Maybe use a different syringe for each bird you are treating so nothing gets passed on or around the flock from using the one syringe, just a suggestion :)
I clean up every few hours and have specific footwear that I wear in the run cleaning up - although impossible to keep it totally clean 24/7. I couldn't leave chicken poop for days and clean the coop daily too.
Not sure about the risk of infection though; I would be more worried about the...