It's obviously not the truth. If someone stricken with Ebola makes it into a crowded location, then the chances of several people getting it will be high. This is obvious given the fact that a single Ebola patient infected a trained worker with protective gear. Extrapolate that out to an Ebola victim in the middle of Times Square, puking, coughing, sweating, etc.
Right now we can trace people back to those that traveled in West Africa so it make identification relatively easy.
But once the disease spreads to the point where people who haven't been to West Africa start getting it, then what are we going to do? Quarantine everyone who starts showing the most basic Ebola symptoms, which are high fever and abdominal pains?
And if Ebola starts spreading, what will we do with people with suspected Ebola? Put them in the same facility as those with Ebola? That will drive people to not report their symptoms, increasing their chances of spreading it to others. It's the exact same thing that happened in Africa, and Americans will behave the same way because it's simple Game Theory. If I have Ebola symptoms, and that means I'm going to get stuck into a facility where other Ebola people are, I'd rather just pretend I don't have it because "chances of infection are so low, I probably don't have it."