Optimal Control of Rumor Spreading Model on Homogeneous Social Network with Consideration of Influence Delay of Thinkers

The internet has emerged as a part of life for modern generations. We use social networks to interact with others instantly and make ourselves up to date. A bunch of information and misinformation propagate quickly through social networks. It is necessary to detect the correctness of the information and create a mechanism to control rumor; otherwise, it may be drastic to the whole world in emergencies. Here, we introduce an optimal control of rumor spreading in a homogeneously mixed population considering influence delay of thinkers. Firstly, we formulate the model and obtain possible steady states of the system with their local stability conditions. It is necessary for the extinction of the rumor that the control influence number should not exceed 1. We also observe that the system bifurcates and Hopf bifurcation occurs when the influence delay of thinkers is higher than its threshold value, which may become a reason for panic in emergencies. Secondly, we recognize the most sensitive parameters for the proposed model. Moreover, using Pontryagin's maximum principle and counter news attack mechanism, we design an optimal rumor control system to minimize the density of rumor adopters and control cost.