We consider a Boltzmann-like model of outgassing and contamination in a three-dimensional region V= V1∪V2∪V3. V1 is the region where the contaminant particles are produced, and V2 is the region where such particles migrate and interact with some inert gas. V3 is where contamination takes place because of the particles emanating from V2. In each of the three regions, the behaviour of the contaminant particles is represented by means of a Boltzmann-like equation. We show that such a problem has a unique positive strict solution, belonging to a suitable L1 Banach space X. Finally, a system of ordinary differential equations is derived which gives the evolution of the total number of contaminant particles in each of the three regions.