<p>With the widespread adoption of online social platforms, multimodal rumours integrating text, images, and audio have become increasingly prevalent, posing severe threats to public opinion and social stability due to their complex propagation dynamics and resistance to debunking. Existing rumour propagation models rarely examine the synergistic interaction between individual and official debunking, nor do they account for the differentiated effects of official reports across user states. To address these limitations, this paper proposes a novel SIOUR rumour propagation model that integrates dual debunking mechanisms, differentiated state transitions, three time-delay factors, and multimodal propagation characteristics. Theoretical analysis derives the basic reproduction number and establishes the local and global stability of the rumour-free equilibrium for both delayed and non-delayed systems. Numerical simulations investigate the influence of key parameters, time delays, multimodal features, and network topologies. By combining mean-field analysis with agent-based modelling, we evaluate the model under heterogeneous network structures and identify the limitations of homogeneous approximations. Finally, comparative experiments on four public rumour events show that the proposed model significantly outperforms existing approaches across multiple evaluation metrics.</p>