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The dynamics of public attention to COVID-19 epidemic is studied. The level of public attention is described by the daily number of search requests in Google made by users from a given country. In the empirical part of the work, data on the number of requests and the number of infected cases for a number of countries are considered. It is shown that in all cases the maximum of public attention occurs earlier than the maximum daily number of newly infected individuals. Thus, for a certain period of time, the growth of the epidemics occurs in parallel with the decline in public attention to it. It is also shown that the decline in the number of requests is described by an exponential function of time. In order to describe the revealed empirical pattern, a mathematical model is proposed, which is a modification of the model of the decline in attention after a one-time political event. The model develops the approach that considers decision-making by an individual as a member of the society in which the information process takes place. This approach assumes that an individual’s decision about whether or not to make a request on a given day about COVID is based on two factors. One of them is an attitude that reflects the individual’s long-term interest in a given topic and accumulates the individual’s previous experience, cultural preferences, social and economic status. The second is the dynamic factor of public attention to the epidemic, which changes during the process under consideration under the influence of informational stimuli. With regard to the subject under consideration, information stimuli are related to epidemic dynamics. The behavioral hypothesis is that if on some day the sum of the attitude and the dynamic factor exceeds a certain threshold value, then on that day the individual in question makes a search request on the topic of COVID. The general logic is that the higher the rate of infection growth, the higher the information stimulus, the slower decreases public attention to the pandemic. Thus, the constructed model made it possible to correlate the rate of exponential decrease in the number of requests with the rate of growth in the number of cases. The regularity found with the help of the model was tested on empirical data. It was found that the Student’s statistic is 4.56, which allows us to reject the hypothesis of the absence of a correlation with a significance level of 0.01.

In the last quarter of the twentieth century, the nature of global demographic and economic development began to change rapidly: the continuously accelerating growth of the main characteristics that took place over the previous two hundred years was replaced by a sharp slowdown. In the context of these changes, the role of a long-term forecast of global dynamics is increasing. At the same time, the forecast should be based not on inertial projection of past trends into future periods, but on mathematical modeling of fundamental patterns of historical development. The article presents preliminary results of research on mathematical modeling and forecasting of global demographic and economic dynamics based on this approach. The basic dynamic equations reflecting this dynamics are proposed, the modification of these equations in relation to different historical epochs is justified. For each historical epoch, based on the analysis of the corresponding system of equations, a phase portrait was determined and its features were analyzed. Based on this analysis, conclusions were drawn about the patterns of world development in the period under review.

It is shown that mathematical description of technology development is important for modeling historical dynamics. A method for describing technological dynamics is proposed, on the basis of which the corresponding mathematical equations are proposed.

Three stages of historical development are considered: the stage of agrarian society (before the beginning of the XIX century), the stage of industrial society (XIX–XX centuries) and the modern era. The proposed mathematical model shows that an agrarian society is characterized by cyclical demographic and economic dynamics, while an industrial society is characterized by an increase in demographic and economic characteristics close to hyperbolic.

The results of mathematical modeling have shown that humanity is currently moving to a fundamentally new phase of historical development. There is a slowdown in growth and the transition of human society into a new phase state, the shape of which has not yet been determined. Various options for further development are considered.

The rheological behavior of aqueous suspensions based on nanoscale silicon dioxide particles strongly depends on the dynamic viscosity, which affects directly the use of nanofluids. The purpose of this work is to develop and validate models for predicting dynamic viscosity from independent input parameters: silicon dioxide concentration SiO₂, pH acidity, and shear rate $\gamma$. The influence of the suspension composition on its dynamic viscosity is analyzed. Groups of suspensions with statistically homogeneous composition have been identified, within which the interchangeability of compositions is possible. It is shown that at low shear rates, the rheological properties of suspensions differ significantly from those obtained at higher speeds. Significant positive correlations of the dynamic viscosity of the suspension with SiO₂ concentration and pH acidity were established, and negative correlations with the shear rate $\gamma$. Regression models with regularization of the dependence of the dynamic viscosity $\eta$ on the concentrations of SiO₂, NaOH, H₃PO₄, surfactant (surfactant), EDA (ethylenediamine), shear rate γ were constructed. For more accurate prediction of dynamic viscosity, the models using algorithms of neural network technologies and machine learning (MLP multilayer perceptron, RBF radial basis function network, SVM support vector method, RF random forest method) were trained. The effectiveness of the constructed models was evaluated using various statistical metrics, including the average absolute approximation error (MAE), the average quadratic error (MSE), the coefficient of determination $R^2$, and the average percentage of absolute relative deviation (AARD%). The RF model proved to be the best model in the training and test samples. The contribution of each component to the constructed model is determined. It is shown that the concentration of SiO₂ has the greatest influence on the dynamic viscosity, followed by pH acidity and shear rate γ. The accuracy of the proposed models is compared to the accuracy of models previously published. The results confirm that the developed models can be considered as a practical tool for studying the behavior of nanofluids, which use aqueous suspensions based on nanoscale particles of silicon dioxide.

