Why do my babies on sims 4 keep disappearing right after birth?

Try resetting your user files as described in the factory reset described in this post: http://answers.ea.com/t5/The-Sims-4/FAQ-Troubleshooting-Steps-amp-Known-Issues/m-p/3621199

You won't loose anything. Just make sure to not delete any files  and make a backup.

Try on a new game with nothing added back first (start new game). If it works, put back your save game and try again. If it still works, you can start putting stuff back.

• Options.ini (or just re-set all your settings manually)
• Tray Folder
• Screenshots/Videos/Custom music
• Rest of the save games (leave out slot_00000001.save which is the autosave)
• Custom content like clothing, hair, skins, make up ....
• Mods (one by one, making sure they all still work)

The rest does not need putting back, cause they are mostly cache files that recreate or log files that are not needed for the game.