Irrational behaviour-induced suppression, enhancement and oscillations of voluntary vaccination: a stochastic model with delays

We investigate how human irrationality and memory shapes voluntary vaccine uptake in the absence of disease. Building on a deterministic imitation game framework, after providing its analytical solution, we show that finite-population stochasticity alone produces either vaccine-free or full-coverage states, but with fixation times so long that the mean-field ODE model remains valid for large population sizes. To capture rapid irrational swings in public sentiment, we model them using white noise, deriving an SDE whose stationary density undergoes noise-induced transitions as noise intensity crosses critical thresholds. Larger noise can switch the population to either disease-free or full-coverage regimes, and very strong noise yields stochastic bistability reminiscent of phase transitions. Numerical simulations reveal very slow transients and rich behaviors when delays and noise interact. In particular, noise-induced quasi-cycles can onset. Generalizing to fat-tailed Cauchy noise, we find that heavy-tailed fluctuations increase the probability of fixation at zero or full coverage. Fluctuations in the imitation rate parameter produce large amplitude cycle distortions, occasionally destroying cyclic patterns. Finally, we show that a public health campaign term with ‘intensity’ exceeding a noise-dependent threshold eliminates the vaccine-free attractor and stabilizes full coverage.