There are different ways, depending on the situation.
There are some diseases, like Newcastle virus, they basically advise people to keep their birds and poultry shielded from wild birds; and they may euthanize an entire farm full of poultry that has become infected. Because it spreads so rapidly that this is necessary. Wild birds will die from it of course, but they will not all die, and the ones with more resistance to the virus will survive and breed. So the outbreak will die out on its own over a period of months.
Now, diseases like rabies, hantavirus, and bubonic plague will be present in small numbers among wild animals; and the government takes samples of wild animals to see whether the disease is present. Rodents can carry these diseases, and if they find a population of wild rodents where there is an outbreak, they will warn people to stay away from the area, and they will kill off the infected population, and keep people out until the disease is gone from there.
Vaccinating feral cats is a good idea; but if the cat is already infected, euthanasia may be necessary. It is usually organizations of cat-lovers who will trap feral cats and spay or neuter and vaccinate them, and either look for homes or release them, preferably in some area where they can help control the rodent population. True feral cats are extremely hard to tame though.