[Fomite]

I'd be surprised if there are any documented transmissions of those pathogens, especially given they need more complicated uptake.

The issue with Staph is that skin-to-skin contact, or the indirect version mediated by a mat, is more than enough for it go to "Yeah, I live here now..."

[peterwwillis]

According to doctors/nurses I've talked to, smooshing a mucous membrane or open sore onto material which came into contact with infected blood can be enough for transmission. Which is why competent BDSM dungeons clean their apparatus with cavicide before/after use, and why they recommend a barrier between a person and the apparatus.

[Fomite]

I'd absolutely suggest it for those environments, but that's because, to be blunt, there's a lot more exposure of mucous membranes in those environments than your average dojo.

The open sore is honestly a risk for both - S. aureus loves it some open wounds.