Edit: Apparently, it is possible for them to drop at any time but it's very rare. So ignore my original comment that they don't drop in early game.
Antibiotics do not drop in loot in early game from what I have seen. On very rare occasions you might find herbals kind of early but they are still not common. Honey is the number one way to deal with infection in early game. Looting to cure infection is a waste of time and effort. Chop stumps and you will find honey easily enough. The drop rate is 20% and even with bad luck, you should have at least one within 10 stumps. Once you get the book to get honey from trees, it is even easier since trees are easier to find. Don't try to find antibiotics in early game.