Looking at my home country (New Zealand), there's quite a few ways that something like the Rash Illness could go. Once the epidemic was realised to be super-serious, international travel via airports and ships would be banned. Since we're effectively a very large, very remote archipelago this would deal with the problem in the short term....assuming our rather small defense force could enforce the quarantine (although we're helped by our position: the Tasman Sea, South Pacific and Southern Ocean are not a group of waters known for being particularly tranquil).
The long term isn't so rosy though: New Zealand has quite high marine mammal populations, and given that at least one study has shown that TB crossed the atlantic in the lungs of seals, it's entirely possible that infected marine life would bring the disease over anyway.
So either the whole country would survive, but infrastructure would collapse and the population would drop due to a lack of food production.....or one or more main islands would collapse. The west coast of the South Island might make it...it's remote enough, has a low population that's particularly no-nonsense and hardy, and isn't connected to the rest of the country except though mountain passes that would be easy to block off with saturation bombing or just knocking a bridge or two. The other two places I'd imagine would survive (in the medium term) would be the Chatham Islands, 800km away from the mainland, and possibly Stewart Island.
And the Ross Dependency would be just fine, but Antarctica is probably not a place you want to be during the apocalypse.
The interesting thing for me (as an ecologist) is that most of the problems with the ecosystems here are caused by introduced mammals. Killing them off might actually lead to some interesting results, although replacing them with ravenous undead monstrosities is not an approach that is likely to win funding if I propose it to anyone.
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