Signs of Rose-Type Immortality in Humans?:

If you're familiar with the work of aging researcher Michael Rose, you'll know that he uses "immortality" to mean no increase in mortality rate with advancing time. He demonstrated that mortality rates in flies, for example, stop rising after a certain age. This is counterintuitive for most people, given our experience with the world - we expect the chance of death on any given day to continue to increase as age-related degeneration piles up.

The flies all still die, of course, because the odds come up sooner or later, but they don't get any more likely to die per unit time whilst in this "immortal phase." You can find a more detailed explanation in Rose's essay in the online version of The Scientific Conquest of Death - it's the first essay in the book.