Crazy eyes

Sometimes you may run into people who are obviously mentally unstable.  This happened to a friend of mine recently.  After awhile, you get trained to recognize these people and avoid eye contact.  My friend had an experience where he was followed by one such person when he crossed his path.  He managed to get away, but it makes you think of the people who don’t escape.  What happens to them?  Could your worst nightmares come true just from accidentally making eye contact with someone on the street who now targets you relentlessly?