This site has info on refunds for American
https://www.refunds.aa.com/help.asp#Extenuating
and they may be able to help you with that or to rebook.
But you may need to purchase the travel insurance and use that
http://www.etravelprotection.com/servlet/WASCPure?accam=F030356&csid=K11
from the AA FAQ on travel protection insurance
" There are many covered reasons for cancellation. The most common reason travelers cancel their trip is due to injury or illness. For example, if a family is planning a summer vacation and a week before the trip, a close relative is admitted to the hospital with a life threatening illness, the family could cancel their trip and file a claim for any prepaid expenses or other costs associated with a change or cancellation of their vacation plans. Restrictions may or may not apply for pre-existing illness; please review your Certificate of Insurance/Policy for details."
Delta refund info here
http://www.delta.com/onlinerefunds/refundsHome.do?null
they also offer trip insurance which may apply
https://www.delta.com/planning_reservations/plan_flight/insurance/index.jsp
It may take some time on the phone but if all else fails call a reservation number and ask to be connected to a person who can help with changing reservation dates due to medical emergency for the traveler.
As you go along try to write down the name and department of everyone you talk to in case you have to write in about this.
Hope this helps you!
Internet Researcher
Adept at Internet Research & I know just where to find information!