Rats spread the plague, just as mosquitoes spread malaria. They don't have to "have" the disease to carry it. Rats carry fleas, which bite you and give you the germs. Then you get infected. Mosquitoes don't have malaria. They carry an internal parasite which gives you malaria when you get a mosquito bite.
A genetic disease carrier is an individual who possesses one copy of a mutation that causes a disease only when two copies are present. Although carriers not affected by the disease unless the gene is codominant with the "normal" gene, two carriers can produce a child who has the disease.