Mice can carry diseases, leaving viruses/microbes in their waste, running through it into areas we'd rather not know about, etc.
Yep. Turning that compost (use a compost fork, aka pitchfork or similar) will roust them out of their cozy nests.
After having cleaned up my in-laws' garage post-rat-waste-infestation of yuck (now that's
an image for you...), I have absolutely no patience with rats/mice/scurrying little rodents, whether in compost piles or anywhere else.
I had to observe hantavirus precautions in Fall 2007 and impress them upon my sister-in-law, who was absolutely convinced that everything in that garage was a treasure, even if it was covered in rat droppings.
I told her that, if she developed a respiratory infection and fever within 72 hours of handling rat-waste items with bare hands or failing to spray a shelf with water (to keep down the dust) before picking something up, I would not call 911 for her.
Can you tell that I was supremely "ticked"
So, yes, mice and rats and all of their little friends may not be ill themselves, but they can certainly bring illnesses to us.
Sunset Zone 17, USDA Zone 9