Is there a term for a disease that is spread ONLY from person to person?

These diseases are known as anthroponoses (adjective form: anthroponotic)

Anthroponoses (Greek “anthrópos” = man, “nosos” = disease) are diseases transmissible from human to human. Examples include rubella, smallpox, diphtheria, gonorrhea, ringworm (Trichophyton rubrum), and trichomoniasis.

NVZ also provided this link to anthroponoses in my comments section.

Issues from Comments Section below (addressed)

1. Janus mentioned this might not be exactly correct in the comments. I want to explain why this is definitely the correct word even if we are being meticulous about what the OP meant exactly. And there is no offense to Janus. None at all. I just want to clear it up.

Zoonoses [that were transmissible to humans] were originally called anthropozoonoses, and the reverse were called, zooanthroponoses [diseases from humans to animals]. There was a consensus that this caused confusion because scientists were mixing the 2 varieties and so they came up with the usage of the word: anthroponoses - link to Google Books

In addition, this CDC Appendix of disease types clearly lists the category: Anthroponoses (of which each one is a human to human only vector). In fact they specifically mention on that page, for those diseases which could cause confusion, that it is of the human genotype only variety. An example: Giardiasis; because it can affect both humans and animals but there is a human genotype that is only transmitted from human to human and this would known as an anthronponotic disease of Giardiasis.

Sources:

CDC: Hubálek Z. Emerging Human Infectious Diseases: Anthroponoses, Zoonoses, and Sapronoses. Emerging Infectious Diseases. 2003;9(3):403-404. doi:10.3201/eid0903.020208.

National Research Council, Division on Earth and Life Studies, Commission on Geosciences, Environment and Resources, Board on Atmospheric Sciences and Climate, Committee on Climate, Ecosystems, Infectious Diseases, and Human Health National Academies Press, May 29, 2001 - Science - 160 pages

CFSPH Technical Fact Sheets, Giardiasis at http://www.cfsph.iastate.edu/DiseaseInfo/default.htm

We also have two other close etymological cousins:

Zoonoses (Greek “zoon” = animal) are diseases transmissible from living animals to humans.

Sapronoses (Greek “sapros” = decaying; “sapron” means in ecology a decaying organic substrate) are human diseases transmissible from abiotic environment (soil, water, decaying plants, or animal corpses, excreta, and other substrata)