Supplementary material from "Vaccinating behaviour guided by imitation and aspiration"

Vaccinating decisions can be influenced by imitation as well as self-evaluation or aspiration. This work endeavours to analyse vaccinating behaviours by coupling both—imitation and aspiration—update rules, adopting an existing setup of mean-field vaccination game. We incorporate the imitation mechanism with several variants of aspiration protocol, encompassing constant and adaptive aspirations. Equations of the combined dynamics have been derived by grouping the population according to—(i) vaccinating strategies and (ii) healthy and infected status within each strategy. If aspiration levels are fixed but differentiated by vaccinating strategies, then vaccinators aspiring less than non-vaccinators are found to ameliorate the vaccination coverage, thereby yielding a less infectious state. The adaptive aspirations maintain a positive correlation with the vaccine efficacy while keeping the opposite relation with vaccination cost. When vaccinating strategies, aspirations and update rules are allowed to evolve synchronously, then either imitation or aspiration process takes over the entire population. If aspiration rules prevail, then vaccinees and non-vaccinees coexist equally (according to (i)) or vaccine uptake follows a non-monotonic trend with the efficacy (according to (ii)). The imitative rule performs better when vaccination is less expensive or cheap, while the aspiration updating safeguards the tenacity of vaccinees despite vaccination being expensive.