Dynamic Analysis of a Reaction–Diffusion Rumor Propagation Model

Abstract

The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. Rumor propagation in social networks has brought new challenges to network security and social stability. This paper, based on partial differential equations (PDEs), proposes a new SIS rumor propagation model by considering the effect of the communication between the different rumor infected users on rumor propagation. The stabilities of a nonrumor equilibrium point and a rumor-spreading equilibrium point are discussed by linearization technique and the upper and lower solutions method, and the existence of a traveling wave solution is established by the cross-iteration scheme accompanied by the technique of upper and lower solutions and Schauder's fixed point theorem. Furthermore, we add the time delay to rumor propagation and deduce the conditions of Hopf bifurcation and stability switches for the rumor-spreading equilibrium point by taking the time delay as the bifurcation parameter. Finally, numerical simulations are performed to illustrate the theoretical results.

Dynamic analysis of a rumor propagation model with Lévy noise

“…Zhao and Zhu introduced the following model: where t >0, x ∈ D =(0, L ) with homogeneous Neumann boundary conditions and where S ( t , x ) and I ( t , x ) represent the densities of the rumor‐susceptible users and the rumor‐infected users with a distance of x at time t , respectively. is a diffusion term, being used to describe the impact of the mobility on the rate of change in the density of users with a distance of x at time t .…”

“…Therefore, the rumor propagation model is different from the usual epidemic model. About the meaning of positive parameters A , β , μ , η , and α , see Zhao and Zhu for more details.…”

“…Zanette first studied the dynamic behavior of rumor propagation in small‐world networks, and similar to disease models, he obtained the spreading threshold. Recently, based on the classical susceptible‐infected‐removed rumor propagation model, Zhao et al developed a new susceptible‐infected‐hibernator‐removed model by adding a direct link from ignorants to stiflers and a new group hibernator, also see Zhao and Zhu . Afassinou, through comparing the behavior of educated individuals and non‐educated individuals, analyzed the impact of education rate on the rumor spreading mechanism.…”

“…Model (1) describes the rumor propagation. Due to the rapid development and wide popularity of the Internet, the way to acquire information for people is being quietly changed.…”

“…Especially, in recent years, with the rise of the mobile communications equipment, such as smartphones and tablet computers, social network rumors have shown a surge in trend. Thus, it is worth studying the model (1). And we remarked that rumor was regarded as collective problem solving.…”