Urban planning is progressively shifting away from car-centric design and towards public transport due to a combination of environmental, societal, and economic factors. One method to encourage public transport usage is the installation of traffic filters – interventions that restrict certain modes of transport (such as cars) from parts of the network while allowing others (such as buses or trams) through. Traffic filters are being increasingly adopted by planning authorities worldwide, making it important to model them and formally study their impact. To capture the nuanced traffic and behavioural dynamics at play, we propose a multi-modal, stochastic user equilibrium model with elastic demands. We prove that an equilibrium always exists in this formulation and demonstrate how it can be found in practice by implementing a gap function-based column generation algorithm. We then use this model to study optimal traffic filter placement under a range of objective functions.