### Abstract

Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the L_{1} boundedness of the orbit. Then the precompactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number R_{0} is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if R_{0} > 1, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

Original language | English |
---|---|

Pages (from-to) | 525-541 |

Number of pages | 17 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 23 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1 2018 |

### Fingerprint

### Keywords

- Asymptotic behaviour
- Lyapunov functions
- Reaction-diffution equations
- Virus dynamics model

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

**Asymptotic behaviour of the solutions to a virus dynamics model with diffusion.** / Sasaki, Toru; Suzuki, Takashi.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems - Series B*, vol. 23, no. 2, pp. 525-541. https://doi.org/10.3934/dcdsb.2017206

}

TY - JOUR

T1 - Asymptotic behaviour of the solutions to a virus dynamics model with diffusion

AU - Sasaki, Toru

AU - Suzuki, Takashi

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the L1 boundedness of the orbit. Then the precompactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number R0 is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if R0 > 1, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

AB - Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the L1 boundedness of the orbit. Then the precompactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number R0 is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if R0 > 1, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

KW - Asymptotic behaviour

KW - Lyapunov functions

KW - Reaction-diffution equations

KW - Virus dynamics model

UR - http://www.scopus.com/inward/record.url?scp=85041042784&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041042784&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2017206

DO - 10.3934/dcdsb.2017206

M3 - Article

AN - SCOPUS:85041042784

VL - 23

SP - 525

EP - 541

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 2

ER -