"In your paper, the people vaccinating or not do so based on perceived risk i.e. perception of vaccine damage, not actual risk."
In order to understand what a Game Theory paper is saying, you need to know a minimum about this subject. GT always considers perceived risk, because it is a study of personal decision making.
"The authors are not saying the reaction is rational"
Again, you need to read a bit of GT to know that it is a study of rational behaviour - i.e. what individuals do, given their goals, personal risk assessments, maximising their own utility functions.
Take a look at the conclusion:
For any perceived relative risk r > 0, the expected vaccine uptake is less than the eradication threshold, i.e., P < p crit (Fig. 1). This finding formalizes an argument that has previously been made qualitatively (8, 14); namely, it is impossible to eradicate a disease through voluntary vaccination when individuals act according to their own interests. In situations where vaccination is perceived to be more risky than contracting the disease (r > 1), one would expect, even without the aid of a model, that no parents would vaccinate their children.*
I hope that is clear now.
"...they are simply showing what happens in a population based on some simple starting rules of individuals' behaviour."
That is what GT does 
"And in addition, people behave as their neighbour does - "if others stop vaccinating the risk must be really high""
Not true at all. Those who just do what everyone else does just go and vaccinate. It is the ones who have read a bit on the controversy or know people with vaccine-damaged DC who then refuse some vaccines, going against what all their neighbours do.