A weighted networked SIRS epidemic model

Abstract

Population mobility in an SIRS epidemic model is considered via graph Laplacian diffusion. We show the existence and uniqueness of solutions to the SIRS model defined on weighed graph. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1. By constructing Lyapunov function, we show that the endemic equilibrium is globally asymptotically stable for the model with the same diffusion rates if the basic reproduction number is greater than 1. With numerical simulations, we apply our generalized weighed graph to Watts-Strogatz network, where the degree of node is illustrated to determine the peak number of infectious population. It also indicates that the network has an impact on small-time behavior of epidemic transmission.