From what I've heard & read, we don't actually know that people are rendered immune — and even if they are, we don't know how long the immunity would last.
Even if it turns out that immunity doesn't happen, I don't think it changes the reaction that we need at this moment. It will change our decisions going forward, of course.
What I think a most plausible explanation is it takes time after symptoms disappear for our immune system to finish wiping out virus. Influenza also has that stretch, though only days long, not weeks.