Analogy is an important part of human thought, so to estimate when aliens will invade Earth, I extrapolate from 2 relevant analogies:

When were Neanderthals invaded by Homo Sapiens?

When were Native Americans invaded by Europeans?

It’s estimated that Neanderthals and Sapiens shared a common ancestor about 500 kya and that Neanderthals colonized Europe 300 kya and then were invaded (in large lasting numbers) by our species, Sapiens, about 42 kya.

Thus it took 200,000 years for future Neanderthals to find Europe since splitting from future Sapiens and then got to live there in peace for 258,000 years before being invaded by their distant relatives. So in this case, they had 1.29 times as much time to enjoy their homeland as they did to emerge in it in the first place.

Meanwhile, it’s estimated Native Americans and Europeans shared a common ancestor 41 kya and that Native Americans colonized the Americas 15 kya and then were invaded by Europeans about 0.5 kya.

Thus it took 26,000 years for future Native Americans to find the Americas since splitting from future Europeans, and they had 14.5 kya to enjoy it before being invaded. Thus 0.56 as much time.

Applying the same logic to humans and aliens is tricky since we don’t share a common ancestor, but we do share a common origin; and assuming there are no intelligent aliens in our solar system, that common origin is the emergence of our galaxy itself about 13.61 billion years ago and 9.91 billion years later, we began living on Earth.

So how long will it take our alien “relatives” to hunt us down and attack us? Using the invasions of Neanderthals and Native Americans as analogies, it should tale between 1.29 and 0.56 of 9.91 billion years so between 12.8 billion and 5.5 billion years.

So given that we’ve been on Earth for 3.7 billion years, that gives us another 9.1 billion to 1.4 billion years to live in peace, with the best estimate being 5.25 billion years.

Thus it makes perfect sense that we haven’t been invaded yet and have more to fear from a nuclear holocaust, a meteor and climate change than we do from alien invasion.

Who needs the Drake equation when we got the Pumpkin equation?