Dynamic Analysis of a Cross-Lingual Coupled Rumor Propagation Model with Response Delay in Online Social Networks

As online social networks (OSNs) evolve into multilingual ecosystems, rumors can cross language boundaries through translation and bilingual re-expression, increasing governance difficulty. To characterize cross-lingual coupling and response delay, this study proposes a time-delay S2LCHR dynamical model for bilingual OSNs, in which the coupled spreader state C describes cross-lingual coupled rumor transmission and a fixed response delay represents cross-lingual comprehension, judgment, and re-expression. The basic reproduction number R0 is derived using the next-generation matrix method. Lyapunov analysis, the Routh–Hurwitz criterion, characteristic-equation analysis, and numerical simulations are combined to examine equilibrium stability, delay-induced Hopf bifurcation, parameter sensitivity, and social-impact indicators. A real-world aggregate trend-fitting case study using English–Spanish COVID-19-related tweet data is further conducted to assess empirical plausibility. The results show that R0 determines the threshold between rumor extinction and persistence in the delay-free system, while an excessive response delay can destabilize the rumor-prevailing equilibrium and induce bounded oscillatory behavior. Sensitivity and social-impact analyses indicate that α and β promote rumor persistence, whereas σ and φ associated with state C are key inhibitory factors. These findings suggest that coupled spreaders should be prioritized in cross-lingual rumor governance.