a, Theoretical prediction for the future total size of anti-vaccination and pro-vaccination support without new interventions (coloured lines with 2σ bands from the simulation). Under the present conditions, it predicts that total anti-vaccination support reaches dominance in around 10 years. b, Top left, our theoretical model predicts that, as observed empirically, many smaller-sized anti-vaccination clusters form, with each cluster having its own nuanced type of narrative (for example, X, Y, Z) that surrounds a general topic (vaccines in this case). Bottom left, the predicted growth profile of individual clusters can be manipulated by altering the heterogeneity to delay the onset and decrease the growth. Bottom middle, pro-vaccination population B is predicted to overcome the anti-vaccination population, or persuade the undecided population, X, within a given network patch in time T by using Fig. 1a to identify and then engage with all the clusters. Bottom right, the link dynamics can be manipulated to prevent the spread of negative narratives. See Supplementary Information for all mathematical details.