We present `wake-cough', an application of wake-word spotting to coughs using a Resnet50 and the identification of coughers using i-vectors, for the purpose of a long-term, personalised cough monitoring system. Coughs, recorded in a quiet (73$\pm$5 dB) and noisy (34$\pm$17 dB) environment, were used to extract i-vectors, x-vectors and d-vectors, used as features to the classifiers. The system achieves 90.02\% accuracy when using an MLP to discriminate between 51 coughers using 2-sec long cough segments in the noisy environment. When discriminating between 5 and 14 coughers using longer (100 sec) segments in the quiet environment, this accuracy improves to 99.78% and 98.39% respectively. Unlike speech, i-vectors outperform x-vectors and d-vectors in identifying coughers. These coughs were added as an extra class to the Google Speech Commands dataset and features were extracted by preserving the end-to-end time-domain information in a trigger phrase. The highest accuracy of 88.58% is achieved in spotting coughs among 35 other trigger phrases using a Resnet50. Thus, wake-cough represents a personalised, non-intrusive cough monitoring system, which is power-efficient as on-device wake-word detection can keep a smartphone-based monitoring device mostly dormant. This makes wake-cough extremely attractive in multi-bed ward environments to monitor patients' long-term recovery from lung ailments such as tuberculosis (TB) and COVID-19.