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The paper considers integro-differential equations of fractional order moisture transfer with the Bessel operator. The studied equations contain the Bessel operator, two Gerasimov – Caputo fractional differentiation operators with different orders $\alpha$ and $\beta$. Two types of integro-differential equations are considered: in the first case, the equation contains a non-local source, i.e. the integral of the unknown function over the integration variable $x$, and in the second case, the integral over the time variable τ, denoting the memory effect. Similar problems arise in the study of processes with prehistory. To solve differential problems for different ratios of $\alpha$ and $\beta$, a priori estimates in differential form are obtained, from which the uniqueness and stability of the solution with respect to the right-hand side and initial data follow. For the approximate solution of the problems posed, difference schemes are constructed with the order of approximation $O(h^2+\tau^2)$ for $\alpha=\beta$ and $O(h^2+\tau^{2-\max\{\alpha,\beta\}})$ for $\alpha\neq\beta$. The study of the uniqueness, stability and convergence of the solution is carried out using the method of energy inequalities. A priori estimates for solutions of difference problems are obtained for different ratios of $\alpha$ and $\beta$, from which the uniqueness and stability follow, as well as the convergence of the solution of the difference scheme to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.

The possibility to simplify the integration of Langevin equation using the variation of correlation between augmentation was researched. The analytical expression for a set of numerical schemes is presented. It’s shown that asymptotic limits for squared velocity depend on step size. The region of convergence and the convergence orders were estimated. It turned out that the incorrect correlation between increments decrease the accuracy down to the level of first-order methods for schemes based on precise solution.

This paper is devoted to the analysis of the effect of additive and parametric noise on the processes occurring in the nerve cell. This study is carried out on the example of the well-known Morris – Lecar model described by the two-dimensional system of ordinary differential equations. One of the main properties of the neuron is the excitability, i.e., the ability to respond to external stimuli with an abrupt change of the electric potential on the cell membrane. This article considers a set of parameters, wherein the model exhibits the class 2 excitability. The dynamics of the system is studied under variation of the external current parameter. We consider two parametric zones: the monostability zone, where a stable equilibrium is the only attractor of the deterministic system, and the bistability zone, characterized by the coexistence of a stable equilibrium and a limit cycle. We show that in both cases random disturbances result in the phenomenon of the stochastic generation of mixed-mode oscillations (i. e., alternating oscillations of small and large amplitudes). In the monostability zone this phenomenon is associated with a high excitability of the system, while in the bistability zone, it occurs due to noise-induced transitions between attractors. This phenomenon is confirmed by changes of probability density functions for distribution of random trajectories, power spectral densities and interspike intervals statistics. The action of additive and parametric noise is compared. We show that under the parametric noise, the stochastic generation of mixed-mode oscillations is observed at lower intensities than under the additive noise. For the quantitative analysis of these stochastic phenomena we propose and apply an approach based on the stochastic sensitivity function technique and the method of confidence domains. In the case of a stable equilibrium, this confidence domain is an ellipse. For the stable limit cycle, this domain is a confidence band. The study of the mutual location of confidence bands and the boundary separating the basins of attraction for different noise intensities allows us to predict the emergence of noise-induced transitions. The effectiveness of this analytical approach is confirmed by the good agreement of theoretical estimations with results of direct numerical simulations.

