(a) Vaccination coverage and measles prevalence. Data are from Switzerland (red), Germany (black) the UK (blue). The dashed lines show the vaccination coverage and the bars show the number of reported cases in each year. (b) Increase in outbreak probability. Each bar of a given colour (i.e. vaccination coverage) represents the increase in outbreak probability relative to the lowest non-zero outbreak probability at this vaccination coverage. The parameter Ω measures the strength of opinion formation: the probability that an individual swaps opinion is given by the frequency of neighbouring individuals who hold the opposite opinion, weighted by Ω, i.e. Ω=0 indicates that no opinion formation occurs. The data given are based on 220 000 simulations of networks of 2000 nodes, and an outbreak was registered when the initial infection led to at least 10 follow-up infections (for a detailed description of the model see §2.2). For each value of Ω, we tested 11 different levels of vaccination coverage from 0.5 to 0.95. (c) Effect of opinion formation on herd immunity. Outbreak probabilities without opinion formation (black bars, Ω=0) and with opinion formation (grey bars, Ω=1).

Visual representation of the opinion formation process. The colours red and blue represent the two possible opinions. After initial assignment of opinions, the dissimilarity index d (i.e. the percentage of neighbouring nodes that are of the opposite opinion) is calculated for each node. An opinion change occurs in the following way. (a) A random node is chosen (in this example, we focus on the central blue node, where d=0.8) and its opinion is changed with probability d×Ω. Note that any non-neighbouring node of the focal node and their edges are omitted for visual clarity. The d values of the neighbouring nodes are chosen randomly as an example and are based on the assumption that every node has five neighbours (in the simulations, the average degree of a node is 10). (b) If an opinion change occurs, the dissimilarity index is recalculated for the focal node and all its neighbouring nodes.