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

The work is devoted to the investigation of virus quasispecies 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 as 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.

Download Full-text

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

Mathematics ◽

10.3390/math8010117 ◽

2020 ◽

Vol 8 (1) ◽

pp. 117 ◽

Cited By ~ 2

Author(s):

Nikolai Bessonov ◽

Gennady Bocharov ◽

Andreas Meyerhans ◽

Vladimir Popov ◽

Vitaly Volpert

Keyword(s):

Viral Evolution ◽

Reaction Diffusion ◽

Periodic Wave ◽

Continuous Variable ◽

Host Cells ◽

Reaction Diffusion Equation ◽

Nonlocal Effects ◽

Reaction Diffusion Model ◽

Species Evolution ◽

Nonlocal Reaction

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.

Download Full-text

Existence and Dynamics of Strains in a Nonlocal Reaction-Diffusion Model of Viral Evolution

SIAM Journal on Applied Mathematics ◽

10.1137/19m1282234 ◽

2021 ◽

Vol 81 (1) ◽

pp. 107-128

Author(s):

Nikolai Bessonov ◽

Gennady Bocharov ◽

Andreas Meyerhans ◽

Vladimir Popov ◽

Vitaly Volpert

Keyword(s):

Diffusion Model ◽

Viral Evolution ◽

Reaction Diffusion ◽

Reaction Diffusion Model ◽

Nonlocal Reaction

Download Full-text

An Accelerating Numerical Computation of the Diffusion Term in a Nonlocal Reaction-Diffusion Equation

Mathematics ◽

10.3390/math8122111 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2111

Author(s):

Mitică CRAUS ◽

Silviu-Dumitru PAVĂL

Keyword(s):

Diffusion Equation ◽

Numerical Approximation ◽

Approximation Scheme ◽

Reaction Diffusion ◽

Reaction Diffusion Equation ◽

Diffusion Term ◽

Reaction Diffusion Model ◽

Computationally Intensive ◽

Nonlocal Reaction ◽

Nonlocal Formulation

In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integral formulation, and thus finding a better way of computing its numerical approximation could be of interest, given that the numerical analysis usually takes place on large input domains having more than one dimension. After introducing the general reaction-diffusion model, we discuss a numerical approximation scheme for the diffusion term, based on a finite difference method. In the next sections we propose two algorithms to solve the numerical approximation scheme, focusing on finding a way to improve the time performance. While the first algorithm (sequential) is used as a baseline for performance measurement, the second algorithm (parallel) is implemented using two different memory-sharing parallelization technologies: Open Multi-Processing (OpenMP) and CUDA. All the results were obtained by using the model in image processing applications such as image restoration and segmentation.

Download Full-text