Instead of 1 doctor covering a 24 hour shift, why not pair them and overlap?
12:00am - 6:00am: Doctor 1 and Doctor 4 are doing everything together.
6:00am - 12:00pm: Doctor 1 and Doctor 2 are doing everything together.
12:00pm - 6:00pm: Doctor 2 and Doctor 3 are doing everything together.
6:00pm - 12:00am: Doctor 3 and Doctor 4 are doing everything together.
This way, all 4 doctors only do 12 hour shifts, and the patient's state is maintained continuously through all 24 hours.