<p>With the advancement of society, rumor propagation has become increasingly complex and persistent due to individual heterogeneity and memory effects, posing serious challenges to social governance. To address this issue, a fractional-order SIR-C rumor propagation model is constructed by introducing a latent group and refining the classification of spreaders, thereby providing a more realistic representation of rumor dynamics. Within this framework, the basic reproduction number is obtained, and the existence as well as the local and global asymptotic stability of the rumor-free and endemic equilibria are rigorously analyzed. In addition, an optimal control problem is formulated and solved using Pontryagin’s Maximum Principle to determine the optimal intervention strategy. Furthermore, a numerical simulation is conducted based on the public opinion event surrounding the claim that “2022 China’s average mathematics score in the college entrance examination hit a record low.” The results show that the fractional-order model with an order of 0.6 more effectively captures memory effects and demonstrates superior predictive performance compared to traditional integer-order models. Moreover, the comparison of three control strategies indicates that multi-strategy interventions are significantly more effective than single measures in curbing rumor spread. These findings underscore the advantages of fractional-order modeling in studying public opinion dynamics and highlight the importance of integrated governance strategies.</p>