The biggest reason why we could eradicate smallpox is because the smallpox virus could only infect humans. Thus, once we immunized the entire human population against it, the virus completely died out. Since there were no more live populations of the smallpox virus in the wild, there were no opportunities for the smallpox virus to mutate to get around our vaccines. Today the only living samples of smallpox are kept in a small number of highly secured research labs.
Most other infectious pathogens like viruses and bacteria can infect other organisms than humans. Ebola naturally infects bats. The flu virus can infect a whole bunch of animals. These populations of viruses infecting animals thus can continue to mutate and reproduce, eventually rendering existing human vaccines useless.
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