American Institute of Mathematical Sciences

2016, 21(8): 2567-2585. doi: 10.3934/dcdsb.2016061

A reaction diffusion system modeling virus dynamics and CTLs response with chemotaxis

 1 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China 2 Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 3k7, Canada

Received  October 2015 Revised  December 2015 Published  September 2016

In this paper, we study the effect of chemotactic movement of CTLs on HIV-1 infection dynamics by a reaction diffusion system with chemotaxis. Choosing a typical chemosensitive function, we find that chemoattractive movement of CTLs due to HIV infection does not change stability of a positive steady state of the model, meaning that the stability/instability of the model remains the same as in the model without spatial effect. However, chemorepulsion movement of CTLs can destabilize the positive steady state as the strength of the chemotactic sensitivity increases. In this case, Turing instability occurs, which may result in Hopf bifurcation or steady state bifurcation, and spatial inhomogeneous pattern forms.
Citation: Xiulan Lai, Xingfu Zou. A reaction diffusion system modeling virus dynamics and CTLs response with chemotaxis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2567-2585. doi: 10.3934/dcdsb.2016061
References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). [2] H. Amann, Dynamcial theory of quasilinear parabolic equations III: Global existence,, Math. Z., 202 (1989), 219. doi: 10.1007/BF01215256. [3] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in: Schmeisser H.J., 133 (1993), 9. doi: 10.1007/978-3-663-11336-2_1. [4] D. M. Brainard et al, Migration of antigen-specific T cells away from CXCR4-binding Human Immunodeficiency Virus Type 1 gp120,, J. Virol., 78 (2004), 5184. [5] D. M. Brainard et al, Decreased CXCR$3^+$ CD8 T cells in Advanced Human Immunodeficiency Virus infection suggest that a homing defect contributes to cytotoxic T-lymphocyte dysfunction,, J. Virol., 81 (2007), 8439. [6] M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1. doi: 10.1016/j.mbs.2005.12.006. [7] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Advances in Appl. Math., 26 (2001), 280. doi: 10.1006/aama.2001.0721. [8] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. [9] Y. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population,, J. Math. Biol., 62 (2011), 543. doi: 10.1007/s00285-010-0346-8. [10] J. D. Murray, Mathematical Biology I: An Introduction, In: Interdisciplinary Applied Mathematics, Vol. 18,, Springer New York, (2002). [11] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, In: Interdisciplinary Applied Mathematics, Vol. 18,, Springer New York, (2003). [12] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Applied Mathematics Quarterly, 10 (2002), 501. [13] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). [14] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal dependent sensivity,, Math. Nachr., 283 (2010), 1664. doi: 10.1002/mana.200810838. [15] D. Wodarz, Killer Cell Dynamics: Mathematical and Computational Approaches to Immunology, Interdisciplinary Applied Mathematics, vol. 32,, Springer Berlin, (2007). doi: 10.1007/978-0-387-68733-9. [16] D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding,, Nonlinear Analysis, 59 (2004), 1293. doi: 10.1016/S0362-546X(04)00327-X. [17] D. Wrzosek, Volume filling effect in modeling chemotaxis,, Math. Model. Nat. Phenom., 5 (2010), 123. doi: 10.1051/mmnp/20105106. [18] F. Vianello, I. T. Olszak and M. C. Poznansky, Fugetaxis: Active movement of leukocytes away from a chemokinetic agent,, J. Mol. Med., 83 (2005), 752. doi: 10.1007/s00109-005-0675-z. [19] P. Yu, Closed-form conditions of bifurcation points for general differential equations,, Int. J. Bifurcation Chaos Appl. Sci. Eng., 15 (2005), 1467. doi: 10.1142/S0218127405012582.

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References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). [2] H. Amann, Dynamcial theory of quasilinear parabolic equations III: Global existence,, Math. Z., 202 (1989), 219. doi: 10.1007/BF01215256. [3] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in: Schmeisser H.J., 133 (1993), 9. doi: 10.1007/978-3-663-11336-2_1. [4] D. M. Brainard et al, Migration of antigen-specific T cells away from CXCR4-binding Human Immunodeficiency Virus Type 1 gp120,, J. Virol., 78 (2004), 5184. [5] D. M. Brainard et al, Decreased CXCR$3^+$ CD8 T cells in Advanced Human Immunodeficiency Virus infection suggest that a homing defect contributes to cytotoxic T-lymphocyte dysfunction,, J. Virol., 81 (2007), 8439. [6] M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1. doi: 10.1016/j.mbs.2005.12.006. [7] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Advances in Appl. Math., 26 (2001), 280. doi: 10.1006/aama.2001.0721. [8] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. [9] Y. Lou and X. Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population,, J. Math. Biol., 62 (2011), 543. doi: 10.1007/s00285-010-0346-8. [10] J. D. Murray, Mathematical Biology I: An Introduction, In: Interdisciplinary Applied Mathematics, Vol. 18,, Springer New York, (2002). [11] J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, In: Interdisciplinary Applied Mathematics, Vol. 18,, Springer New York, (2003). [12] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Applied Mathematics Quarterly, 10 (2002), 501. [13] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). [14] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal dependent sensivity,, Math. Nachr., 283 (2010), 1664. doi: 10.1002/mana.200810838. [15] D. Wodarz, Killer Cell Dynamics: Mathematical and Computational Approaches to Immunology, Interdisciplinary Applied Mathematics, vol. 32,, Springer Berlin, (2007). doi: 10.1007/978-0-387-68733-9. [16] D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding,, Nonlinear Analysis, 59 (2004), 1293. doi: 10.1016/S0362-546X(04)00327-X. [17] D. Wrzosek, Volume filling effect in modeling chemotaxis,, Math. Model. Nat. Phenom., 5 (2010), 123. doi: 10.1051/mmnp/20105106. [18] F. Vianello, I. T. Olszak and M. C. Poznansky, Fugetaxis: Active movement of leukocytes away from a chemokinetic agent,, J. Mol. Med., 83 (2005), 752. doi: 10.1007/s00109-005-0675-z. [19] P. Yu, Closed-form conditions of bifurcation points for general differential equations,, Int. J. Bifurcation Chaos Appl. Sci. Eng., 15 (2005), 1467. doi: 10.1142/S0218127405012582.
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