Bifurcation Analysis of a Reaction–Diffusion Rumor Spreading Model with Nonsmooth Control

Rumors may cause serious harm to social productivity and life. Studying the dynamics of rumor propagation can help provide corresponding countermeasures to control communication effectively. Based on the classic [Formula: see text] infectious disease model, this paper studies a rumor spreading model with a time delay caused by communication delay between rumor communicators and nonsmooth threshold propagation function in reality. In this model, we solve non-negative equilibrium points firstly. By analyzing the characteristic equations of smooth and nonsmooth rumor systems, we establish local stability at non-negative equilibrium points and find the existence of Hopf bifurcation from two aspects of diffusion coefficient and delay, respectively. In addition, considering the discontinuity of rumor system, we separate it into two systems and get the condition of Hopf bifurcation when the system is discontinuous. We further verify the feasibility of the theoretical results by simulation results.