Stochastic models for two‐mode compartmental systems and their asymptotic reduction

We consider a class of population models, which will be named “two‐mode” and is suitable to represent phenomena characterized by the interaction of two compartment groups. The two groups, both containing the same number of compartments, may exhibit some difference in their behaviour (“mode”), described in terms of transition rates. We conventionally denote them as “active” and “inactive” compartments. Exchanges between compartments are modelled by a multidimensional jump Markov process. Within both active and inactive subsystems the exchanges of individuals may occur in all possible ways. On the other hand, each active compartment communicates in both directions with just one inactive compartment. Such exchanges have indeed to be considered as transitions of “status” (mode) for the individuals; with this interpretation, a special feature of our model is that they occur much faster than exchanges within each subsystem. Possible non‐linear interactions among compartments (for instance due to crowding effects) are included in the model by letting the exchange rates depend on the total numbers of individuals in the pair of corresponding active and inactive compartments. Birth/reproduction and death phenomena are also accounted for. We show that, as the rate of mode transition diverges, an instantaneous relationship (in the mean value sense) links the number of individuals Xakt and Xikt in each pair of active and inactive compartments. A stronger result is proved to hold for the “pooled variables” Xakt+ Xakt, which converge weakly (in the sense of distributions on the path space) to the solution of a reduced n‐dimensional model. In the paper, Section 1 is devoted to discussing the peculiar features of the model and to motivate its relevance in a variety of possible applications fields. Section 2 contains a formal presentation of the original and reduced models, along with an intermediate model (equivalent to the original one) which helps in clarifying connections between the above‐mentioned ones. The aim of Section 2 is also to ascertain that these models are well defined. In Section 3 the convergence results are proved.