Calcium is a major nutrient regulating metabolism in a plant. Deficiency of calcium results in a growth decline of plant tissues. Ca may be lost from forest soils due to acidic atmospheric deposition and tree harvesting. Plant-available calcium compounds are in the soil cation exchange complex and soil waters. Model of soil calcium dynamics linking it with the model of soil organic matter dynamics ROMUL in forest ecosystems is developed. ROMUL describes the mineralization and humification of the fraction of fresh litter which is further transformed into complex of partially humified substance (CHS) and then to stable humus (H) in dependence on temperature, soil moisture and chemical composition of the fraction (nitrogen, lignin and ash contents, pH). Rates of decomposition and humification being coefficients in the system of ordinary differential equations are evaluated using laboratory experiments and verified on a set of field experiments. Model of soil calcium dynamics describes calcium flows between pools of soil organic matter. Outputs are plant nutrition, leaching, synthesis of secondary minerals. The model describes transformation and mineralization of forest floor in detail. Experimental data for calibration model was used from spruсe forest of Bulgaria.

An effective regional policy in order to stabilize production is impossible without an analysis of the dynamics of economic processes taking place. This article focuses on developing a mathematical model reflecting the interaction of several economic agents with regard to their interests. Developing such a model and its study can be considered as an important step in solving theoretical and practical problems of managing growth.

To research dynamics of inhomogeneous polynucleotide DNA chain the Y-model with no dissipation term was used. Basing on this model using numerical methods calculations were carried out, which have shown the behaviour of nonlinear conformational excitation (kink), spreading along the inhomogeneous polynucleotide chain, consisting of two different homogeneous nucleotide sequences. As numerical analysis shows there are three ways of behaviour of the nonlinear kink excitation spreading along the DNA chain. After reaching the interface between two homogeneous sequences consisting of different types of bases kink can a) reflect, b) pass the interface with acceleration (increase its velocity), c) pass the interface with deceleration (decrease its velocity).

This paper is devoted to investigation of the swirl flows. Such flows are widely used in various industrial processes. Swirl flows can be accompanied by time-dependent effects, for example, precession of the vortex core. In turn, the large-scale fluctuations due to the precession of the vortex can cause damage of structures and reduce of equipment reliability. Thus, for engineering calculations approaches that sufficiently well described such flows are required. This paper presents the technique of swirl flows calculation, tested for CFD packages Fluent and SigmaFlow. A numerical simulation of several swirl flow test problems was carried out. Obtained results are compared with each other and with the experimental data.

We study the stochastic dynamics of the two-dimensional Hindmarsh–Rose model in the parametrical zone of coexisting stable equilibria and limit cycles. The phenomenon of noise-induced transitions between the attractors is investigated. Under the random disturbances, equilibrium and periodic regimes combine in bursting regime: the system demonstrates an alternation of small fluctuations near the equilibrium with high amplitude oscillations. This effect is analysed using the stochastic sensitivity function technique and a method of estimation of critical values for noise intensity is proposed.

Protection of aquatic biological resources in the coastal area has significant features (a large number of small fishing vessels, the dynamism of the situation, the use of coastal protection), by virtue of which stands in a class of applications. A mathematical model of protection designed for the determination of detection equipment and means of violators of the situation in order to ensure the function of deterrence of illegal activities. Resolves a tactical game-theoretic problem - find the optimal line patrol (parking) means of implementation (guard boats) and optimal removal of seats from the shore fishing violators. Using the methods of the theory of experimental design, linear regression models to assess the contribution of the main factors affecting the results of the simulation.

In order to enhance the sustainability and adequacy of the model is proposed to use the mechanism of rankings means of protection, based on the borders and the rank and Pareto allows to take into account the principles of protection and further means of protection. To account for the variability of the situation offered several scenarios in which it is advisable to perform calculations.

It is known that the sound speed in medium that contain highly compressible inclusions, e.g. air pores in an elastic medium or gas bubbles in the liquid may be significantly reduced compared to a homogeneous medium. Effective nonlinear parameter of medium, describing the manifestation of nonlinear effects, increases hundreds and thousands of times because of the large differences in the compressibility of the inclusions and the medium. Spatial change in the concentration of such inclusions leads to the variable local sound speed, which in turn calls the spatial-temporal redistribution of acoustic energy in the wave and the distortion of its temporal profiles and cross-section structure of bounded beams. In particular, focal areas can form. Under certain conditions, the sound channel is formed that provides waveguide propagation of acoustic signals in the medium with similar inclusions. Thus, it is possible to control spatial-temporal structure of acoustic waves with the introduction of highly compressible inclusions with a given spatial distribution and concentration. The aim of this work is to study the propagation of acoustic waves in a rubberlike material with non-uniform spatial air cavities. The main objective is the development of an adequate theory of such structurally inhomogeneous media, theory of propagation of nonlinear acoustic waves and beams in these media, the calculation of the acoustic fields and identify the communication parameters of the medium and inclusions with characteristics of propagating waves. In the work the evolutionary self-consistent equation with integro-differential term is obtained describing in the low-frequency approximation propagation of intense acoustic beams in a medium with highly compressible cavities. In this equation the secondary acoustic field is taken into account caused by the dynamics of the cavities oscillations. The method is developed to obtain exact analytical solutions for nonlinear acoustic field of the beam on its axis and to calculate the field in the focal areas. The obtained results are applied to theoretical modeling of a material with non-uniform distribution of strongly compressible inclusions.