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In this work, we develop a model of the immune response to respiratory viral infections taking into account some particular properties of the SARS-CoV-2 infection. The model represents a system of ordinary differential equations for the concentrations of epithelial cells, immune cells, virus and inflammatory cytokines. Conventional analysis of the existence and stability of stationary points is completed by numerical simulations in order to study dynamics of solutions. Behavior of solutions is characterized by large peaks of virus concentration specific for acute respiratory viral infections.

At the first stage, we study the innate immune response based on the protective properties of interferon secreted by virus-infected cells. On the other hand, viral infection down-regulates interferon production. Their competition can lead to the bistability of the system with different regimes of infection progression with high or low intensity. In the case of infection outbreak, the incubation period and the maximal viral load depend on the initial viral load and the parameters of the immune response. In particular, increase of the initial viral load leads to shorter incubation period and higher maximal viral load.

In order to study the emergence and dynamics of cytokine storm, we consider proinflammatory cytokines produced by cells of the innate immune response. Depending on parameters of the model, the system can remain in the normal inflammatory state specific for viral infections or, due to positive feedback between inflammation and immune cells, pass to cytokine storm characterized by excessive production of proinflammatory cytokines. Furthermore, inflammatory cell death can stimulate transition to cytokine storm. However, it cannot sustain it by itself without the innate immune response. Assumptions of the model and obtained results are in qualitative agreement with the experimental and clinical data.

The study of the Volterra model describing the competition of three types is carried out. The corresponding system of first-order differential equations with a quadratic right-hand side, after a change of variables, reduces to a system with eight parameters. Two of them characterize the growth rates of populations; for the first species, this parameter is taken equal to one. The remaining six coefficients define the species interaction matrix. Previously, in the analytical study of the so-called symmetric model [May, Leonard, 1975] and the asymmetric model [Chi, Wu, Hsu, 1998] with growth factors equal to unity, relations were established for the interaction coefficients, under which the system has a one-parameter family of limit cycles. In this paper, we carried out a numerical-analytical study of the complete system based on a cosymmetric approach, which made it possible to determine the ratios for the parameters that correspond to families of equilibria. Various variants of oneparameter families are obtained and it is shown that they can consist of both stable and unstable equilibria. In the case of an interaction matrix with unit coefficients, a multicosymmetry of the system and a two-parameter family of equilibria are found that exist for any growth coefficients. For various interaction coefficients, the values of growth parameters are found at which periodic regimes are realized. Their belonging to the family of limit cycles is confirmed by the calculation of multipliers. In a wide range of values that violate the relationships under which the existence of cycles is ensured, a slow oscillatory establishment, typical of the destruction of cosymmetry, is obtained. Examples are given where a fixed value of one growth parameter corresponds to two values of another parameter, so that there are different families of periodic regimes. Thus, the variability of scenarios for the development of a three-species system has been established.

The development of a viral infection in the organism is a complex process which depends on the competition race between virus replication in the host cells and the immune response. To study different regimes of infection progression, we analyze the general mathematical model of immune response to viral infection. The model consists of two ODEs for virus and immune cells non-dimensionalized concentrations. The proliferation rate of immune cells in the model is represented by a bell-shaped function of the virus concentration. This function increases for small virus concentrations describing the antigen-stimulated clonal expansion of immune cells, and decreases for sufficiently high virus concentrations describing down-regulation of immune cells proliferation by the infection. Depending on the virus virulence, strength of the immune response, and the initial viral load, the model predicts several scenarios: (a) infection can be completely eliminated, (b) it can remain at a low level while the concentration of immune cells is high; (c) immune cells can be essentially exhausted, or (d) completely exhausted, which is accompanied (c, d) by high virus concentration. The analysis of the model shows that virus concentration can oscillate as it gradually converges to its equilibrium value. We show that the considered model can be obtained by the reduction of a more general model with an additional equation for the total viral load provided that this equation is fast. In the case of slow kinetics of the total viral load, this more general model should be used.

