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Abstract

We model the motion of a substance in a channel of a network that consists of chain of (i) nodes of the network and (ii) edges that connect the nodes and form the way for motion of the substance. The nodes of the channel can have different "leakage", i.e., some amount of the substance can leave the channel at a node and the rate of leaving can be different for the different nodes of the channel. The nodes close to the end of the channel for some (design or other) reason may be more "attractive" for the substance in comparison to the nodes around the incoming node of the channel. We discuss channels containing infinite or finite number of nodes. The main outcome of the model is the distribution of the substance along the nodes. Two regimes of functioning of the channels are studied: stationary regime and non-stationary regime. The distribution of the substance along the nodes of the channel for the case of stationary regime is a distribution with a very long tail that contains as particular case the Waring distribution (for channel with infinite number of nodes) or the truncated Waring distribution (for channel with finite number of nodes). In the non-stationary regime of functioning of the channel one observes an exponential increase or exponential decrease of the amount of substance in the nodes. However the asymptotic distribution of the substance among the nodes of the channel in this regime remains stationary.
The studied model is applied to the case of migration of humans through a migration channel consisting of chain of countries. In this case the model accounts for the number of migrants entering the channel through the first country of the channel; permeability of the borders between the countries; possible large attractiveness of some countries of the channel; possibility for migrants to obtain permission to reside in a country of the channel. The main outcome of the model is the distribution of migrants along the countries of the channel. We discuss the conditions for concentration of migrants in selected country of the channel. Finally two scenarios of changes of conditions of the functioning of the channel are discussed. It is shown that from the point of view of decreasing of the number of migrants in the countries of the channel it is more effective to concentrate efforts on preventing the entrance of migrants in the first country of the channel when compared to concentration of efforts on decrease of permeability of the borders between the countries of the channel. (C) 2017 Elsevier B.V. All rights reserved.