Stability and spatial patterns of an epidemic-like rumor propagation model with diffusions

Social networks have recently become an effective and innovative channel for rumor propagation, and they influence not only people's living style but also people's thinking and values. Unlike previous articles that research rumor propagation by empirical approaches or ordinary differential equations, in this paper, we propose a delayed reaction-diffusion rumor propagation model based on partial differential equations. By taking the delay as the bifurcation parameter, the local stabilities of the equilibria points are investigated and the conditions of Hopf bifurcation are obtained. Numerical simulations are included to illustrate our theoretical results. Moreover, we successfully reveal that the delay leads to stable and unstable spiral patterns and the spatial diffusion can change the structure of spiral patterns by numerical simulations.