To be on the safe side, you have to count half the duration of the fast. So at least a week for a 15 days fast.
By the way, I read on another forum that you also contemplated switching to a "normal" diet after the fast. I would be very careful with that. If by "eat like a normal human being", you mean something like the standard American diet with lots of Sugar and processed foods, then I don't think that's a good idea. But even a more generally thought of "healthy" vegetarian/vegan diet with lots of fruit and low in fat may, depending on you metabolic type, not be very beneficial to you and make you gain weight. If you're used to Atkins, then the maintenance phase looks like a good choice to me: you still exclude the bad carbs, but you incorporate more of the acceptable variety such as fruits and whole grains. I would not go over 100 grams of carbs per day though, unless you really get a lot of exercise.