A new modelling approach, based on finite mixtures of multiple discrete distributions of different multiplicities, is proposed to fit data with a lot of periodic spikes in certain values. An EM algorithm is provided in order to ensure the models' ease-of-fit and then a simulation study is presented to show its efficiency. A numerical application with a real data set involving the length, measured in days, of inability to work after an accident occurs is treated. The main finding is that the model provides a very good fit when working week, calendar week and month multiplicities are taken into account.