I understand any extra data gives you an advantage, but I think I'm missing a step where they can steal a solution while simultaneously keeping that solution for themselves (assuming stealing means copying/sneaking out information others have already gathered).
Maybe a slight headstart if they throw extra research money on it.
It's not okay to keep solutions for only their citizens, but it would still be in your best interest to share what you know with their citizens so that you have a lower priority of getting infected by their citizens who may travel to your place.
Humans are humans. The virus doesn't care for the politics, it will spread without regard to human-invented political borders, so the preventative measures shouldn't stop at political borders, either. That would be contradictory to the apolitical nature of the virus.
(In any case, if you shared something with them and it is actually useful, there's a good chance you're pretty close to the solution yourself, anyway.)
Maybe a slight headstart if they throw extra research money on it.