TITLE:
Optimal Control and Dynamic Analysis of a New Caputo Fractional Rumor Propagation Prediction Model
AUTHORS:
Lijie You, Sijie Yu
KEYWORDS:
Caputo Fractional Derivative, Basic Reproduction Number, Local Stability, Global Stability, Optimal Control
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.5,
May
29,
2026
ABSTRACT: In recent years, controlling the spread of rumors has emerged as an issue of significant public concern. Mathematical modeling provides a powerful tool for conducting quantitative analysis of this problem and can offer critical support for decision-making. This paper addresses the control of rumor propagation in online social networks by introducing a novel fractional-order rumor spreading model. The model categorizes the population into seven compartments and simultaneously incorporates three types of intervention measures-educational mechanisms, memory and forgetting mechanisms, and rumor-refutation mechanisms to better reflect real-world rumor mitigation scenarios. We first establish the existence, non-negativity, uniqueness, and boundedness of the solutions to the proposed model. Similar to epidemiological modeling, a basic reproduction number for rumor spread is defined, and the existence and stability of the model’s equilibrium points are analyzed. On this basis, an optimal control problem is formulated with the aim of balancing intervention effectiveness and implementation cost. By applying Pontryagin’s Maximum Principle, we derive the necessary conditions for optimal control and obtain the corresponding optimal strategy. Numerical simulations demonstrate that the synergistic use of the three intervention mechanisms significantly reduces the peak prevalence and final impact of rumors, outperforming any single intervention, which confirms the validity and practical utility of our model.