From personal experience I can say it's very hard to make logic control your fear. However that being said, knowledge is power.
If you're worried about a brain parasite epidemic that would make a mass population centre into something akin to zombies there are some things you can look up. I'd start with how the parasites are transmitted and once you have that it should give you an idea of how hard it would be to 'infect' others. There's always the timeline to take into consideration as well, how long does it take a parasite to 'take over' the host. TV shows and movies have it happen at a rate I don't think would be possible.
Now look at the medical profession. How long would it take to discover an epidemic, what are the precautions they take, etc.
Finally, in the case of 'true zombies' you need to realize, without higher brain functions the human body is incapable of sustaining itself long term. Your brain stem is only capable of autonomic functions such as breathing, reflexes, etc. So while the brain dead would be physically alive, they wouldn't be walking or running anywhere (or trying to bite you).
Like I said, the facts will not necessarily stop the fear... But it will help you identify that it is in fact just a fear and not something that can actually happen. So when the fear is stirring just recall all the facts on why it (A Zombie Apocalypse) could never happen. I know it will be hard but try and keep your breathing calm, that will help you maintain a level of "cool" as you recite the reasons to yourself.