The green staff is the staff of Asclepius
(according to Greek mythology, the first healer, the son of Apollo, the sun god); and green was a color associated with the Medical Corps during the last half of the 19th century.
The World Health Organization and most professional health organisations use, as the traditional symbol of healing and medicine, the staff of Asclepius
, which is depicted as a single serpent encircling a staff.
The representation of the staff of the Messenger is not the staff of Asclepius
, the god of medicine.