Stability Analysis of an SVEIR Rumor Spreading Model in a Complex Heterogeneous Social Network

Rumors are relentlessly pervasive in social and organizational settings, enduring through time. They grab attention, ignite emotions, compel participation, shape attitudes and behaviors, and are omnipresent. As electronic technology advances and network platforms become more diverse, there is growing interest in conducting comprehensive research on disseminating rumors. The dissemination of rumors within complex social networks presents significant challenges to the protection of social stability and the maintenance of public order. It is crucial to es-tablish models for the dynamics of rumor propagation and implement appropriate control measures accordingly. Many current studies either neglect the diversity of complex social networks or fail to consider the costs associated with implementing control measures. Driven by these concerns, our paper presents a rumor propagation model based on complex and heterogeneous networks that integrate different control measures while accounting for network diversity. We assess the stability and existence of both rumor-free and rumor-prevailing equilibrium solutions in our proposed model. Initially, we computed the basic reproduction number which determines the local and global stability of the rumor-free equilibrium. Using the Lyapunov function and the LaSalle invariance principle, we demonstrate the global stability of the rumor-prevailing equilibrium under certain conditions. In addition, we introduce three control strategies that aim to curb rumor propagation. Our objective is to minimize the spread of rumors. By applying Pontryagin's maximum principle, we derive optimal control strategies that ensure the complete eradication of a rumor for a particular amount of time at the lowest cost. Finally, we use some numerical examples to demonstrate the validity of the derived results. We numerically study the effects of three psychological behaviors in humans that significantly affect the spread of a rumor: the forgetting mechanism, the emotion mechanism, and the awareness of vigilance. We compare the numerical simulations of our proposed model with and without control and prove the effectiveness of our control strategies.