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This is a disease that is pretty uncommon in pets today: rabies. This dog is showing what is sometimes called "dumb" rabies which is characterized by progressive paralysis, including a slack jaw and drooling. This is in contrast to "furious rabies," which is associated with the aggressive behavior most often associated with the disease.
Rabies is easily preventable with vaccinations and as such pets that receive regular medical care, including yearly (or less frequently as prescribed by a vet) vaccines, will generally not contract the disease. However, rabies continues to be a disease that is carried and transmitted through wildlife. It is transmitted through saliva, and as such being bitten is the usual way in which people and unvaccinated pets can contract it. However, this is not always the case. Major carries of the disease include racoons, coyotes, foxes, and skunks.
When someone contracts rabies, the first symptoms are flu-like. If you are bitten by a wild animal, especially one of the major carriers, you should contact your doctor immediately. A series of injections will prevent the disease from taking hold. Once a person starts showing symptoms, however, it is unlikely they will survive. So, fast action is important.
There are numerous nations that have been declared rabies-free, most of which are islands as it is easier to control the disease in an island setting.
Main sources are the Centers for Disease Control and Prevention and Missouri Department of Conservation.