The term for the spread of disease is "epidemiology." This field studies how diseases are distributed in populations and the factors that influence their spread. Another related term is "transmission," which specifically refers to the process by which a disease spreads from one individual to another.
It is your ability to spread disease from one person to another.
An epidemic in a regional area, if spread around the world it is called a pandemic.
A contagion is either a disease spread by contact, the spread of such a disease, or the spread of anything harmful, as if it were a disease.
cancer
An Epidemiologist studies the spread of disease.
A disease which is carried and spread by an agent (animal or microorganism) is a vector spread disease. Eg. Mosquitoes are the vectors for malaria.
Lyme disease is spread by ticks. Lyme disease can be pretty much everywhere.
(noun or noun adjunct) "The disease spread rapidly and there was no cure." "Measles is a viral disease that usually affects children." "The disease organism in many of the cases was a mutated bacteria."
The likely terms are contagious, infectious, communicable, or transmissible.
Epidemiologist is, I believe, the word you were looking for.
Rats that were infected by the disease and have died on the side of the street to easily spread the disease
No. It does not spread by ameba. Ameba is a protozoa. Lyme disease is a tick born disease.