Lyapunov functions and global stability for a spatially diffusive SIR epidemic model

This paper deals with the problem of global asymptotic stability for equilibria of a spatially diffusive SIR epidemic model with homogeneous Neumann boundary condition. By discretizing the model with respect to the space variable, we first construct Lyapunov functions for the corresponding ODEs model, and then broaden the construction method into the PDEs model in which either susceptible or infective populations are diffusive. In both cases, we obtain the standard threshold dynamical behaviors, that is, if R-0 <= 1, then the disease-free equilibrium is globally asymptotically stable and if R-0 > 1, then the (strictly positive) endemic equilibrium is so. Numerical examples are given to illustrate the effectiveness of the theoretical results.