Problem of describing the function of a GPR source

In this paper, we consider the problem of determining the source h(t)δ(x) of electromagnetic waves from GPR data. The task of electromagnetic sensing is to find the pulse characteristic of the medium r(t) and consists in calculating the response of the medium to the pulse source of excitation δ(t) (Dirac Delta function). To determine the analytical expression of the impulse response of a homogeneous medium r(t), we use the method proposed in [1-2]. To determine h(t), the inverse problem is reduced to a system of Volterra integral equations. The source function h(τ ), is defined as the solution of the Volterra integral equation of the first kind, f(t) = ∫t0 r(t − τ )h(τ )dτ in which f(t) is the data obtained by the GPR at the observation points. The problem of calculating the function of the GPR source h(τ ) consists in numerically solving the inverse problem, in which the function of the source h(τ ) is unknown, and the electromagnetic parameters of the medium are known: the permittivity ε; the conductivity σ; the magnetic permeability µ and the response of the medium to a given excitation h(τ )

MATHEMATICAL MODEL FOR MEDIUM-TERM COVID-19 FORECASTS IN KAZAKHSTAN

In this paper has been formulated and solved the problem of identifying unknown parameters of the mathematical model describing the spread of COVID-19 infection in Kazakhstan, based on additional statistical information about infected, recovered and fatal cases. The considered model, which is part of the family of modified models based on the SIR model developed by W. Kermak and A. McKendrick in 1927, is presented as a system of 5 nonlinear ordinary differential equations describing the variational transition of individuals from one group to another. By solving the inverse problem, reduced to solving the optimization problem of minimizing the functional, using the differential evolution algorithm proposed by Rainer Storn and Kenneth Price in 1995 on the basis of simple evolutionary problems in biology, the model parameters were refined and made a forecast and predicted a peak of infected, recovered and deaths among the population of the country. The differential evolution algorithm includes the generation of populations of probable solutions randomly created in a predetermined space, sampling of the algorithm’s stopping criterion, mutation, crossing and selection

Development of a mathematical model for signal processing using laboratory data

In this paper, we consider a mathematical model for the interpretation of the radarograms which obtained by GPR systems. As noted in [1–3], in addition to testing the algorithms, it is necessary to compare the calculated data of the mathematical model with the real data obtained from the GPR. One of the reasons preventing the spread of GPR technologies is the complexity of data interpretation, which requires the involvement of highly qualified specialists. In connection with this research as a mathematical model and a comparison with the real data of the GPR in an ideal layered medium, will provide a method for interpreting radarograms. We have conducted a series of experimental studies using the Loza – A georadar at the newly created laboratory ground. A distinctive feature of these studies is the choice of several localized objects in the form of iron sheets placed in an ideal layered medium, namely in clean dry sand. The choice of such an environment is necessary for testing the algorithms, the mathematical models developed by us for determining the depth of localized several objects. A series of experimental studies were conducted using georadar and a number of radarograms were obtained to study the depth of objects. A cycle of calculations was carried out to verify the conformity of the results of mathematical modeling with real georadar data. Key words: electrodynamics equation, magnetic permeability, dobeshi wavelets, medium conductivity, dielectric permeability, Maxwell equation

Identifiability of mathematical models in medical biology

Analysis of biological data is a key topic in bioinformatics, computational genomics, molecular modeling and systems biology. The methods covered in this article could reduce the cost of experiments for biological data. The problem of identifiability of mathematical models in physiology, pharmacokinetics and epidemiology is considered. The processes considered are modeled using nonlinear systems of ordinary differential equations. Math modeling of dynamic processes is based on the use of the mass conservation law. While addressing the problem of estimation of the parameters characterizing the process under the study, the question of nonuniqueness arises. When the input and output data are known, it is useful to perform an a priori analysis of the relevance of these data. The definition of identifiability of mathematical models is considered. Methods for analysis of identifiability of dynamic models are reviewed. In this review article, the following approaches are considered: the transfer function method applied to linear models (useful for analysis of pharmacokinetic data, since a large class of drugs is characterized by linear kinetics); the Taylor series expansion method applied to nonlinear models; a method based on differential algebra theory (the structure of this algorithm allows this to be run on a computer); a method based on graph theory (this method allows for analysis of the identifiability of the model as well as finding a proper reparametrization reducing the initial model to an identifiable one). The need to perform a priory identifiability analysis before estimating parameters characterizing any process is demonstrated with several examples. The examples of identifiability analysis of mathematical models in medical biology are presented