The repressor is the first genetic regulatory network in synthetic biology, which was artificially constructed in 2000. It is a closed network of three genetic elements — $lacI$, $\lambda cI$ and $tetR$, — which have a natural origin, but are not found in nature in such a combination. The promoter of each of the three genes controls the next cistron via the negative feedback, suppressing the expression of the neighboring gene. In this paper, the nonlinear dynamics of a modified repressilator, which has time delays in all parts of the regulatory network, has been studied for the first time. Delay can be both natural, i.e. arises during the transcription/translation of genes due to the multistage nature of these processes, and artificial, i.e. specially to be introduced into the work of the regulatory network using synthetic biology technologies. It is assumed that the regulation is carried out by proteins being in a dimeric form. The considered repressilator has two more important modifications: the location on the same plasmid of the gene $gfp$, which codes for the fluorescent protein, and also the presence in the system of a DNA sponge. In the paper, the nonlinear dynamics has been considered within the framework of the deterministic description. By applying the method of decomposition into fast and slow motions, the set of nonlinear differential equations with delay on a slow manifold has been obtained. It is shown that there exists a single equilibrium state which loses its stability in an oscillatory manner at certain values of the control parameters. For a symmetric repressilator, in which all three genes are identical, an analytical solution for the neutral Andronov–Hopf bifurcation curve has been obtained. For the general case of an asymmetric repressilator, neutral curves are found numerically. It is shown that the asymmetric repressor generally is more stable, since the system is oriented to the behavior of the most stable element in the network. Nonlinear dynamic regimes arising in a repressilator with increase of the parameters are studied in detail. It was found that there exists a limit cycle corresponding to relaxation oscillations of protein concentrations. In addition to the limit cycle, we found the slow manifold not associated with above cycle. This is the long-lived transitional regime, which reflects the process of long-term synchronization of pulsations in the work of individual genes. The obtained results are compared with the experimental data known from the literature. The place of the model proposed in the present work among other theoretical models of the repressilator is discussed.

The paper presents a review of recent developments of hybrid discrete-continuous models in cell population dynamics. Such models are widely used in the biological modelling. Cells are considered as individual objects which can divide, die by apoptosis, differentiate and move under external forces. In the simplest representation cells are considered as soft spheres, and their motion is described by Newton’s second law for their centers. In a more complete representation, cell geometry and structure can be taken into account. Cell fate is determined by concentrations of intra-cellular substances and by various substances in the extracellular matrix, such as nutrients, hormones, growth factors. Intra-cellular regulatory networks are described by ordinary differential equations while extracellular species by partial differential equations. We illustrate the application of this approach with some examples including bacteria filament and tumor growth. These examples are followed by more detailed studies of erythropoiesis and immune response. Erythrocytes are produced in the bone marrow in small cellular units called erythroblastic islands. Each island is formed by a central macrophage surrounded by erythroid progenitors in different stages of maturity. Their choice between self-renewal, differentiation and apoptosis is determined by the ERK/Fas regulation and by a growth factor produced by the macrophage. Normal functioning of erythropoiesis can be compromised by the development of multiple myeloma, a malignant blood disorder which leads to a destruction of erythroblastic islands and to sever anemia. The last part of the work is devoted to the applications of hybrid models to study immune response and the development of viral infection. A two-scale model describing processes in a lymph node and other organs including the blood compartment is presented.

The paper presents a mathematical model that describes the process of obtaining biogas from livestock waste. This model describes the processes occurring in a biogas plant for mesophilic and thermophilic media, as well as for continuous and periodic modes of substrate inflow. The values of the coefficients of this model found earlier for the periodic mode, obtained by solving the problem of model identification from experimental data using a genetic algorithm, are given.

For the model of methanogenesis, an optimal control problem is formulated in the form of a Lagrange problem, whose criterial functionality is the output of biogas over a certain period of time. The controlling parameter of the task is the rate of substrate entry into the biogas plant. An algorithm for solving this problem is proposed, based on the numerical implementation of the Pontryagin maximum principle. In this case, a hybrid genetic algorithm with an additional search in the vicinity of the best solution using the method of conjugate gradients was used as an optimization method. This numerical method for solving an optimal control problem is universal and applicable to a wide class of mathematical models.

In the course of the study, various modes of submission of the substrate to the digesters, temperature environments and types of raw materials were analyzed. It is shown that the rate of biogas production in the continuous feed mode is 1.4–1.9 times higher in the mesophilic medium (1.9–3.2 in the thermophilic medium) than in the periodic mode over the period of complete fermentation, which is associated with a higher feed rate of the substrate and a greater concentration of nutrients in the substrate. However, the yield of biogas during the period of complete fermentation with a periodic mode is twice as high as the output over the period of a complete change of the substrate in the methane tank at a continuous mode, which means incomplete processing of the substrate in the second case. The rate of biogas formation for a thermophilic medium in continuous mode and the optimal rate of supply of raw materials is three times higher than for a mesophilic medium. Comparison of biogas output for various types of raw materials shows that the highest biogas output is observed for waste poultry farms, the least — for cattle farms waste, which is associated with the nutrient content in a unit of substrate of each type.

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.