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When you get a wound that can be infected (that is, not a bruise, crack, or missing part), the game runs a countdown between 15000 and 45000 ticks (determined by RNG). Once the timer is up, it runs a percentage check based on the following:
If it's a burn, frostbite, or a bite wound, the base chance is 25%. Other wounds, not including bruises, cracks and missing parts, have a 10% base chance.
An untreated wound is 100% chance multiplier. A treated wound scales linearly from 85% multiplier at 0% tend quality to 5% at 100% tend quality.
Room cleanliness at the time of tending gets taken into consideration, scaling from 32% chance multiplier with a clean sterile (0.6 cleanliness) room, to 50% in a clean floored (0 cleanliness) room. I'm not sure if it's a linear curve, but it would make sense, making a 100% chance (the likely cap of this multiplier) at a cleanliness level of roughly -1.64. Note that tending outdoors or in a caravan is always a 100% multiplier.
Finally, if the pawn in question is an animal, an additional 20% multiplier applies.
Other noteworthy additions to keep in mind:
Blood stacks on the same tile. This will not make the tile, and as such the room, any dirtier. 1x blood is the exact same as 3x blood so long as it's on the same tile.
A large enough hospital will have so many clean tiles that blood and dirt will barely impact the overall cleanliness in most circumstances.
Items do not affect cleanliness, even if left on the floor, although chunks do. Medicine is fine to leave on the floor.