The key to understanding this is the lack of article. 'Disease', without an article, is an uncountable noun. It's a general term for infection, 'bugs', bacteria/virus/pathogens e.g. Rats carry disease. 'A disease' is specific - it refers to a particular disorder. If you are not vaccinated, you may get a disease.

We use different collocations: "get sick", but "contract a disease". "Get disease" sounds like a learner error.

carry disease 或 carry dieseases = 病症传染. get = 得. 正確語法 get a disease 或 get diseases. 原句是单數 因此不能用 "get" .

"carry a disease" means that the thing (animal, person) is contagious and can give the disease to others. So, they're taking the animals to the hospitals so the animals (which already have the disease) do not spread it around.