<![CDATA[Misinformation can persist in digital interactions even after corrections have been provided, highlighting the need for an analytical measure of intervention intensity capable of changing its dynamical regime. This study develops a three-state mathematical model consisting of individuals who do not believe misinformation and are not undergoing active correction, individuals who believe misinformation, and individuals who have received corrective information. Fact-checking intervention is modeled as a constant additional correction rate. The analysis encompasses conservation and positive invariance, equilibria, local stability, bifurcation, center manifold analysis, sensitivity analysis, and numerical simulations using the fourth-order Runge–Kutta method. The model yields a persistence index that decreases with increasing fact-checking intensity and an explicit intervention threshold. Under the assumed simulation parameters, this threshold is 0.20. Below the threshold, increasing intervention intensity reduces the equilibrium proportion of individuals believing misinformation but does not eliminate its persistence. At the threshold, two equilibrium branches exchange stability through a transcritical bifurcation, and the system exhibits critical slowing down. Simulations up to time 3000 show that the scaled values of misinformation prevalence, the corrected state, and the distance from the misinformation-free equilibrium approach asymptotic constants of 0.5, 2, and 2.5, respectively. Above the threshold, the misinformation-free equilibrium becomes stable. Numerical verification using step sizes of 0.02 and 0.01 yields conservation errors on the order of 10-14 and differences between numerical trajectories on the order of 10-13. These findings distinguish subcritical interventions, which merely reduce misinformation persistence, from supercritical interventions, which induce a change in the dynamical regime. All model parameters and initial conditions are assumed for simulation purposes and are not estimated from empirical data on digital platform users.]]>
