Vector-borne diseases represent a threat to human and wildlife populations and mathematical models provide a means to understand and control epidemics involved in complex host-vector systems. The disease model studied here is a host-vector system with a relapsing class of host individuals, used to investigate tick-borne relapsing fever (TBRF). Equilibrium analysis is performed for models with increasing numbers of relapses and multiple hosts and the disease reproduction number, R0, is generalized to establish relationships with parameters that would result in the elimination of the disease. We show that host relapses in a single competent host-vector system is needed to maintain an endemic state. We show that the addition of an incompetent second host with no relapses increases the number of relapses needed for maintaining the pathogen in the first competent host system. Further, coupling of the system with hosts of differing competencies will always reduce R0, making it more difficult for the system to reach an endemic state.