Guaranteed Prediction Estimates of Solving Systems of Differential Equations with Gompertz Dynamics under Observations at Discrete Time Instants

Abstract

The mathematical model of information spread in a social medium is analyzed. It is assumed that in a sociocommunicative space there spread n types of information messages different in content. The number of individuals spreading one of the types of information messages is a key indicator of the model dynamics. Information messages spread through internal (interpersonal communication) and external (media influence) flows. The model is presented in the form of the system of n Gompertz nonlinear differential equations. It is appropriate to apply such models in practical problems of analyzing an information spread in a social medium dynamics of which is fast growing in time. Having a nonlinear right part such models claim to be an adequate representation of processes in a subject area. One of the practical important problems which occur while analyzing processes of information spread in a social medium is the problem of finding prediction estimates of such processes dynamics. For the systems of Gompertz differential equations this problem becomes nontrivial due to natural logarithms in the right-hand sides of these equations. The problem of finding the guaranteed prediction estimates of vectors is formulated. For a particular case of this problem with discrete observations there were proposed the efficient algorithms for finding guaranteed and approximate guaranteed prediction estimates of state and error vectors of prediction guaranteed estimates. As example there are presented results of finding the guaranteed prediction estimates of dynamics of mathematical model of one form information spread in a social medium. Results of numerical computer experiment demonstrate practical opportunities of this scheme. The proposed technique can be used for development of decision support systems for analyzing processes in sociocommunicative space.