Stability and Hopf Bifurcation of a Delayed Fractional-Order ISR Rumor Propagation Model with Saturation Incidence

With the widespread adoption of internet technology globally, the propagation of rumors on social networks has attracted significant scholarly attention. This paper discusses a type of fractional-order rumor propagation model incorporating a saturation effect. In this model, it is assumed that the state of an ignorant individual will invariably change upon contact with a spreader, while the influence of time delays is also considered. Subsequently, we focus on analyzing the existence of a Hopf bifurcation arising from the time delay between ignorant individuals and spreaders, along with the dynamic behavior of the model. Stability analysis is conducted using methods such as the fractional-order stability criterion. Finally, a series of numerical simulations validate our findings and reveal that increasing the fractional order reduces the critical bifurcation value, thereby shrinking the stability region. Therefore, the stability region of the model can be regulated by adjusting the fractional order.