You're assuming that viruses don't have natural reservoirs. For example, viral hemorrhagic fevers, particularly filovirus VHFs, are both highly transmissible and highly deadly -- accounting for their history of rapid, fast-growing, and quickly-diminishing epidemics amongst humans -- but have stable zoonotic reservoirs that allow them to circulate freely across those populations. (Bats' unique immune system, like camelids, makes them a particularly good evolutionary proving ground for viral evolution, so they serve as a reservoir and a source of ongoing viral mutations.)
