Sleeping sickness (gambiensehuman African trypanosomiasis, gHAT) is a vector-borne disease targeted for global elimination of transmission (EoT) by 2030. There are, however, unknowns that have the potential to hinder the achievement and measurement of this goal. These include asymptomatic gHAT infections (inclusive of the potential to self-cure or harbour skin-only infections) and whether gHAT infection in animals can contribute to the transmission cycle in humans. Using modelling we explore how cryptic (undetected) transmission impacts the monitoring of progress towards as well as the achievement of the EoT goal. We have developed gHAT models that include either asymptomatic or animal transmission, and compare these to a baseline gHAT model without either of these transmission routes, to explore the potential role of cryptic infections on the EoT goal. Each model was independently calibrated using available historic human case data for 2000-2020 (obtained from the World Health Organization's HAT Atlas) which includes routine data from active and passive screening for five different health zones in the Democratic Republic of the Congo (DRC). Our results suggest that when matched to past case data, we estimated similar numbers of new human infections between model variants, although human infections were slightly higher in the models with cryptic infections. We simulated the continuation of screen-confirm-and-treat interventions and found that forward projections from the animal and asymptomatic transmission models produced lower probabilities of EoT than the baseline model. Simulation of a (as yet to be available) screen-and-treat strategy found that removing a parasitological confirmation step was predicted to have a more noticeable benefit to transmission reduction under the asymptomatic model compared to the others. Our simulations suggest vector control could greatly impact all transmission routes in all models, although this resource-intensive intervention should be carefully prioritised.