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The second part presents numerical studies of the parameters of the lower ionosphere at altitudes of 40–90 km when heated by powerful high-frequency radio waves of various frequencies and powers. The problem statement is considered in the first part of the article. The main attention is paid to the interrelation between the energy and kinetic parameters of the disturbed $D$-region of the ionosphere in the processes that determine the absorption and transformation of the radio beam energy flux in space and time. The possibility of a significant difference in the behavior of the parameters of the disturbed region in the daytime and at nighttime, both in magnitude and in space-time distribution, is shown. In the absence of sufficiently reliable values of the rate constants for a number of important kinetic processes, numerical studies were carried out in stages with the gradual addition of individual processes and kinetic blocks corresponding at the same time to a certain physical content. It is shown that the energy thresholds for inelastic collisions of electrons with air molecules are the main ones. This approach made it possible to detect the effect of the emergence of a self-oscillating mode of changing parameters if the main channel for energy losses in inelastic processes is the most energy-intensive process — ionization. This effect may play a role in plasma studies using high-frequency inductive and capacitive discharges. The results of calculations of the ionization and optical parameters of the disturbed $D$-region for daytime conditions are presented. The electron temperature, density, emission coefficients in the visible and infrared ranges of the spectrum are obtained for various values of the power of the radio beam and its frequency in the lower ionosphere. The height-time distribution of the absorbed radiation power is calculated, which is necessary in studies of higher layers of the ionosphere. The influence on the electron temperature and on the general behavior of the parameters of energy losses by electrons on the excitation of vibrational and metastable states of molecules has been studied in detail. It is shown that under nighttime conditions, when the electron concentration begins at altitudes of about 80 km, and the concentration of heavy particles decreases by two orders of magnitude compared to the average $D$-region, large-scale gas-dynamic motion can develop with sufficient radio emission power The algorithm was developed based on the McCormack method and two-dimensional gas-dynamic calculations of the behavior of the parameters of the perturbed region were performed with some simplifications of the kinetics.

The COVID-19 pandemic has caused increased interest in mathematical models of the epidemic process, since only statistical analysis of morbidity does not allow medium-term forecasting in a rapidly changing situation.

Among the specific features of COVID-19 that need to be taken into account in mathematical models are the heterogeneity of the pathogen, repeated changes in the dominant variant of SARS-CoV-2, and the relative short duration of post-infectious immunity.

In this regard, solutions to a system of differential equations for a SIR class model with a heterogeneous duration of post-infectious immunity were analytically studied, and numerical calculations were carried out for the dynamics of the system with an average duration of post-infectious immunity of the order of a year.

For a SIR class model with a heterogeneous duration of post-infectious immunity, it was proven that any solution can be continued indefinitely in time in a positive direction without leaving the domain of definition of the system.

For the contact number $R_0 \leqslant 1$, all solutions tend to a single trivial stationary solution with a zero share of infected people, and for $R_0 > 1$, in addition to the trivial solution, there is also a non-trivial stationary solution with non-zero shares of infected and susceptible people. The existence and uniqueness of a non-trivial stationary solution for $R_0 > 1$ was proven, and it was also proven that it is a global attractor.

Also, for several variants of heterogeneity, the eigenvalues of the rate of exponential convergence of small deviations from a nontrivial stationary solution were calculated.

It was found that for contact number values corresponding to COVID-19, the phase trajectory has the form of a twisting spiral with a period length of the order of a year.

This corresponds to the real dynamics of the incidence of COVID-19, in which, after several months of increasing incidence, a period of falling begins. At the same time, a second wave of incidence of a smaller amplitude, as predicted by the model, was not observed, since during 2020–2023, approximately every six months, a new variant of SARS-CoV-2 appeared, which was more infectious than the previous one, as a result of which the new variant replaced the previous one and became dominant.

The paper constructs and studies a generalized model describing two-component systems of reaction – diffusion type with power nonlinearity, considering the influence of external noise. A methodology has been developed for analyzing the generalized model, which includes linear stability analysis, nonlinear stability analysis, and numerical simulation of the system’s evolution. The linear analysis technique uses basic approaches, in which the characteristic equation is obtained using a linearization matrix. Nonlinear stability analysis realized up to third-order moments inclusively. For this, the functions describing the dynamics of the components are expanded in Taylor series up to third-order terms. Then, using the Novikov theorem, the averaging procedure is carried out. As a result, the obtained equations form an infinite hierarchically subordinate structure, which must be truncated at some point. To achieve this, contributions from terms higher than the third order are neglected in both the equations themselves and during the construction of the moment equations. The resulting equations form a set of linear equations, from which the stability matrix is constructed. This matrix has a rather complex structure, making it solvable only numerically. For the numerical study of the system’s evolution, the method of variable directions was chosen. Due to the presence of a stochastic component in the analyzed system, the method was modified such that random fields with a specified distribution and correlation function, responsible for the noise contribution to the overall nonlinearity, are generated across entire layers. The developed methodology was tested on the reaction – diffusion model proposed by Barrio et al., according to the results of the study, they showed the similarity of the obtained structures with the pigmentation of fish. This paper focuses on the system behavior analysis in the neighborhood of a non-zero stationary point. The dependence of the real part of the eigenvalues on the wavenumber has been examined. In the linear analysis, a range of wavenumber values is identified in which Turing instability occurs. Nonlinear analysis and numerical simulation of the system’s evolution are conducted for model parameters that, in contrast, lie outside the Turing instability region. Nonlinear analysis found noise intensities of additive noise for which, despite the absence of conditions for the emergence of diffusion instability, the system transitions to an unstable state. The results of the numerical simulation of the evolution of the tested model demonstrate the process of forming spatial structures of Turing type under the influence of additive noise.

