Global stability of a network-based SIRS epidemic model with nonmonotone incidence rate

This paper studies the dynamics of a network-based SIRS epidemic model with vaccination and a nonmonotone incidence rate. This type of nonlinear incidence can be used to describe the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals on heterogeneous networks is getting larger. Using the analytical method, epidemic threshold $R_0$ is obtained. When $R_0$ is less than one, we prove the disease-free equilibrium is globally asymptotically stable and the disease dies out, while $R_0$ is greater than one, there exists a unique endemic equilibrium. By constructing a suitable Lyapunov function, we also prove the endemic equilibrium is globally asymptotically stable if the inhibitory factor $\alpha$ is sufficiently large. Numerical experiments are also given to support the theoretical results. It is shown both theoretically and numerically a larger $\alpha$ can accelerate the extinction of the disease and reduce the level of disease.