More importantly, if man had no immune system before the fall why would God suddenly create one for him?
Coming in late to the game as I have, my response may already have been made by another. I don't doubt that man was created with an immune system in place. Before they chose to follow the serpent, they may not have needed one but even that we do not have scripture to know. What we do have is that there were two trees in the middle of the garden; the tree of the knowledge of good and evil and the tree of life. They were only told not to touch the one.
After they ate of the forbidden tree, Genisis 3:22 says:
And the LORD God said, "The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever."
So Adam and Eve were put out of the garden.
Certainly not a conclusive answer to your OP, but might shed a bit of light in that direction.
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