Wiping out rats is possible. While not exactly the same situation, the Canadian province of Alberta is rat-free due to an aggressive program to prevent and contain infestations [0].
[0] http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/agd...
[0] http://www1.agric.gov.ab.ca/$department/deptdocs.nsf/all/agd...
New Zealand is vastly more welcoming to the invasive species it has. In a decade or three, they'll call it "good enough" on the mainland islands and spend a great deal of effort in perpetuity to keep pest levels under a modicum of control.