I doubt the bull sticks caused a problem...I believe they are sterilized so that nothing can be passed on. My dog loves them and has never had a problem.
However, there are several diseases that can be transmitted from cats to humans. Keep in mind that they are in and out, and digging in their litter boxes, or walking on all sorts of things outdoors.
See this link for furter info and see if anything fits your symptoms. Make sure to scroll all the way down.