Yeah, you could use either word for both. "Eliminate" is a complex synonym of "remove". However, I wouldn't recommend it for the second sentence. "To remove" is not a synonym of "to entirely destroy", but "to eliminate" can be just that. By using this second definition, it becomes clear that the doctors want to completely and irreparably annihilate whatever's causing the epidemic. "Eliminate", due to this alternative definition, is typically used in more severe cases than "remove". You can remove your facial hair, but you don't typically say that you're going to eliminate your facial hair. You can remove a rug out from someone's feet; you can't eliminate it out from someone's feet. "To eliminate" is a bit more permanent than "to remove".

"Remove" and "eliminate" can be interchangeable, especially in the examples you gave above. The words are more of context-based words and can depend on what degree you mean. Remove-to take something off of another thing or subject and pay no mind to it or set it somewhere else. Eliminate (similar to terminate)-to remove something and pay no mind to it afterwards OR to destroy something or narrow something out. In short, "remove" is just taking something away while "eliminate" means to get rid of it or destroy it. I hope this helps.

The words are fairly interchangeable, although I would say eliminate is stronger and more final than remove.