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Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains

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Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, 199178 Saint Petersburg, Russia
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Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, 199333 Moscow, Russia
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Key Center of Excellence on Experimental immunophysiology and immunochemistry, Ural Federal University, 620002 Ekaterinburg, Russia
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ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain
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Infection Biology Laboratory, Universitat Pompeu Fabra, 08003 Barcelona, Spain
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Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, 117198 Moscow, Russia
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Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France
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INRIA Team Dracula, INRIA Lyon La Doua, 69603 Villeurbanne, France
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 117; https://doi.org/10.3390/math8010117
Received: 2 December 2019 / Revised: 8 January 2020 / Accepted: 9 January 2020 / Published: 12 January 2020
(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)
This work is devoted to the investigation of virus quasi-species evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction–diffusion equation for the virus density depending on the genotype considered to be a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described.
Keywords: virus density distribution; genotype; virus infection; immune response; resistance to treatment; nonlocal interaction; quasi-species diversification virus density distribution; genotype; virus infection; immune response; resistance to treatment; nonlocal interaction; quasi-species diversification
MDPI and ACS Style

Bessonov, N.; Bocharov, G.; Meyerhans, A.; Popov, V.; Volpert, V. Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains. Mathematics 2020, 8, 117.

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