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 course and outcome of the battle is largely dependent on the morale of the troops, characterized by the percentage of loss in killed and wounded, in which the troops still continue to fight. Every fight is a psychological act of ending his rejection of one of the parties. Typically, models of battle psychological factor taken into account in the decision of Lanchester equations (the condition of equality of forces, when the number of one of the parties becomes zero). It is emphasized that the model Lanchester type satisfactorily describe the dynamics of the battle only in the initial stages. To resolve this contradiction is proposed to use a modification of Lanchester's equations, taking into account the fact that at any moment of the battle on the enemy firing not affected and did not abandon the battle fighters. The obtained differential equations are solved by numerical method and allow the dynamics to take into account the influence of psychological factor and evaluate the completion time of the conflict. Computational experiments confirm the known military theory is the fact that the fight usually ends in refusal of soldiers of one of the parties from its continuation (avoidance of combat in various forms). Along with models of temporal and spatial dynamics proposed to use a modification of the technology features of the conflict of S. Skaperdas, based on the principles of combat. To estimate the probability of victory of one side in the battle takes into account the interest of the maturing sides of the bloody casualties and increased military superiority.

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 paper reviews possibilities to predict buckling of thin cylindrical shells with non-destructive techniques during operation. It studies shallow shells made of high strength materials. Such structures are known for surface displacements exceeding the thickness of the elements. In the explored shells relaxation oscillations of significant amplitude can be generated even under relatively low internal stresses. The problem of the cylindrical shell oscillation is mechanically and mathematically modeled in a simplified form by conversion into an ordinary differential equation. To create the model, the researches of many authors were used who studied the geometry of the surface formed after buckling (postbuckling behavior). The nonlinear ordinary differential equation for the oscillating shell matches the well-known Duffing equation. It is important that there is a small parameter before the second time derivative in the Duffing equation. The latter circumstance enables making a detailed analysis of the obtained equation and describing the physical phenomena — relaxation oscillations — that are unique to thin high-strength shells.

It is shown that harmonic oscillations of the shell around the equilibrium position and stable relaxation oscillations are defined by the bifurcation point of the solutions to the Duffing equation. This is the first point in the Feigenbaum sequence to convert the stable periodic motions into dynamic chaos. The amplitude and the period of relaxation oscillations are calculated based on the physical properties and the level of internal stresses within the shell. Two cases of loading are reviewed: compression along generating elements and external pressure.

It is highlighted that if external forces vary in time according to the harmonic law, the periodic oscillation of the shell (nonlinear resonance) is a combination of slow and stick-slip movements. Since the amplitude and the frequency of the oscillations are known, this fact enables proposing an experimental facility for prediction of the shell buckling with non-destructive techniques. The following requirement is set as a safety factor: maximum load combinations must not cause displacements exceeding specified limits. Based on the results of the experimental measurements a formula is obtained to estimate safety against buckling (safety factor) of the structure.

Rugose spiraling whitefly (RSW) is one of the major pests which affects the coconut trees. It feeds on the tree by sucking up the water content as well as the essential nutrients from leaves. It also forms sooty mold in leaves due to which the process of photosynthesis is inhibited. Biocontrol of pest is harmless for trees and crops. The experimental results in literature reveal that Pseudomallada astur is a potential predator for this pest. We investigate the dynamics of predator, Pseudomallada astur’s interaction with rugose spiralling whitefly, Aleurodicus rugioperculatus in coconut trees using a mathematical model. In this system of ordinary differential equation, the pest-predator interaction is modeled using Holling type III functional response. The parametric values are calculated from the experimental results and are tabulated. An approximate analytical solution for the system has been derived. The homotopy analysis method proves to be a suitable method for creating solutions that are valid even for moderate to large parameter values, hence we employ the same to solve this nonlinear model. The $\hbar$-curves, which give the admissible region of $\hbar$, are provided to validate the region of convergence. We have derived the approximate solution at fifth order and stopped at this order since we obtain a more approximate solution in this iteration. Numerical simulation is obtained through MATLAB. The analytical results are compared with numerical simulation and are found to be in good agreement. The biological interpretation of figures implies that the use of a predator reduces the whitefly’s growth to a greater extent.

