In digital health applications, speech offers advantages over other physiological signals, in that it can be easily collected, transmitted, and stored using mobile and Internet of Things (IoT) technologies. However, to take full advantage of this positioning, speech-based machine learning models need to be deployed on devices that can have considerable memory and power constraints. These constraints are particularly apparent when attempting to deploy deep learning models, as they require substantial amounts of memory and data movement operations. Herein, we test the suitability of pruning and quantisation as two methods to compress the overall size of neural networks trained for a health-driven speech classification task. Key results presented on the Upper Respiratory Tract Infection Corpus indicate that pruning, then quantising a network can reduce the number of operational weights by almost 90 %. They also demonstrate the overall size of the network can be reduced by almost 95 %, as measured in MB, without affecting overall recognition performance.