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Spatial Modelling Using Reaction–Diffusion Systems

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Mathematical Immunology of Virus Infections
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Abstract

Mathematical immunology is dealing with increasingly complex models of immune phenomena formulated with ODEs or DDEs. Except for few studies, mathematical models of the immune response against virus infections conventionally consider the infected whole organism as a single homogenous compartment.

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Notes

  1. 1.

    Material of Sects. 6.1 and 6.2 uses the results of our studies from Bocharov et al., Spatiotemporal dynamics of virus infection spreading in tissues, PlosOne, 2016, 11(12):e0168576.

  2. 2.

    Material of Sect. 6.4 uses the results from Mathematical Modelling of Natural Phenomena, Vol. 6, Bocharov et al., Reaction-Diffusion Modelling of Interferon Distribution in Secondary Lymphoid Organs, Pages 13–26, Copyright \(\copyright \) 2011, with permission from EDP Sciences.

References

  1. G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, V. Volpert. Spatiotemporal dynamics of virus infection spreading in tissues. PlosOne, 2016 Dec 20;11(12):e0168576.

    Google Scholar 

  2. Bocharov G., Danilov A., Vassilevski Yu., Marchuk G.I., Chereshnev V.A., Ludewig B. (2011) Reaction-Diffusion Modelling of Interferon Distribution in Secondary Lymphoid Organs. Math. Model. Nat. Phenom., 6(7): 13–26.

    Google Scholar 

  3. Mikael Boulle, Thorsten G. Muller, Sabrina Dahling, Yashica Ganga, Laurelle Jackson, Deeqa Mahamed, Lance Oom, Gila Lustig, Richard A. Neher, Alex Sigal. HIV Cell-to-Cell Spread Results in Earlier Onset of Viral Gene Expression by Multiple Infections per Cell. PLOS Pathogens, https://doi.org/10.1371/journal.ppat.1005964 November 3, 2016.

  4. Walther Mothes, Nathan M. Sherer, Jing Jin, Peng Zhong. Virus Cell-to-Cell Transmission. Journal of Virology, Sept. 2010, Vol. 84, No. 17, 83608368.

    Google Scholar 

  5. A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull. Moscow Univ., Math. Mech., 1:6 (1937), 1–26. In: Selected Works of A.N. Kolmogorov, Vol. 1, V.M. Tikhomirov, Editor, Kluwer, London, 1991.

    Google Scholar 

  6. R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937), 355–369.

    Google Scholar 

  7. A. Volpert, Vit. Volpert, Vl. Volpert. Traveling wave solutions of parabolic systems. Translation of Mathematical Monographs, Vol. 140, Amer. Math. Society, Providence, 1994.

    Google Scholar 

  8. V. Volpert. Elliptic partial differential equations. Volume 2. Reaction-diffusion equations. Birkhäuser, 2014.

    Google Scholar 

  9. Ya. B. Zeldovich, D. A. Frank-Kamenetskii. A theory of thermal propagation of flame, Acta Physicochim. USSR 9 (1938), 341–350; Zhurnal Fizicheskoi Himii, 9 (1939), no. 12, 1530 (Russian)

    Google Scholar 

  10. Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, G. M. Makhviladze. The mathematical theory of combustion and explosion. Plenum Publishing Co., New York, 1985.

    Google Scholar 

  11. Fletcher CV, Staskus K, Wietgrefe SW, Rothenberger M, Reilly C, Chipman JG, Beilman GJ, Khoruts A, Thorkelson A, Schmidt TE, Anderson J, Perkey K, Stevenson M, Perelson AS, Douek DC, Haase AT, Schacker TW. Persistent HIV-1 replication is associated with lower antiretroviral drug concentrations in lymphatic tissues. Proc Natl Acad Sci U S A. 2014;111(6):2307–12

    Google Scholar 

  12. Rose R, Lamers SL, Nolan DJ, Maidji E, Faria NR, Pybus OG, Dollar JJ, Maruniak SA, McAvoy AC, Salemi M, Stoddart CA, Singer EJ, Mcgrath MS. 2016. HIV maintains an evolving and dispersed population in multiple tissues during suppressive combined antiretroviral therapy in individuals with cancer. J Virol 90:89848993. https://doi.org/10.1128/JVI.00684-16.

  13. Lorenzo-Redondo R, Fryer HR, Bedford T, Kim EY, Archer J, Pond SLK, Chung YS, Penugonda S, Chipman J, Fletcher CV, Schacker TW, Malim MH, Rambaut A, Haase AT, McLean AR, Wolinsky SM. Persistent HIV-1 replication maintains the tissue reservoir during therapy. Nature. 2016;530(7588):51–56. https://doi.org/10.1038/nature16933.

  14. G. Bocharov, R. Zust, L. Cervantes-Barragan, T. Luzyanina, E. Chiglintcev, V.A. Chereshnev, V. Thiel, B. Ludewig. A systems immunology approach to plasmacytoid dendritic cell function in cytopathic virus infections. PLoS Pathogens, 6(7) (2010), e1001017. https://doi.org/10.1371/journal.ppat.1001017, 1–14.

  15. A.A. Danilov. Unstructured tetrahedral mesh generation technology. Comput. Math. Math. Phys., 50 (2010), 146–163.

    Google Scholar 

  16. A.A. Danilov, Yu.V. Vassilevski. A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes. Russ. J. Numer. Anal. Math. Modelling, 24 (2009), 207–227.

    Google Scholar 

  17. T. Junt, E. Scandella, B. Ludewig. Form follows function: lymphoid tissues microarchitecture in antimicrobial immune defense. Nature Rev. Immunol., 8 (2008), 764–775.

    Google Scholar 

  18. J. Keener, J. Sneyd. Mathematical physiology. Springer-Verlag, New York, 1998.

    Google Scholar 

  19. T. Lammermann, M. Sixt. The microanatomy of T cell responses. Immunol. Reviews, 221 (2008), 26–43.

    Google Scholar 

  20. G.I. Marchuk. Methods of Numerical Mathematics. Springer-Verlag, New York, 1982.

    Google Scholar 

  21. F. Pfeiffer, V. Kumar, S. Butz, D. Vestweber, B.A. Imhof, J.V. Stein, B. Engelhardt. Distinct molecular composition of blood and lymphatic vascular endothelial cell junctions establishes specific functional barriers within the peripheral lymph node. Eur. J. Immunol., 38 (2008), 2142–2155.

    Google Scholar 

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Correspondence to Gennady Bocharov .

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Bocharov, G., Volpert, V., Ludewig, B., Meyerhans, A. (2018). Spatial Modelling Using Reaction–Diffusion Systems. In: Mathematical Immunology of Virus Infections. Springer, Cham. https://doi.org/10.1007/978-3-319-72317-4_6

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