Uniform disease persistence is investigated for the time evolution of bluetongue, a viral disease in sheep and cattle that is spread by midges as vectors. The model is a system of several delay differential equations. As in many other infectious disease models, uniform disease persistence occurs if the basic disease reproduction number for the whole system, R_0, exceeds one. However, since bluetongue affects sheep much more severely than cattle, uniform disease persistence can occur in two different scenarios which are distinguished by the disease reproduction number for the cattle-midge-bluetongue system without sheep, ~ R_0. If R_0 > 1 and ~ R_0 >1, bluetongue persists in cattle and midges even though it may eradicate the sheep relying on cattle as a reservoir. If R_0 > 1 > ~ R_0, bluetongue and all host and vector species coexist, and bluetongue does not eradicate the sheep because it cannot persist on midges and cattle alone. The two scenarios require different use of dynamical systems persistence theory.