Shevchuk I. Development and analysis of population models of information spreading process

The dissertation is devoted to such subject area as information and communication space, in particular, the information spreading process in the society. For systems of ordinary differential equations, is analyzed of solutions. The conditions of the boundedness and non-negative of functions that model the information spreading process are presented in cases where analytical search of an analytical solution is impossible. The method of the small parameter of J. Poincare can be used for individual cases of models of information spreading process and allows to found approximate analytical solutions. In the dissertation, this approach is used for systems of differential equations with stationary parameters and for systems with non-stationary parameters. Information on the dynamics of processes in the information and communicative space is important to the problems of practice and obtained through the theory of stability. The stability analysis is carried out at the first approximation in the special points because the nonlinearity of the models that were selected for research in the dissertation. In particular, the necessary and for the special cases of the basis models, sufficient conditions of stability for the first approximation in the neighbourhood of special points were formulated for the systems of differential equations with stationary parameters and for the systems with disturbing non-stationary parameters that. The obtained results allow us to determine the admissible areas for the model parameters, the values of which will guarantee the asymptotic the root mean square stability at the first approximation in the neighbourhood of stationary points. Particular attention in the dissertation is paid to the evaluation of the parameters of systems of differential equations. Algorithms for constructing optimal and guaranteed estimates of parameters were formulated for the cases of continuous observations and discrete observations of the dynamics of the information spreading process. Similar results were obtained for systems of difference equations. These approaches for constructing optimal estimates bases on Bellman function and Kalman-Bussi filter. Another important task in the analysis of models of information spreading process is the estimation of external influences. This problem is investigated in the dissertation paper for the model of spreading two types of information messages and for special cases which characterized by the known data analysis of system parameters. The statements of the problems of finding the estimates of external influence are given for the case of observation by the number of supporters of one information flow and known parameters of the system, for the case of observing the number of supporters of both information flows and known system parameters for one equation and for the case of observing the number of supporters of both flows and known parameters of the system. Also, algorithms for constructing predictive estimates for the special case of a general model are proposed in this work. It allows predicting with a certain accuracy the behavior of the dynamics of the process of spreading information messages. The algorithms for obtaining optimal root mean square predictive estimation and guaranteed forecast estimation are presented. The example of finding the optimal root mean square estimation is given for the case of the propagation of one type of information. The algorithm for finding guaranteed predictive estimates is considered for an special case of representing the set of possible observational errors. The mathematical model for the spreading of one kind of information is researched. It is presented as a system of nonlinear differential equations with stationary parameters and is includes the dissemination of information by external and internal channels. The proposed model also contains mechanisms that affect the process of disseminating information: forgetting, two-step coverage of information and division of society into two homogeneous subgroups. The algorithm for obtaining averaged optimal mean square predictive estimation is presented for this model.