There are a number of ways to induce various maladies to create animal models of diseases. Some ways are genetic, some environmental, some through selective breeding, some through nutrition, and many other ways. Most animal research however doesn’t use primates (aside from humans). Mice are often the go to for mammalian research. Their genome is well understood and relatively easily modifiable. For instance, we can modify a gene that makes them less sensitive to insulin and thereby they effectively have type II diabetes. Sometimes some animals naturally have genetic abnormalities that make them good animal models for human diseases. Golden retrievers actually experience something very similar to Duchenne Muscular Dystrophy, and through selective breeding, a colony of dystrophic dogs was managed to study the impacts of treatments that might be viable for humans. Sometimes we also find animals in the wild too. There’s a species of monkeys, I think in Central or South America, that are spontaneously hypertensive and they’ve been used to better understand high blood pressure in humans. For most common human diseases though, you can almost guarantee an animal model has been found or developed. Just type in the disease followed by “animal model” in Google and you’ll find fascinating stuff.