Spread rate would also need to stay in a certain range for a virus to stick around. Spread too fast and there won't be a second wave. The herd immunity effect gives this system some self-balancing properties, but there's surely a tipping point where even that won't help a speeding virus to survive.
In general the less deadly viruses spread better as dead people are not able to spread the virus around as well as living people. The general trend is for viruses to become less deadly over time, but it can take a long time for this to happen.