It depends on a lot of factors -- how many people will catch the disease, assuming no one is vaccinated, per infected individual is the primary one). I think most livestock diseases are aerosols (foot and mouth disease, etc.), and so are a lot of human diseases, but thankfully not AIDS.

The problem is the standard models of herd immunity depend on some simplifying assumptions which may be true in livestock, but are not true in humans -- specifically, that it's a homogenous population with equal chance of giving any other individual the disease, and that vaccines will be distributed fairly randomly as well.

If you have backwater villages which never get vaccinated, and are highly interconnected, you can end up with persistent reservoirs of disease, even if it's only 1% of your overall population. Then, you can have periodic epidemics when an individual from that area travels to an unvaccinated area elsewhere (because compliance rates on vaccination go down once the disease is "cured" in a population).

(I know about this from drinking with some of the UN/WHO vaccination people in Afghanistan)