According to a friend that became a doctor, that disease pathway went both ways. (That should NOT be a surprise)
As taught in one of his classes (epidemia studies, or something like that), historians were able to trace the locations of the Columubus crew members after they returned to Europe by tracking the early outbreaks of Syphilis. The native Americans had that disease, and it was "communicated" to the European crew members. That was particularly nasty because modern antibiotics capable of curing the disease would not be discovered/invented for hundreds of years, (penicillin first became widely available in the 1940s according to Wikipedia)
Sidenote: There were, politely put, ghastly treatments before then, but those treatments were both painful, and unreliable as a cure.
If you consider this little bit of info in the context of another movie, the Mel Gibson version of MUTINY ON THE BOUNTY, (taking place well after 1492), you should now have an understanding of why Captain Bligh's request for Mr. Christian to "entertain" the Chief's daughter was something one gentleman (officer) would not normally even suggest to another officer It was not a big deal for the ordinary crew members (who didn't matter), but a HUGE request of an officer, and NOT for ethical/moral reasons the movie showed. (i.e. officers had to be aware of the risk of getting incurably sick, and all that could mean for their future and the remainder of their lives)
Maybe that contributed to Mr. Christian's involvement in the mutiny?
You can imagine why entertainment movies would not mention, or dwell on this, so simply showing the rat was not meant as a metaphor, but a reminder, or a trigger for discussion.

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Less metaphorical even, it's the rats that helped to bring disease which lead to the deaths of tens-of-millions of Native Americans. I think even the population that Columbus first met got decimated soon after by diseases that came with his ships.

I think the rat was a sort of metaphor for the insidious things that Columbus and his party inadvertently brought to the New World.