>However I'm curious how long before we get to zero cases.
When will influenza or the common cold get to zero cases? Never. Viruses mutate. They change. If the rate of change is fast enough and the hosts population large enough then they never burnout. But they tend to evolve to the optimal level of virulence for their transmission mechanism. Convergent evolution. That's why there are some >200 different viruses that we call "the common cold". This will soon become just another one.