  The containment of epidemic spreading is a major challenge in science.
Vaccination, whenever available, is the best way to prevent the spreading,
because it eventually immunizes individuals. However, vaccines are not perfect,
and total immunization is not guaranteed. Imperfect immunization has driven the
emergence of anti-vaccine movements that totally alter the predictions about
the epidemic incidence. Here, we propose a mathematically solvable mean-field
vaccination model to mimic the spontaneous adoption of vaccines against
influenza-like diseases, and the expected epidemic incidence. The results are
in agreement with extensive Monte Carlo simulations of the epidemics and
vaccination co-evolutionary processes. Interestingly, the results reveal a
non-monotonic behavior on the vaccination coverage, that increases with the
imperfection of the vaccine and after decreases. This apparent counterintuitive
behavior is analyzed and understood from stability principles of the proposed
mathematical model.
