Those illnesses are as wide ranging as the species you choose to look at.
Typically all will have internal and external parasites.
Canine types in the US may have parvo, distemper, rabies, heartworm etc and also skeletal issues such as hip dysplasia. Animals with bad structure or birth defects typically do not make it long with the herbivores becoming prey at a young age and the carnivores not being able to compete for food.
Other animals susceptible to distemper (for one example of a viral disease) include racoons, skunks, weasels. Feline diseases can be seen in feral cats and in wild types of cats.
Around here deer can suffer from brain worm and we see die offs of predators from rabies or distemper.
Right now leptospirosis is locally an issue and seems to have reduced the rodent population in a time when carnivores have already been killed by rabies so you'd expect the rodents to be booming.
You may have read about the bats dying in the US of some sort of contagious fungal disease.
http://www.newsday.com/news/local/wire/newyork/ny-bc-ny--batdie-off0719jul19,0,4160026.story
Wild animals are not particularly healthier than tame ones its just the wild ones that are not healthy end up as someone's lunch.
Hope this helps you!
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