What do you do when your duck is sick?

Honestly, you probably wouldn't know that the duck had an avian influenza virus. In birds, the flu is an intestinal infection while as in humans the flu tends to be a respiratory infection. Ducks do not appear to be as susceptible to the various influenza viruses as chickens are - in general they either have a transient subclinical infection or it makes them very sick and rapidly kills them.

You would be very unlikely to contract the flu from the duck, but to be safe you should wash your hands with soap and warm water for 30 seconds after holding the duck. This is good advice when handling any animals, actually.