I don't think there is a separate "historical epidemiology" field.
Major epidemics are chronicled just like wars - written record is the main source of information. Death records - both papers (when preserved) and cemetery headstones - may provide additional statistical evidence.
For prehistorical deceases, the main sources of information are the genetic analyses of pathogens and the human immune systems. Epidemics in the modern sense did not occur because humans lived in mostly isolated communities.
Recurring epidemics, like flu, are subject to intensive medical research; e.g.:
In June 2010, a team at the Mount Sinai School of Medicine reported
the 2009 flu pandemic vaccine provided some cross-protection against
the 1918 flu pandemic strain.
Decease transfer by colonizers
As for the Europeans carrying diseases to the Americas, you might want to take a look at Guns, Germs, and Steel.
Basically, the European conquest of America is sufficiently recent for ample written record to exist, including the famous episode when the Europeans attempted to pass smallpox-infected blankets to the Indians (the transfer of blankets was in the summer, the smallpox outbreak among the Indians - in the spring. I am not a physician, but the connection seems tenuous).