Stochastic Model of Emotional Stress Development in the Educational Process
Authors: Solodov A. A. , Solodova E. A. , Trembach T. G. 
Published: November 2022

The aim of the study is the development of mathematical models that describe the reaction of the individual to the occurrence of stresses of various nature, including those that appear during the implementation of the educational process. The complexity of developing such a model is confirmed by the lack of theoretical results to substantiate the classic experiments of Holmes and Rae on the effect of stress on personality. In this regard, the task is to develop such a mathematical model that would allow not only to give a theoretical explanation of the experimental results used in the Holmes and Rae stress calculator, but also to become a tool for studying the effect of stress on a person in other conditions, including in the process of educational activities of both the teacher and the learner.

The research method consists in a mathematical description of the process of occurrence of stresses that develop over time, and it is believed that stresses occur at random times and are characterized by relative stress values indicated in the classical table of Holmes and Rae. The need to involve these results is that they allow us to confirm the correspondence of the developed theoretical mathematical models to already known practical results. The following main assumptions are accepted in the paper. It is believed that a person is exposed to stresses that can occur at random localized points in time and are interpreted as a sequence of points on the time axis, the number and location of which is random. The response of a person to a particular stress is described by a decreasing exponential function of three arguments – current time, random time of stress occurrence, and stress magnitude. The reaction of the individual to a sequence of stresses is the sum of the responses of the person to individual stresses, i.e. it is assumed that the personality exhibits the properties of linearity. In the process of developing a mathematical model, the distribution of the number of random stresses is substantiated according to the Poisson law, which is used to describe the occurrence of random events with a distinct discreteness. The paper introduces one of the key indexes - the coefficient of emotional load, equal to the ratio of the mathematical expectation of everyday stressful background and the mathematical expectation of the sum of this background and additional stress. The response of a person to a particular stress is described by an exponential response function widely used in natural science applications. The total relative value of processed, experienced, random relative values of stress, as well as their non-random mathematical expectations, is introduced into consideration.

The new results of the study are: – development of a stochastic mathematical model for the development of stresses over time, depending on the parameters included in the model. It is shown that the behavior of a person’s reaction to stress, predicted by a mathematical model, corresponds to the previously mentioned experimental results. – a study of the behavior in time of a person’s reaction to stress for a situation that was not previously considered and in which the person is affected by a single stress of great intensity, as well as regular emotional imposition.

In conclusion, it is noted that the developed model not only makes it possible to theoretically explain the experimental data, but also significantly expand the scope of the studied effect of stress on personality. Thus, it turned out to be possible to predict the impact of a single impact, as well as to indicate a way to account for periodic intentional exposure (emotional suppression). In addition, the results can be used in the study of emotional stresses in the educational process in order to predict and consider them in practical activities.