Ive worked in psych/mental health nursing for approx 15 years. I have never experienced a patient chasing after me. Mental health/psych nursing teaches you much about observation and you develop fabulous peripheral vision (I can be talking with a patient directly in front of me and be aware of whats going on over at the left or right side of the ward). You spend your shift with your patients and observe their behaviour throughout. Because of this, you pick up on any deviations in their behaviour and their "affect" as we call it. As such, early intervention to address this is essential - to curb any escalation/deterioration in behaviour/mental state/aggression.
Just to finish, nurses that work in this field struggle to "medicalise" and "normalise" mental illness - there is so much stigmatisation, even within the health care sector. These patients are not to be feared....they are just patients with an illness, same as anyone else...