This is the Rod of Asclepius*, an ancient symbol of medicine. Likewise, the Staff of Aaron, and the Nehushtan from the Old Testament - the latter was supposed to cure the effects of the Fiery Serpents that plagued the wandering Israelites.
Some scholars suggest that the symbol once represented a worm wrapped around a rod - and thus we get to the parasite of the day - the guinea worm!
After Smallpox, Dracunculus medinensis is the ONLY parasite that we are even close to deliberately wiping out. The Human Flea is becoming quite rare, and people are actively trying to save the Rhino Maggot, Polio eradication is being held up by religion and politics AGAIN, but a worldwide drive of eradication has reduced cases of Guinea Worm from 3.5 million cases in 1986 to approximately 11,500 reported cases in 2005. This is incredibly successful, and if we can finish the job will be greeted with a sigh of relief from most of mankind.
Guinea worms are NOT NICE. An adult female guinea worm is up to 3 feet long, and the first you'll know about it is when she crawls down thru your living flesh into your lower legs or feet, or toes, and vomits up a few babies. The result is excruciating fiery pain (Dracunculus means Little Dragon) and an ulcer, that can only be soothed by bathing the wound in water. Which is exactly what the worm has been waiting for, and she vomits up more babies that swim off into the waterhole with little cries of "Wheee!" and look for a freshwater copepod to parasitise.
Later on, some unfortunate bastard drinks a glassful of copepods, the larvae fail to be digested, drill out into your body cavity, have lots of sex, and it all begins again. Dracunculiasis used to be rife across any part of Africa, Europe, and Asia that had dry seasons. Because that made it easily for the cycle to continue since everybody used the same waterhole to drink from and bath gaping vomit-baby wounds.
The only way to safely remove a guinea worm was to very slowly wind it out onto a stick, over several days. Do it too fast and you risk snapping the worm in half and getting a nice 3-foot-long ulcer, assuming you dont get gangrene etc from the original ulcer anyway.
This is where some scholars think the Rod, and the Fiery Serpent legends, come from. It is also the source of a horrible practical joke a parasitologist played on his university audience, after telling them all about guinea worms, then saying "As it happens I was in Africa last year.." and winding up his pants-leg to reveal a matchstick he glued there earlier. One time he heard a "THUMPthumpthumpthump" and raised the auditorium lights to see that a girl up near the back had passed out and was rolling down the stairs.
But it is easy to break the cycle, by filtering drinking water thru fine muslin and keeping infected victims at home. And thus, humanity is on the verge of eradicating a unique scourge (altho hopefully they somehow keep a few around for posterity's sake, and indeed there's a volunteer organisation attempting to do so.)
Asclepius was said to have learned the art of healing from the centaur Chiron and served as a surgeon on the ship, the Argo. Asclepius was so skilled in the medical arts, that he was reputed to have brought patients back from the dead. For this, he was punished, according to Greek mythology, and placed in the heavens as the constellation Ophiuchus. (the 13th sign of the zodiac that astrologers never seen to mention - but WTF? Punished???? By being turned into a constellation??? Still not my favorite bit of astromythology tho - that one goes to the Coma Berenices. Berenice apparently being a pharoah's daughter so beautiful that a lock of her hair was kept in a temple, from which it was promptly nicked by some ancient Egyptian hair-fetishist. The guards were for the chop of course, until one of the priests piped up with "no, no, it wasn't stolen, it was turned into that patch of stars. that one, the fuzzy faint ones nobody has ever bothered to notice before. Honest. You can believe me, I'm a priest")