Outbreaks of Rift Valley fever (RVF) in eastern Africa have previously occurred following specific rainfall dynamics and flooding events that appear to support the emergence of large numbers of mosquito vectors. As such, transmission of the virus is considered to be sensitive to environmental conditions and therefore changes in climate can impact the spatiotemporal dynamics of epizootic vulnerability. Epidemiological information describing the methods and parameters of RVF transmission and its dependence on climatic factors are used to develop a new spatio-temporal mathematical model that simulates these dynamics and can predict the impact of changes in climate. The Liverpool RVF (LRVF) model is a new dynamic, process-based model driven by climate data that provides a predictive output of geographical changes in RVF outbreak susceptibility as a result of the climate and local livestock immunity. This description of the multi-disciplinary process of model development is accessible to mathematicians, epidemiological modellers and climate scientists, uniting dynamic mathematical modelling, empirical parameterisation and state-of-the-art climate information.