Louie K, Roberts M G, Wake G C
Department of Mathematics, Massey University, Palmerston North, New Zealand.
IMA J Math Appl Med Biol. 1993;10(3):207-26. doi: 10.1093/imammb/10.3.207.
A simple two-class (susceptibles and infectives) model describing the dynamics of a fatal disease in a variable-size population is presented and analysed. Spatial dependence is introduced into the model by considering two different mechanisms for the geographic spread of the disease: nonlocal interaction between susceptibles and infectives, and migratory spread of the animals. The steady states and their stability for these spatially dependent models are deduced; no spatially heterogeneous steady states were possible. For nonlocal interaction, there were two spatially uniform steady states: the trivial state (no infectives or susceptibles), which was unstable, and the endemic state (constant proportion of the population infected), which was locally asymptotically stable. With migratory spread, the number of spatially uniform steady states was dependent on the boundary conditions imposed. With hostile (Dirichlet) boundary conditions, only the trivial steady state was possible and its local stability found to depend on the rate of diffusion of the total population. With no-flux (Neumann) boundary conditions, the steady states are the trivial and endemic states; these were unstable and locally asymptotically stable, respectively.
提出并分析了一个简单的两类（易感者和感染者）模型，该模型描述了可变规模种群中一种致命疾病的动态变化。通过考虑疾病地理传播的两种不同机制，将空间依赖性引入模型：易感者与感染者之间的非局部相互作用以及动物的迁移传播。推导了这些具有空间依赖性模型的稳态及其稳定性；不存在空间异质稳态。对于非局部相互作用，存在两个空间均匀稳态：平凡态（无感染者或易感者），它是不稳定的；以及地方病态（感染个体占种群的恒定比例），它是局部渐近稳定的。对于迁移传播，空间均匀稳态的数量取决于所施加的边界条件。在敌对（狄利克雷）边界条件下，只可能存在平凡稳态，并且发现其局部稳定性取决于总人口的扩散速率。在无通量（诺伊曼）边界条件下，稳态是平凡态和地方病态；它们分别是不稳定的和局部渐近稳定的。
