Nonlinear dynamic analysis and optimum control of reaction-diffusion rumor propagation models in both homogeneous and heterogeneous networks

Abstract The spread of rumors will make a big influence on people's life. This paper aims to study the dynamic behavior of rumor propagation. We consider the effects of transmission rates between different groups and establish 2ISR rumor propagation models with diffusions and time delay in both homogeneous and heterogeneous networks. In the homogeneous network model, we gain the basic reproduction number through the next generation matrix. The existence of two equilibrium points are discussed according to the basic reproduction number. Besides, based on the LaSalle's invariance principle, we obtain the global stability of two equilibrium points. In the heterogeneous network model, we further discuss the global stability of the rumor-free equilibrium point. Moreover, a set of useful rumor control strategies are put forward respectively in two systems. Finally, some necessary numerical simulations are displayed, which also suggest the effectiveness of theories.