The Best Way to Head Off an Epidemic Is to Cut Off Travel to Affected Countries.
Hollywood loves a good epidemic (see "Outbreak" , "The Crazies" , "Contagion" , etc.). They usually involve a highly contagious, fast-killing strain of some sort that forces the government to quarantine large numbers of people — healthy and sick — in an effort to stop a biological apocalypse.
In reality, experts rarely recommend such extreme measures, as demonstrated by the 2014-2015 Ebola outbreak in West Africa. While American politicians from both parties called for travel restrictions, health officials were busy explaining all the reasons why that was actually a bad idea.
For one, such restrictions really don't work. In 2009 several countries banned flights to and from Mexico in response to a swine flu outbreak, but a subsequent study found this just delayed the arrival of the virus by about three days.
Not only are travel restrictions largely ineffective, but they might even make things worse. Such limits would make it really hard to get doctors, nurses and supplies to the affected countries. They would also complicate contact tracing, in which health workers seek out and test those who have come in contact with infected individuals. People would likely cross borders anyway by avoiding airports or lying about where they came from, making it nearly impossible for officials to retrace their movements.
The best way to contain an outbreak? Stop it at the source. Not too exciting, but hey, we'll leave that to the movies.