The Ebola virus disease is caused by the Ebola virus which belongs to the filoviridae virus family. The 2014 outbreaks were estimated to have caused over 11,000 fatalities. In this paper, we formulate and analyze a system of ordinary differential equations which incorporates disease relapse and reinfection. The Ebola model with disease relapse and reinfection is locally-asymptotically stable when the basic reproduction number is less than unity. The model exhibits in the presence of disease reinfection, the phenomenon of backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. The feasibility of backward bifurcation occurring increases with increasing values of both relapse and reinfection. The total number of new cases of Ebola-infected individuals increases with increasing values of the relapse and reinfection parameters. Further simulations show that Ebola transmission models that do not incorporate relapse and reinfection may under-estimate disease burden in the community. Similar under-estimation is observed in models that include only one infected and recovered classes. Using results obtained from sensitivity analysis indicates that Ebola (given disease relapse and reinfection) can be effectively curtailed in the community by using control measures with a high-effectiveness level. This strategy is more effective than either the moderate- or low-effectiveness levels.