Answer is March 15th.

Assume Joseph knows the month. Joseph says "I know Roi doesn't know". Joseph knows that Roi only knows the day. The only way Roi could know the birthday is if Roi knew that it was 17th or 18th (as there is only one month for those feb and jan respectively). So for Joseph to know that Roi doesn't know, we know that Joseph must know that the month isn't Jan or Feb.

Hence it can only be March 13th, 15th, Dec 13th, 14th or 16th.

Roi says "I didn't know, but now I know" because he didn't know before if it was Jan or Feb or not, but now he knows it's not Jan or feb.

We now know it's not March 13th or Dec 13th, because otherwise Roi wouldn't know if it was March or Dec.

hence it can only be march 15th, dec 14th or dec 16th.

Josef then says "I now know too"

Thus it can only be March 15th, because Josef knows the month, and if he knew it was dec, he still wouldn't know if it was dec 14th or 16th, but as he knows, he must know it's march and there is only one march left march 15th.