I wish I could believe we can simply eliminate the virus. In some place for some time sure. China has managed quite well, New Zealand is famous for having eliminated it in democracy and rewarding the government for that... But the problem is that it should be out of everywhere at some time, and that's very difficult.
Not to talk of P, NP and worse, but we don't even have the resources to assess whether the goal is attained. Let alone the resources to achieve the goal. We can't test the whole world population simultaneusly a few times. And even if we could, the virus could stay in (other) animals. It is reported that the virus was present in people around the Mediterranean already in summer 2019 if not before, and nobody knew until december 2019 in Wuhan. If we ever got rid of the virus, how could we even know ?
And in practice there'll always be asymptomatic carriers that will import it any time some territory thinks it got it rid of it. I think the solution will come more from immuntiy (herd immunity or helped by vaccines, sooner or later) and with time the knowledge on the disease will improve and the treatments too. Once it runs out of vulnerable humans, it will reduce to a disease with fewer cases, and since most are light cases or asymptomatic, the severe cases and deaths will be even fewer, hospitals won't be overwhelmed anymore and we'll have more problematic diseases to care about. It may disappear from the media (possibly deservedly), but I doubt it will ever disappear completely. The Plague is unfortunately still with us, but we don't even vaccinate everyone against it, because it is no longer the problem it was in the Middle Ages. It's no longer killing one or two thirds of the population but 0,0000001% (cases, not deaths. which is still too many, but not at all so bad as once was).