You can make up anything you like. It could be caused by a bite or come swallowing like you said, or it could be a reaction to drinking a potion made from a ground-up, mystical root in your universe. It could be in response to being knotted, or being raised amongst omegas. It could be in response to an omega hunt gone wrong; you could tell us what happens if your alpha catches another alpha, instead of the intended omega! - 🐸