Dynamical analysis of rumor spreading model with impulse vaccination and time delay

Rumor cause unnecessary conflicts and confusion by misleading the cognition of the public, its spreading has largely influence on human affairs. All kinds of rumors and people's suspicion are often caused by the lack of official information. Hence, the official should take a variety of channels to deny the rumors. The promotion of scientific knowledge is implemented to improve the quality of the whole nation, reduce the harm caused by rumor spreading. In this paper, regarding the process of the science education that official deny the rumor many times as periodic impulse, we propose a XWYZ rumor spreading model with impulse vaccination and time delay, and analyze the global dynamics behaviors of the model. By using the discrete dynamical system determined by the comparison theory and Floquet theorem, we show that there exists a rumor-free periodic solution. Further, we show that the rumor-free periodic solution is globally attractive under appropriate conditions. We also obtain a sufficient condition for the permanence of model. Finally, with the numerical simulation, our results indicate that large vaccination rate, short impulse period or long latent period is sufficient condition for the extinction of the rumors.