Laser interference microscopy was used to study morphology and intracellular dynamics of erythrocytes, neurons and mast cells. We have found that changes of the local refractive index (RI) of cells have regular components that relate to the cooperative processes in the cellular submembrane and centre regions. We have shown that characteristic frequencies of RI dynamics differ for various cell types and can be used as markers of specific cellular processes.

The dynamics of coupled elements’ ensembles are investigated in the context of description of spatio-temporal processes in the myocardium. Basic element is map-based model constructed by simplification and reduction of Luo-Rudy model. In particular, capabilities of the model in replication of different regimes of cardiac activity are shown, including excitable and oscillatory regimes. The dynamics of 1D and 2D lattices of coupled oscillatory elements with a random distribution of individual frequencies are considered. Effects of cluster synchronization and transition to global synchronization by increasing of coupling strength are discussed. Impulse propagation in the chain of excitable cells has been observed. Analysis of 2D lattice of excitable elements with target and spiral waves have been made. The characteristics of the spiral wave has been analyzed in depending on the individual parameters of the map and coupling strength between elements of the lattice. A study of mixed ensembles consisting of excitable and oscillatory elements with a gradient changing of the properties have been made, including the task for description of normal and pathological activity of the sinoatrial node.

An approach to numerical analysis of thermo-hydraulic processes proceeding in a fast-neutron reactor is described in the given article. The description covers physical models, numerical schemes and geometry simplifications accepted in the computational model. Steady-state and dynamic regimes of reactor operation are considered. The steady-state regimes simulate the reactor operation at nominal power. The dynamic regimes simulate the shutdown reactor cooling by means of the heat-removal system.

Simulation of thermo-hydraulic processes is carried out in the FlowVision CFD software. A mathematical model describing the coolant flow in the first loop of the fast-neutron reactor was developed on the basis of the available geometrical model. The flow of the working fluid in the reactor simulator is calculated under the assumption that the fluid density does not depend on pressure, with use a $k–\varepsilon$ turbulence model, with use of a model of dispersed medium, and with account of conjugate heat exchange. The model of dispersed medium implemented in the FlowVision software allowed taking into account heat exchange between the heat-exchanger lops. Due to geometric complexity of the core region, the zones occupied by the two heat exchangers were modeled by hydraulic resistances and heat sources.

Numerical simulation of the coolant flow in the FlowVision software enabled obtaining the distributions of temperature, velocity and pressure in the entire computational domain. Using the model of dispersed medium allowed calculation of the temperature distributions in the second loops of the heat exchangers. Besides that, the variation of the coolant temperature along the two thermal probes is determined. The probes were located in the cool and hot chambers of the fast-neutron reactor simulator. Comparative analysis of the numerical and experimental data has shown that the developed mathematical model is correct and, therefore, it can be used for simulation of thermo-hydraulic processes proceeding in fast-neutron reactors with sodium coolant.

By numerical simulation methods the interaction processes of oscillating soliton (breather) with a 180-degree Neel domain wall in the framework of a (2 + 1)-dimensional supersymmetric O(3) nonlinear sigma model is studied. The purpose of this paper is to investigate nonlinear evolution and stability of a system of interacting localized dynamic and topological solutions. To construct the interaction models, were used a stationary breather and domain wall solutions, where obtained in the framework of the two-dimensional sine-Gordon equation by adding specially selected perturbations to the A3-field vector in the isotopic space of the Bloch sphere. In the absence of an external magnetic field, nonlinear sigma models have formal Lorentz invariance, which allows constructing, in particular, moving solutions and analyses the experimental data of the nonlinear dynamics of an interacting solitons system. In this paper, based on the obtained moving localized solutions, models for incident and head-on collisions of breathers with a domain wall are constructed, where, depending on the dynamic parameters of the system, are observed the collisions and reflections of solitons from each other, a long-range interactions and also the decay of an oscillating soliton into linear perturbation waves. In contrast to the breather solution that has the dynamics of the internal degree of freedom, the energy integral of a topologically stable soliton in the all experiments the preserved with high accuracy. For each type of interaction, the range of values of the velocity of the colliding dynamic and topological solitons is determined as a function of the rotation frequency of the A3-field vector in the isotopic space. Numerical models are constructed on the basis of methods of the theory of finite difference schemes, using the properties of stereographic projection, taking into account the group-theoretical features of constructions of the O(N) class of nonlinear sigma models of field theory. On the perimeter of the two-dimensional modeling area, specially developed boundary conditions are established that absorb linear perturbation waves radiated by interacting soliton fields. Thus, the simulation of the interaction processes of localized solutions in an infinite two-dimensional phase space is carried out. A software module has been developed that allows to carry out a complex analysis of the evolution of interacting solutions of nonlinear sigma models of field theory, taking into account it’s group properties in a two-dimensional pseudo-Euclidean space. The analysis of isospin dynamics, as well the energy density and energy integral of a system of interacting dynamic and topological solitons is carried out.

