Disease should be able to spread through infected items to anyone that uses that item in a similar way that it spreads though infected people.
Rats would be still be infected after you killed them.
Any item that is crafted with an infected item is infected.
If an infected item is composted the compost bin becomes infected and all humus it produces is infected.
Anything grown with infected humus or fertilized with a infected item is infected.
If turkeys are feed infected items they will become infected and so will their eggs, offspring, and droppings.
If bugs are fed infected items they become infected.
If worms are fed infective items they become infected and any humus or worms produced in their bin are also infected.
Any item left in a container next to an infected item becomes infected after a random number of hours.
Infected artifacts that are slotted into clothing infects the clothing.
Any infected food eaten, inspirational studied, or clothing worn will infect the character, if you have the berry juice buff then there is a chance you will not be infected.
Infected items should be able to be cleansed through washing with soap, cooking or boiling.
Diseased should not show up intermediately but only after 6 to 48 hours.