Complex dynamic behavior of a rumor propagation model with spatial-temporal diffusion terms

• A spatial-temporal rumor propagation model with discrete and non-local delays is regarded.
• Stability and Hopf bifurcation are proved in a strict mathematical way.
• Sensitivity analysis is applied to search active parameters.
• Hopf bifurcation causes an oscillation phenomenon in this model.
• Numerical simulations provide a new insight into rumor spreading in online social networks.

Rumor propagation as a typical form of social communication in online social networks has had a significant negative impact on a harmonious and stable society. With the rapid development of mobile communication equipments, traditional rumor propagation models, which depend on ordinary differential equations (ODE), may not be suitable for describing rumor propagation in an online social network. In this paper, based on reaction-diffusion equations, we propose a novel epidemic-like model with both discrete and nonlocal delays for investigating the spatial-temporal dynamics of rumor propagation. By analyzing the corresponding characteristic equations of this model, the local stability conditions of a boundary equilibrium point and a positive equilibrium point are established. By applying the linear approximation method of nonlinear systems, sufficient conditions are derived for the existence of Hopf bifurcation at the above two kinds of equilibrium points. Moreover, a sensitivity analysis method based on the density of spreading users is proposed, and then in theoretical and experimental aspect we identify some sensitive parameters in the process of rumor propagation. Finally, numerical simulations are performed to illustrate the theoretical results.