How do I stop patients from leaving the hospital

Remove all of the doors and hold them captive.

Just kidding. I'm having the same problem so I started renovating the hospital and adding a few more patient beds. This helped my issue because I noticed that a lot of the sims were getting off the patient beds to let another sim on. When I added more beds, I noticed that the sims would just move to another. I don't know if this will help you, but it certainly helped my situation.