Only taking into consideration the interplay between processes occurring at different levels of a country can provide the complete social and geopolitical plot of its urban system. We study the interaction of the administrative structure and the geographical connectivity between cities with the help of a multiplex network approach. We found that a spatially-distributed geo-network imposes its own ranking to the hierarchical administrative network, while the latter redistributes the shortest paths between nodes in the geographical layer. Using both real demographic data of population censuses of the Republic of Kazakhstan and theoretical models, we show that in a country-scale urban network and for each specific city, the geographical neighbouring with highly populated areas is more important than its political setting. Furthermore, the structure of political subordination is instead crucial for the wealth of transportation network and communication between populated regions of the country.
Optimal control problems for mathematical models describing HIV infection dynamics in a human body are considered in the paper. An overview of current approaches to solution of control problems for models of HIV dynamics is presented for techniques related to construction of optimal programme (open loop) or positional (feedback) controls for various criteria of control process quality and is based on the Pontryagin's maximum principle and the Bellman's theory of dynamic programming, respectively. In the framework of the theory of positional di erential games of Krasovskii, there are constructive and e cient methods of synthesis of controls for di erent classes of dynamical systems. In this paper we present a formalization of a control problem for HIV infection considered as a corresponding positional di erential game. The control algorithm based on the method of extremal shift is applied to the ODEs model of HIV infection. The numerical implementation of the extremal shift method is developed to construct the control taking the system to a neighbourhood of a given trajectory using the information on the dynamics of the guides or leaders.Keywords: Optimal open-loop control, feedback control, positional di erential games, extremal shift method, mathematical model, treatment of the HIV infection. MSC 2010: 49J15,49N90,95B05,92C50
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