The neglected tropical disease (NTD) visceral leishmaniasis (VL) has been targeted by the WHO for elimination as a public health problem on the Indian subcontinent by 2017 or earlier. To date there is a surprising scarcity of mathematical models capable of capturing VL disease dynamics, which are widely considered central to planning and assessing the efficacy of interventions. The few models that have been developed are examined, highlighting the necessity for better data to parameterise and fit these and future models. In particular, the characterisation and infectiousness of the different disease stages will be crucial to elimination. Modelling can then assist in establishing whether, when, and how the WHO VL elimination targets can be met.