Dynamical analysis for a diffusive SVEIR epidemic model with nonlinear incidences

Abstract

In this article, we are concerned with a diffusive SVEIR epidemic model with nonlinear incidences. We first obtain the well-posedness of solutions for the model. Then, the basic reproduction number R_{0} and the local basic reproduction number {overline{R}}_{0}(x) are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between R_{0} and {overline{R}}_{0}(x) as well as the asymptotic properties of R_{0} when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators L_{1} and L_{2}. Using the theory of monotone dynamical systems and the persistence theory of dynamical systems, we show that the disease-free equilibrium is globally asymptotically stable when R_{0}<1, while the disease is uniformly persistent when R_{0}>1. Furthermore, in the spatially homogeneous case, by using the Lyapunov functions method and LaSalle’s invariance principle, we completely obtain that the disease-free equilibrium is globally asymptotically stable if R_{0}le 1, and the endemic equilibrium is globally asymptotically stable if R_{0}>1 and an additional condition is satisfied.