I’d say that you have at least a minimal duty to them until they’re 18, no matter what. Even if they’re in prison for killing your other child, for a really extreme example, you have a duty to ensure that they’re safe and their needs are being met as well as possible. You don’t have to shelter them from the police or anything, and I can understand if you no longer feel love for them, but you’re responsible for getting them to adulthood as best you can.

Of course, I wouldn’t judge anyone actually in that situation for disowning their child (or any number of other reactions that I would consider ethically wrong, but that’s because they’re in a truly awful situation and it makes sense to go a little crazy), but it’s the moral ideal I would shoot for.