Dynamical analysis for a diffusive SVEIR epidemic model with nonlinear incidences

• Published in: Zeitschrift für angewandte Mathematik und Physik 2023-10, Vol.74 (5), Article 173
• Main Authors: Zhou, Pan, Wang, Jianpeng, Teng, Zhidong, Wang, Kai
• Format: Article
• Language: English
• Subjects: Asymptotic properties; Dynamical systems; Engineering; Epidemics; Equilibrium; Liapunov functions; Linear operators; Mathematical Methods in Physics; Theoretical and Applied Mechanics
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-023-02057-y

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 R ¯ 0 ( x ) are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between R 0 and 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 ≤ 1 , and the endemic equilibrium is globally asymptotically stable if R 0 > 1 and an additional condition is satisfied.