Could you create an event in the calender giving the exact hour, (or in blocks of 5 or 15 min) and day so that once you've created the event (being your appointment with the doctor) others can't take that place any more?
The calender should also not allow and make clear that appointments can only bee made on certain days and between certain hours?
Maybe some of those good calender modules just need minor changes in order to get these effects?
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