Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment
type: Item
instance of: scholarly article
title: Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment
zbMATH Open document ID: 1258.35033
author: Naveen K. Vaidya
author: Feng-Bin Wang
author: Xingfu Zou
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2012-10-26T00:00:00Z
review text: A reaction-diffusion model for the avian influenza dynamics in wild birds with bird mobility and heterogeneous environment is considered. Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain representing the habitat of a bird species which is a host for avian influenza virus. Let \(u_1,u_2,u_3\) be the populations of susceptible, infected, and recovered birds in \(\Omega\) respectively and let \(u_4\) be the virus concentration in water. Then the model can be written as \[ \begin{cases} \partial u_1/\partial t=D\Delta u_1+\lambda-\beta_Iu_1u_2-\beta_Vu_1u_4-du_1+\eta u_3, \\ \partial u_2/\partial t=D\Delta u_2+\beta_Iu_1u_2+\beta_Vu_1u_4-(\gamma+d)u_2, \\ \partial u_3/\partial t=D\Delta u_3+\gamma u_2-(\eta+d)u_3, \\ \partial u_4/\partial t=\alpha u_2-c(x)u_4, \\ \end{cases} \] for \((x,t)\in \Omega\times (0,\infty)\) with the homogeneous Neumann boundary condition \[ \partial u_i/\partial \nu=0, \;(x,t)\in \partial \Omega\times (0,\infty), \;i=1,2,3 \] and initial condition \[ u_i(x,0)=u_i^0(x), \;x\in \Omega, \;i=1,2,3,4. \] Here \(\beta_I\) and \(\beta_V\) represent the direct bird-to-bird transmission rate and the indirect fecal-to-oral transmission rate of the virus respectively. The parameters \(\lambda,d,\gamma,\eta\) represent the rates of recruitment of susceptible birds, of natural death, of recovery from infection, and of immunity loss, respectively. \(c(x)\) is the spatially varying viral decay rate due to environment effects such as temperature, pH and salinity. The lack of a diffusion term in the last equation is due to the fact that the diffusion of virus particles is negligible compared to the birds' mobility. The authors first show that the system has a unique solution. Then they show that a basic reproductive number \({\mathcal R}_0\) is a threshold index for disease persistence. That is, if \({\mathcal R}_0<1\), then the disease free equilibrium \((w^*,0,0,0)\) is globally attractive, while if \({\mathcal R}_0>1\), then the system admits at leas…
reviewed by: Yaping Liu
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 37N25
Mathematics Subject Classification ID: 92D30
Mathematics Subject Classification ID: 35K51
Mathematics Subject Classification ID: 35K58
zbMATH DE Number: 6099259
zbMATH Keywords: basic reproductive number
zbMATH Keywords: spectral radius
zbMATH Keywords: principal eigenvalues
zbMATH Keywords: threshold dynamics
zbMATH Keywords: homogeneous Neumann condition
zbMATH Keywords: disease persistence
MaRDI profile type: Publication
full work available at URL: dcdsb.2012.17.2829
OpenAlex ID: W2003750775
DOI: 10.3934/DCDSB.2012.17.2829
description: scientific article; zbMATH DE number 6099259