From Greek mythology:
https://www.livescience.com/33104-why-is-the-medical-symbol-a-snake-on-a-stick.html
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From The Bible:
God told Moses to make a brazen serpent, to put it on a pole, and to instruct the people who had been bitten with deadly serpents that whoever looked to that serpent on the pole would be healed.
Christ told Nicodemus, “And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up: that whosoever believeth in him should not perish, but have eternal life” (John:3:14-15).
Nicodemus, like the Jews in John:12:31-34, knew that to be “lifted up” meant to be crucified. Christ was telling Nicodemus that, as it was in the case of the uplifted brazen serpent, all those who would look in faith to Him lifted up on the cross would be saved.
But why would Christ, the Lamb of God, the fulfillment of the Old Testament sacrifices for sin, liken His lifting up to that of the brazen serpent upon the pole? Although Satan is “that old serpent, called the Devil” (Revelation:12:9), the brazen serpent was not a symbol of Satan. The “fiery serpents” were sent among the people because they had sinned grievously (Numbers:21:5-7). The serpents were God’s judgment upon it—but more than that, the fact that through judgment there would be salvation. Paul writes, “For he [God] hath made him [Christ] to be sin for us, [he] who knew no sin; that we might be made the righteousness of God in him” (2 Corinthians:5:21).
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As I have postulated before, I believe that God was preparing the Gentiles for Christianity even back as far as Moses time. The Greek and Roman mythologies and even the Egyptian to a lesser extent, were replete with stories of people who were ‘sons’ of gods, so they had little trouble accepting Jesus as the Son of God, which the Jews would not accept. The Snake on a Stick symbology for healing may have been a part of this.
On a side note, when I was a small child in rural Mississippi, it was a common belief that a snake when killed would not completely die until ‘sundown’.
People would hang a killed snake on a fence or limb of a tree, where it would writhe for hours, finally ceasing movement at about sundown. Of course the movement was due to nerve and muscle spasms in the dead snakes body and not some mysterious power the snakes had.............