You have to distinguish between viruses in abstract and a concrete disease. Many viral diseases we are fighting with are human only. So if we erradicate it in humanity, it is gone. This is how smallpox went away. We vaccinated enough people to make it go away once and for ever. Measels have no animal host either, as far as I am aware, so we could have a good chance to erradicate it too.

The corona virus sars-cov-2, as well as the first sars virus jumped from animals to humans. There are several animal species which carry variations of the sars virus, but fortunately, they don't make the transition to humans regularly. If we managed to prevent further transmissions of sars-cov-2 between people, there is a good chance, this virus could go away entirely, unless it manages to jump to some common animal.