A Deterministic and Stochastic Fractional-Order ILSR Rumor Propagation Model Incorporating Media Reports and a Nonlinear Inhibition Mechanism

Nowadays, rumors spread more rapidly than before, leading to more panic and instability in society. Therefore, it is essential to seek out propagation law in order to prevent rumors from spreading further and avoid unnecessary harm. There is a connection between rumor models and symmetry. The consistency of a system or model is referred to as the level of symmetry under certain transformations. For this purpose, we propose a fractional-order Ignorant-Latent-Spreader-Remover (ILSR) rumor propagation model that incorporates media reports and a nonlinear inhibition mechanism. Firstly, the boundedness and non-negativeness of the solutions are derived under fractional differential equations. Secondly, the threshold is used to evaluate and illustrate the stability both locally and globally. Finally, by utilizing Pontryagin's maximum principle, we obtain the necessary conditions for the optimal control in the fractional-order rumor propagation model, and we also obtain the associated optimal solutions. Furthermore, the numerical results indicate that media reports can decrease the spread of rumors in different dynamic regions, but they cannot completely prevent rumor dissemination. The results are also exhibited and corroborated by replicating the model with specific hypothetical parameter values. It can be inferred that fractional order yields more favorable outcomes when rumor permanence in the population is higher. The presented method facilitates the acquisition of profound insights into the dissemination dynamics and subsequent consequences of rumors within a societal network.