Abstract
A susceptible-infectious (SIS) epidemic model on heterogeneous dynamical networks with graph Laplacian diffusion is proposed to investigate the impact of population mobility on the asymptotic behavior of infectious diseases. First we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. Next the monotone iterative sequence approach is employed to show that the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity under the condition that the diffusion rates are equal. Finally some numerical simulations are carried out to demonstrate that solutions of the networked SIS epidemic model converge to a unique heterogeneous endemic equilibrium when the basic reproduction number is greater than unity. It is also shown that the social media has an impact on the magnitude of the endemic equilibrium state. We find that the graph Laplacian diffusion can turn a spatially homogeneous equilibrium into a spatially inhomogeneous one, while the classical Laplacian diffusion cannot change that under the same parameters.