My uneducated guess is that the Europeans lived in much larger communities (in Europe), which is a much better breeding ground for disease. This meant they were much more full of disease than the native Americans.

Another uneducated guess is that this is why your pet <insert species name here> doesn't often get sick - they have much less contact with other <insert species name here>s than you do with other humans, contributing to less and slower spread of disease.

Please could a pathologist jump in and enlighten me?