Let's say that black has been eradicated. Can yellow come into the black cities and infect?
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Yes. Eradication affects the color of cubes, not the color of cities.
The color of the cities tells you what color is on the corresponding infection and player cards, so cities will certainly tend to have only cubes of that color, but they can potentially have cubes of any color.
Eradicating a given color disease simply stops cubes of that color from being added to the board:
But the rules never say that eradication stops, say, the yellow disease spilling over into black from an outbreak. This isn't terribly likely (you have to have an outbreak on the border) and also isn't usually that awful (unless you're running out of that color of cubes), but it can happen.
Eradicating a disease means that new cubes of that color won't get onto the board. It doesn't stop outbreaks of other colors from spreading.
Just because you cure Ebola doesn't mean people become immune to West Nile Virus...