Global attractor for a SEIRS model with diffusion and nonlinear incidence rates

Abstract

We study the existence and asymptotic behavior of solutions to an epidemiological model of SEIRS type with diffusion and nonlinear incidence rates. We first prove the existence and uniqueness of nonnegative global weak solutions to the system by using the Galerkin approximate method and the quasi-positivity of the nonlinear term. Then, we prove the existence of a global attractor for the continuous semigroup generated by the problem. Some properties of the global attractor are investigated such as the upper bound for fractal dimension and the upper semicontinuity with respect to the rate constant for loss of immunity. We also give sufficient conditions for the existence, uniqueness and global exponential stability of nonnegative stationary solutions to the system, which particularly implies that the global attractor becomes a trivial one, i.e., it consists of only a unique nonnegative stationary solution.