Dynamical analysis for a diffusive SVEIR epidemic model with nonlinear incidences

Abstract

AbstractIn 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$$R_{0}$$R0and the local basic reproduction number$${\overline{R}}_{0}(x)$$R¯0(x)are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between$$R_{0}$$R0and$${\overline{R}}_{0}(x)$$R¯0(x)as well as the asymptotic properties of$$R_{0}$$R0when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators$$L_{1}$$L1and$$L_{2}$$L2. 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$$R_{0}<1$$R0<1, while the disease is uniformly persistent when$$R_{0}>1$$R0>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$$R_{0}\le 1$$R0≤1, and the endemic equilibrium is globally asymptotically stable if$$R_{0}>1$$R0>1and an additional condition is satisfied.