In the age of information globalization, research on the mechanism of propagation will help mitigate the bad influence of rumors. Based on the classical rumor propagation model, this paper further analyzes the internal mechanism of the stochastic rumor propagation model incorporating media coverage with white noise. We investigate the existence of a unique global positive solution to the model and study the dynamic properties of the solutions around the rumor-free and local equilibrium points of the deterministic model. Furthermore, we establish sufficient conditions for the existence of traversal static distribution in the model. Numerical simulation shows that the role of media coverage is crucial to reduce the rumor propagation scale. The larger the coverage rate is, the smaller the rumor propagation scale is.
K E Y W O R D Sergodic process, media coverage, rumor propagation, stochastic process
With the development of information technology, rumors propagate faster and more widely than in the past. In this paper, a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise is proposed. The global positivity of the solution process is proved, and further the basic reproductive number R
0 is obtained. When R
0 < 1, the dynamical process of system with Lévy jump tends to the rumor-free equilibrium point of the deterministic system, and the rumor tends to extinction; when R
0 > 1, the rumor will keep spreading and the system will oscillate randomly near the rumor equilibrium point of the deterministic system. The results show that the oscillation amplitude is related to the disturbance of the system. In addition, increasing media coverage can effectively reduce the final spread of rumors. Finally, the above results are verified by numerical simulation.
The appearance of rumors intensifies people’s panic and affects social stability. How to control the spread of rumors has become an important issue which is worth studying. In order to more accurately reflect the actual situation in the real world, a stochastic model incorporating media coverage and Lévy noise is proposed to describe the dynamic process of rumor propagation. By introducing two control strategies of popular science education and media coverage in an emergency event, an near-optimal control problem that minimizes the influence and control cost of rumor propagation is proposed. Sufficient conditions for near-optimal control of the model are established by using a Hamiltonian function. Then the necessary conditions for near-optimal control are obtained by using the Pontryagin maximum principle. Finally, the effect of popular science education, media coverage and Lévy noise on rumor propagation process control is verified by numerical simulation.
The rapid development of the Internet has broadened the channels of information dissemination, which makes rumors spread faster, more widely and cause more negative impact. In this paper, an improved ISR-WV rumor propagation model integrating multi-channels is proposed by considering the system time delay, and the influence of different propagation channels on the dynamic process is further analyzed. Moreover, the basic reproduction number R
0, rumor-free equilibrium, and rumor-prevailing equilibrium, as well as their stability, are deduced. Then, an optimal control problem with pulse vaccination is designed. Finally, the validity of the model and theoretical results are verified by numerical simulations and a practical application. The results show that the rumor propagation threshold R
0 is more sensitive to the propagation rate of the information base channel. The shorter the thinking time τ
1 required for the ignorant to react after obtaining the information, the larger the final propagation scale, under this condition, the time delay τ
2 generated by the spreader producing the video is negatively related to the final propagation scale; conversely, a longer τ
1 implies that people tend to be calmer, which can suppress the spread of rumors, under this condition, τ
2 has little effect on the final propagation scale. In addition, the results also prove that timely implementation of the pulse vaccination control strategy of popular science education can effectively control the rumor propagation and reduce its negative impact.
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