Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment
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 least one positive steady state that is bounded away from zero by a positive constant. When the environment factor \(c(x)\) is a constant \(c\), an explicit formula can be found for \({\mathcal R}_0\). Numerical simulation is performed to explore the impact of heterogeneous environment on the disease dynamics.
scientific article; zbMATH DE number 6099259