  Voluntary vaccination is effective to prevent infectious diseases from
spreading. Both vaccination behavior and cognition of the vaccination risk play
important roles in individual vaccination decision making. However, it is not
clear how the co-evolution of the two shapes the population-wide vaccination
behavior. We establish a coupled dynamics of epidemic, vaccination behavior and
perceived vaccination risk with three different time scales. We assume that the
increase of vaccination level inhibits the rise of perceived vaccination risk,
and the increase of perceived vaccination risk inhibits the rise of vaccination
level. It is shown that the resulting vaccination behavior is similar to the
stag-hunt game, provided that the basic reproductive ratio is moderate and that
the epidemic dynamics evolves fast. This is in contrast with the previous view
that vaccination is a snowdrift like game. Furthermore, we find that epidemic
breaks out repeatedly and eventually leads to vaccine scares if these three
dynamics evolve on a similar time scale. And we propose some ways to promote
vaccination behavior, such as controlling side-effect bias and perceived
vaccination costs. Our work sheds light on epidemic control via vaccination by
taking into account the co-evolutionary dynamics of cognition and behavior.