The short-term prediction of road traffic condition is one of the main tasks of transportation modelling. The main purpose of which are traffic control, reporting of accidents, avoiding traffic jams due to knowledge of traffic flow and subsequent transportation planning. A number of solutions exist — both model-driven and data driven had proven to be successful in capturing the dynamics of traffic flow. Nevertheless, most space-time models suffer from high mathematical complexity and low efficiency. Artificial Neural Networks, one of the prominent datadriven approaches, show promising performance in modelling the complexity of traffic flow. We present a neural network architecture for traffic flow prediction on a real-world road network graph. The model is based on the combination of a recurrent neural network and graph convolutional neural network. Where a recurrent neural network is used to model temporal dependencies, and a convolutional neural network is responsible for extracting spatial features from traffic. To make multiple few steps ahead predictions, the encoder-decoder architecture is used, which allows to reduce noise propagation due to inexact predictions. To model the complexity of traffic flow, we employ multilayered architecture. Deeper neural networks are more difficult to train. To speed up the training process, we use skip-connections between each layer, so that each layer teaches only the residual function with respect to the previous layer outputs. The resulting neural network was trained on raw data from traffic flow detectors from the US highway system with a resolution of 5 minutes. 3 metrics: mean absolute error, mean relative error, mean-square error were used to estimate the quality of the prediction. It was found that for all metrics the proposed model achieved lower prediction error than previously published models, such as Vector Auto Regression, LSTM and Graph Convolution GRU.

The conjugate problem of disk movement into gas-filled volume of the spring-type safety valve is solved. The questions of determining the physically correct value of the disk initial lift are considered. The review of existing approaches and methods for solving of such type problems is conducted. The formulation of the problem about the valve actuation when the vessel pressure rises and the mathematical model of the actuation processes are given. A special attention to the binding of physical subtasks is paid. Used methods, numerical schemes and algorithms are described. The mathematical modeling is performed on basе the fundamental system of differential equations for viscous gas movement with the equation for displacement of disk valve. The solution of this problem in the axe symmetric statement is carried out numerically using the finite volume method. The results obtained by the viscous and inviscid models are compared. In an inviscid formulation this problem is solved using the Godunov scheme, and in a viscous formulation is solved using the Kurganov – Tadmor method. The dependence of the disk displacement on time was obtained and compared with the experimental data. The pressure distribution on the disk surface, velocity profiles in the cross sections of the gap for different disk heights are given. It is shown that a value of initial drive lift it does not affect on the gas flow and valve movement part dynamic. It can significantly reduce the calculation time of the full cycle of valve work. Immediate isotahs for various elevations of the disk are presented. The comparison of jet flow over critical section is given. The data carried out by two numerical experiments are well correlated with each other. So, the inviscid model can be applied to the numerical modeling of the safety valve dynamic.

The article reports the findings of an investigation into pollution migration processes at the municipal solid waste (MSW) landfill located in the water protection zone of Lake Seliger (Tver Region). The distribution of pollutants is investigated and migration parameters are determined in field and laboratory conditions at the landfill site. A mathematical model describing physical and chemical processes of substance migration in soil strata is constructed. Pollutant migration is found to be due to a variety of factors. The major ones, having a significant impact on the migration of MSW ingredients and taken into account mathematically, include convective transport, diffusion and sorption processes. A modified mathematical model differs from its conventional counterparts by considering a number of parameters reflecting the decrease in the concentration of ammonium and nitrate nitrogen ions in ground water (transpiration by plant roots, dilution with infiltration waters, etc.). An analytical solution to assess the pollutant spread from the landfill is presented. The mathematical model provides a set of simulation models helping to obtain a computational solution of specific problems, vertical and horizontal migration of substances in the underground flow. Numerical experiments, analytical solutions, as well as field and laboratory data was studied the dynamics of pollutant distribution in the object under study up to the lake. A long-term forecast for the spread of landfill pollution is made. Simulation experiments showed that some zones of clean groundwater interact with those of contaminated groundwater during the pollution migration from the landfill, each characterized by a different pollutant content. The data of a computational experiments and analytical calculations are consistent with the findings of field and laboratory investigations of the object and give grounds to recommend the proposed models for predicting pollution migration from a landfill. The analysis of the pollution migration simulation allows to substantiate the numerical estimates of the increase in $NH_4^+$ and $NO_3^-$ ion concentration with the landfill operation time. It is found that, after 100 years following the landfill opening, toxic filtrate components will fill the entire pore space from the landfill to the lake resulting in a significant deterioration of the ecosystem of Lake Seliger.