Search Advanced

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5134-5146. doi: 10.3934/mbe.2020277
Previous Article Next Article
Research article Special Issues

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

  • College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
  • 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.


    加载中


    [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.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(3494) PDF downloads(190) Cited by(1)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog