AIDS is a chronic disease.

Incurable : (of a sick person or a disease) not able to be cured.
infectious : (of a disease or disease-causing organism) liable to be transmitted to people, organisms, etc. through the environment.
Contagious : (of a disease) spread from one person or organism to another, typically by direct contact.
Persistent : continuing to exist or occur over a prolonged period.
Chronic : (of an illness) persisting for a long time or constantly recurring.

Synonym of Chronic is Persistent