This work is devoted to the study of the problem of modeling and analyzing complex oscillatory modes, both regular and chaotic, in systems of interacting populations in the presence of random perturbations. As an initial conceptual deterministic model, a Volterra system of three differential equations is considered, which describes the dynamics of prey populations of two competing species and a predator. This model takes into account the following key biological factors: the natural increase in prey, their intraspecific and interspecific competition, the extinction of predators in the absence of prey, the rate of predation by predators, the growth of the predator population due to predation, and the intensity of intraspecific competition in the predator population. The growth rate of the second prey population is used as a bifurcation parameter. At a certain interval of variation of this parameter, the system demonstrates a wide variety of dynamic modes: equilibrium, oscillatory, and chaotic. An important feature of this model is multistability. In this paper, we focus on the study of the parametric zone of tristability, when a stable equilibrium and two limit cycles coexist in the system. Such birhythmicity in the presence of random perturbations generates new dynamic modes that have no analogues in the deterministic case. The aim of the paper is a detailed study of stochastic phenomena caused by random fluctuations in the growth rate of the second population of prey. As a mathematical model of such fluctuations, we consider white Gaussian noise. Using methods of direct numerical modeling of solutions of the corresponding system of stochastic differential equations, the following phenomena have been identified and described: unidirectional stochastic transitions from one cycle to another, trigger mode caused by transitions between cycles, noise-induced transitions from cycles to the equilibrium, corresponding to the extinction of the predator and the second prey population. The paper presents the results of the analysis of these phenomena using the Lyapunov exponents, and identifies the parametric conditions for transitions from order to chaos and from chaos to order. For the analytical study of such noise-induced multi-stage transitions, the technique of stochastic sensitivity functions and the method of confidence regions were applied. The paper shows how this mathematical apparatus allows predicting the intensity of noise, leading to qualitative transformations of the modes of stochastic population dynamics.

The article deals with the nonlinear boundary-value problem of hydrogen permeability corresponding to the following experiment. A membrane made of the target structural material heated to a sufficiently high temperature serves as the partition in the vacuum chamber. Degassing is performed in advance. A constant pressure of gaseous (molecular) hydrogen is built up at the inlet side. The penetrating flux is determined by mass-spectrometry in the vacuum maintained at the outlet side.

A linear model of dependence on concentration is adopted for the coefficient of dissolved atomic hydrogen diffusion in the bulk. The temperature dependence conforms to the Arrhenius law. The surface processes of dissolution and sorptiondesorption are taken into account in the form of nonlinear dynamic boundary conditions (differential equations for the dynamics of surface concentrations of atomic hydrogen). The characteristic mathematical feature of the boundary-value problem is that concentration time derivatives are included both in the diffusion equation and in the boundary conditions with quadratic nonlinearity. In terms of the general theory of functional differential equations, this leads to the so-called neutral type equations and requires a more complex mathematical apparatus. An iterative computational algorithm of second-(higher- )order accuracy is suggested for solving the corresponding nonlinear boundary-value problem based on explicit-implicit difference schemes. To avoid solving the nonlinear system of equations at every time step, we apply the explicit component of difference scheme to slower sub-processes.

The results of numerical modeling are presented to confirm the fitness of the model to experimental data. The degrees of impact of variations in hydrogen permeability parameters (“derivatives”) on the penetrating flux and the concentration distribution of H atoms through the sample thickness are determined. This knowledge is important, in particular, when designing protective structures against hydrogen embrittlement or membrane technologies for producing high-purity hydrogen. The computational algorithm enables using the model in the analysis of extreme regimes for structural materials (pressure drops, high temperatures, unsteady heating), identifying the limiting factors under specific operating conditions, and saving on costly experiments (especially in deuterium-tritium investigations).