I think I know what the OP is getting at, but I am just talking about pets... I had one cat for 19 years, and have had dogs all my life. They very rarely get "sick" the way we get sick (coughing, diarrhea, vomiting, fever, etc). Whenever they do, it seems to be because they "got into something" they shouldn't have. I think there are a few main reasons for this... 1) Lack of exposure to "the outside world" reduces the opportunity to "catch" something. People are "out in the world" touching objects that hundreds of people have touched in just the past 24 hours. We are within a few feet of ten or twenty people per day, maybe even 100-200 people if you work in a retail/hospitality industry or take mass transit daily. 2) Diet. Most pets eat the same food every day. They eat processed food that was prepared under controlled conditions and not handled by people, rather than prepared in small kitchens at restaurants and handled by multiple people, and they don't eat raw fruits and vegetables from fields like we do. 3) We also "let them die" more often, possibly causing evolution to "work better" than it has in humans, since (in the developed world) we save people at virtually any cost, people with weak immune systems are able to grow up and reproduce... Pets definitely get things like cancer though, quite frequently I think...
I am by no means an expert in this field, I'm just throwing a few ideas out there - feel free to shoot it down! :)