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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 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 article presents the numerical modeling and study of the kinetic model of propane pyrolysis. The study of the reaction kinetics is a necessary stage in modeling the dynamics of the gas flow in the reactor.

The kinetic model of propane pyrolysis is a nonlinear system of ordinary differential equations of the first order with parameters, the role of which is played by the reaction rate constants. Math modeling of processes is based on the use of the mass conservation law. To solve an initial (forward) problem, implicit methods for solving stiff ordinary differential equation systems are used. The model contains 60 input kinetic parameters and 17 output parameters corresponding to the reaction substances, of which only 9 are observable. In the process of solving the problem of estimating parameters (inverse problem), there is a question of non-uniqueness of the set of parameters that satisfy the experimental data. Therefore, before solving the inverse problem, the possibility of determining the parameters of the model is analyzed (analysis of identifiability).

To analyze identifiability, we use the orthogonal method, which has proven itself well for analyzing models with a large number of parameters. The algorithm is based on the analysis of the sensitivity matrix by the methods of differential and linear algebra, which shows the degree of dependence of the unknown parameters of the models on the given measurements. The analysis of sensitivity and identifiability showed that the parameters of the model are stably determined from a given set of experimental data. The article presents a list of model parameters from most to least identifiable. Taking into account the analysis of the identifiability of the mathematical model, restrictions were introduced on the search for less identifiable parameters when solving the inverse problem.

The inverse problem of estimating the parameters was solved using a genetic algorithm. The article presents the found optimal values of the kinetic parameters. A comparison of the experimental and calculated dependences of the concentrations of propane, main and by-products of the reaction on temperature for different flow rates of the mixture is presented. The conclusion about the adequacy of the constructed mathematical model is made on the basis of the correspondence of the results obtained to physicochemical laws and experimental data.