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The gambiense form of sleeping sickness is a neglected tropical disease, which is presumed to be anthroponotic. However, the parasite persists in human populations at levels of considerable rarity and as such the existence of animal reservoirs has been posited. Clarifying the impact of animal host reservoirs on the feasibility of interrupting sleeping sickness transmission through interventions is a matter of urgency. We developed a mathematical model allowing for heterogeneous exposure of humans to tsetse, with animal populations that differed in their ability to transmit infections, to investigate the effectiveness of two established techniques, screening and treatment of at-risk populations, and vector control. Importantly, under both assumptions, an integrated approach of human screening and vector control was supported in high transmission areas. However, increasing the intensity of vector control was more likely to eliminate transmission, while increasing the intensity of human screening reduced the time to elimination. Non-human animal hosts played important, but different roles in HAT transmission, depending on whether or not they contributed as reservoirs. If they did not serve as reservoirs, sensitivity analyses suggested their attractiveness may instead function as a sink for tsetse bites. These outcomes highlight the importance of understanding the ecological and environmental context of sleeping sickness in optimizing integrated interventions, particularly for moderate and low transmission intensity settings.