There are a number of illnesses that are interspecies, including colds, rabies, West Nile virus, Lyme's disease, plague, bird flu, hanta virus etc. There is a disease common among parrots and some other birds, that I can't remember the name of, that people sometimes get from their parrots. Leprosy exists in wild populations of armadillos down in the American southwest, and in Mexico. People occasionally catch it from road-killed wild armadillos, sometimes from trying to help them, and sometimes from eating freshly killed ones. That's why one of our country's two leprosariums (leprosy hospitals) is located in Texas. The other one is in Hawaii. Fortunately, there is a cure for that disease now. Usually, a few people a year catch plague from wild rodents in the American west. There is a cure for that, too, but it doesn't always work. A few people a year catch the hanta virus from wild rodents, too, mostly out in the American west, but it is also present in other parts of the country.
So yes, we can make animals sick, and they can make us sick, but most of the time it doesn't happen, so don't stress about it.
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If a man does not keep pace with his companions, perhaps it is because he hears a different drummer.
Let him step to the music which he hears, however measured, or far away.--Henry David Thoreau