Thresholds and stability analysis of models for the spatial spread of a fatal disease

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.