Stability and Hopf Bifurcation Analyses of a Delayed Rumor Spreading Model and Its Discontinuous Control

Abstract

In this paper, a delayed rumor propagation model with a discontinuous control mechanism is investigated. First, we investigate the model’s existence and boundedness. Then, conditions for the existence of the rumor-endemic equilibrium are derived utilizing differential inclusions. We examine the local asymptotic stability and Hopf bifurcation of the rumor-endemic equilibrium when the discontinuous control function is differential at the rumor-endemic equilibrium, considering the delay as the bifurcating parameter. When the discontinuous control function is not differential at the rumor-endemic equilibrium, we investigate the global asymptotically stable rumor-free and rumor-endemic equilibria using the Lyapunov functional technique. We conclude by providing two examples to validate our theoretical predictions. It is demonstrated that delay is a critical system parameter that can result in both Hopf bifurcation and chaos.