Individually, cockroaches do not present many health risks. However, in large numbers, the danger of infection and sicknesses increase greatly. In large numbers, roaches can give you cholera, Typhoid fever, dysentery, and plague.
In large numbers, you can develop allergies to roaches. These allergies will lead to skin rashes, coughing, wheezing, ear infections, and sinus infections. In extreme cases, the allergy can make people allergic to pre-ground coffee. The amount of dust produced by roaches will trigger asthma attacks if there are enough of them.