiple sclerosis characterized by a discrete concentrically layered lesion in the cerebal white matter. Khonsari and Calvez (Plos ONE. 2(2007)) proposed a parabolic-elliptic-ODE chemotaxis model for BCS which describes the evolution of the densities of activated macrophages, cytokine and apoptotic oligodendrocytes. Because "classically activated" M1 microglia can produce cytotoxicity, we introduce a linear production term from the activated microglia in the ODE for pro-inflammatory cytotoxic. For the new BCS chemotaxis model, we first investigate the uniform boundedness and global existence of classical solutions, and then get a range of the chemosensitive rate χ where the unique positive equilibrium point is exponentially asymptotically stable.


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    [1] O. Marburg, Die sogenannte akute multiple sklerose, J. Psychiatrie Neurol., 27 (1906), 211-312.
    [2] J. Baló, Encephaliyies periaxialis concentrica, Arch. Neur. Psych., 19 (1928), 242-264.
    [3] C. B. Courville, Concentric sclerosis, in Multiple Sclsrosis and Pther Demyelinating Dieases (eds. P. J. Vinken and G. W. Bruyn), Amsterdam: North Holland, (1970), 51-437.
    [4] Y. Kuroiwa, Concentric sclerosis, in Demyelinating Dieases (eds. J. C. Koetaier), Amsterdam: Elsevier Science Publishers, (1985), 17-409.
    [5] S. Christine, S. Ludwin, T. Tabira, A. Guseo, C. F. Lucchinetti, L. Leel-ssy, et al., Tissue preconditioning may explain concentric lesions in Balós type of multiple sclerosis, Brain, 128 (2005), 979-987.
    [6] R. H. Khonsari, V. Calvez, The origins of concentric demyelination: Self-organization in the human brain, Plos One, 2 (2007), e150.
    [7] V. Calveza, R. H. Khonsarib, Mathematical description of concentric demyelination in the human brain: Self-organization models from Liesegang rings to chemotaxis, Math. Comput. Modell., 47 (2008), 726-742.
    [8] L. Peferoen, D. Vogel, K. Ummenthum, M. Breur, P. Heijnen, W. H. Gerritsen, et al., Activation status of human microglia is dependent on lesion formation stage and remyelination in multiple sclerosis, J. Neur. Exp. Neurol., 74 (2015), 48-63.
    [9] T. Hillen, K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
    [10] K. Painter, T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543.
    [11] Z. A. Wang, T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108.
    [12] P. Zheng, C. L. Mu, X. G. Hu, Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete. Contin. Dyn. Syst. Ser. A., 35 (2015), 2299-2323.
    [13] M. J. Ma, C. H. Ou, Z. A. Wang, Stationary solutions of a volume-filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math., 72 (2012), 740-766.
    [14] Y. Z. Han, Z. F. Li, J. C. Tao, M. J. Ma, Pattern formation for a volume-filling chemotaxia model with logistic growth, J. Math. Anal. Appl., 448 (2017), 885-907.
    [15] M. J. Ma, M. Y. Gao, R. Carretero-González, Pattern formtion for a two-dimensional reaction-diffusion model with chemotaxis, J. Math. Anal. Appl., 475 (2019), 1883-1909.
    [16] M. Burger, M. D. Francesco, Y. Dolak-Struss, The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion, SIAM J. Math. Anal., 38 (2006), 1288-1315.
    [17] H. J. Guo, S. I. Zheng, B. Liang, Asympotic behaviour of solutions to the Keller-Segel model for chemotaxis with prevention of overcrowding, Nonlinearity., 26 (2013), 405-416.
    [18] M. Winkler, K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
    [19] Z. A. Wang, M. Winkler, D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297.
    [20] X. L. Hu, S. M. Fu, S. B. Ai, Global asymptotic behavior of solutions for a parabolic-parabolic-ODE chemotaxis system modeling multiple sclerosis, J. Diff. Equ., 269 (2020), 6875-6898.
    [21] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Springer Berlin, New York, 1981.
    [22] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for the solutions of elliptic differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727.
    [23] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for the solutions of elliptic differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math., 17 (1964), 35-92.
    [24] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Diff. Equ., 248 (2010), 2889-2905.
    [25] X. L. Bai, M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
    [26] H. L. Jin, Y. J. Kim, Z. A. Wang, Boundedness, stabilization, and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 78 (2018), 1632-1657.
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