  The aim of this paper is to study the dynamics of a reaction--diffusion SIS
(susceptible-infectious-susceptible) epidemic model with a nonlinear incidence
rate describing the transmission of a communicable disease between individuals.
We prove that the proposed model has two steady states under one condition. By
analyzing the eigenvalues and using the Routh--Hurwitz criterion and an
appropriately constructed Lyapunov functional, we establish the local and
global asymptotic stability of the non negative constant steady states subject
to the basic reproduction number being greater than unity and of the
disease--free equilibrium subject to the basic reproduction number being
smaller than or equal to unity in ODE case. By applying an appropriately
constructed Lyapunov functional, we identify the condition of the global
stability in the PDE case. Finally, we present some numerical examples
illustrating and confirming the analytical results obtained throughout the
paper.
