How was Danielle able to give birth to Alex without dying?

I began to ponder on the subject of pregnancy on the island. We all know that people who get pregnant on the island die near the end of their second trimester, unless you get pregant off of the island, then you should be fine. Claire is able to give birth to Aaron on the island because she got pregnant off the island with her old boyfriend Thomas. The only thing that the others do is inject Claire with a syrum of some sort through Ehtan so she is able to give birth to her baby on the island. Sun got pregnant on the island, but the only reason she survived to give birth to her child was that she got off the island, so she was fine. I am just wondering why Danielle didn't die when she was giving birth to Alex. I know she got pregnant off of the island too, but wouldn't she need some sort of syrum as well? I also thought maybe whatever let the pregnant women on the island allowed to have their babies hadn't happened yet like Sawyer said in Lafluer. I just don't know though. Any ideas?

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