Couple of months ago my GF attended the hospital to see if she could gather any forensic evidence from a dead guy who had been stabbed in the neck.
He was merely making his way home with his wife on the bus,when some guy decided to plunge a knife right through his neck. His life snuffed out in a few minutes,and horrifically witnessed by his partner.
The murderer was caught a few days after,and "yes" he had mental health issues and was known to the authorities.
So various other crimes may be commited by people with mental health issues.