YES it's unhealthy to live with cockroaches.
Cockroach disease is something that many people are worried about when
they see a cockroach in their home. Cockroaches are known to carry
disease because they can live on just about anything, meaning they
often live on unsanitary substances.
Cockroaches are potential carriers of disease.
Cockroach disease can affect humans. Cockroaches are known to carry
diseases like dysentery, typhoid and poliomyelitis, as well as
gastroenteritis. Cockroaches can live just about anywhere, and can
live on fermenting products, septic dressings, hair, leather,
wallpaper, feces, rotting food, and more.
Because many of the products that cockroaches feed on are spoiled or
simply unsanitary, it's easy to see where cockroach disease comes
from. Many of the substances that they feed on are already
contaminated, and as the cockroach moves from one location to another
it spreads the disease along the way.
Not only can people get sick from the diseases that cockroaches
contaminate human living space with, many humans are allergic to
cockroach feces. Of course, no one exposes themselves to cockroach
disease and feces on purpose, but it can happen by inhaling particles
of dust in the air that has been left behind by the cockroach.
Reason why they carry disease is that they eat all sorts of unsanitary things. This can then spread to humans, if they are living together with them. some people are also allergic to cockroach feces.