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

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

Sensitivity and identifiability analysis of COVID-19 pandemic models

The paper presents the results of sensitivity-based identif iability analysis of the COVID-19 pandemic spread models in the Novosibirsk region using the systems of differential equations and mass balance law. The algorithm is built on the sensitivity matrix analysis using the methods of differential and linear algebra. It allows one to determine the parameters that are the least and most sensitive to data changes to build a regularization for solving an identif ication problem of the most accurate pandemic spread scenarios in the region. The performed analysis has demonstrated that the virus contagiousness is identif iable from the number of daily conf irmed, critical and recovery cases. On the other hand, the predicted proportion of the admitted patients who require a ventilator and the mortality rate are determined much less consistently. It has been shown that building a more realistic forecast requires adding additional information about the process such as the number of daily hospital admissions. In our study, the problems of parameter identif ication using additional information about the number of daily conf irmed, critical and mortality cases in the region were reduced to minimizing the corresponding misf it functions. The minimization problem was solved through the differential evolution method that is widely applied for stochastic global optimization. It has been demonstrated that a more general COVID-19 spread compartmental model consisting of seven ordinary differential equations describes the main trend of the spread and is sensitive to the peaks of conf irmed cases but does not qualitatively describe small statistical datasets such as the number of daily critical cases or mortality that can lead to errors in forecasting. A more detailed agent-oriented model has been able to capture statistical data with additional noise to build scenarios of COVID-19 spread in the region

In celebration of the 60th birthday of Professor Alemdar Hasanoğlu (Hasanov)

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Inverse and Ill-posed Problems: Theory and Applications

The text demonstrates the methods for proving the existence (if et all) and finding of inverse and ill-posed problems solutions in linear algebra, integral and operator equations, integral geometry, spectral inverse problems, and inverse scattering problems. It is given comprehensive background material for linear ill-posed problems and for coefficient inverse problems for hyperbolic, parabolic, and elliptic equations. A lot of examples for inverse problems from physics, geophysics, biology, medicine, and other areas of application of mathematics are included

Mathematical Modeling of the Wuhan COVID-2019 Epidemic and Inverse Problems

Abstract: Mathematical models for transmission dynamics of the novel COVID-2019 coronavirus, an outbreak of which began in December, 2019, in Wuhan are considered. To control the epidemiological situation, it is necessary to develop corresponding mathematical models. Mathematical models of COVID-2019 spread described by systems of nonlinear ordinary differential equations (ODEs) are overviewed. Some of the coefficients and initial data for the ODE systems are unknown or their averaged values are specified. The problem of identifying model parameters is reduced to the minimization of a quadratic objective functional. Since the ODEs are nonlinear, the solution of the inverse epidemiology problems can be nonunique, so approaches for analyzing the identifiability of inverse problems are described. These approaches make it possible to establish which of the unknown parameters (or their combinations) can be uniquely and stably recovered from available additional information. For the minimization problem, methods are presented based on a combination of global techniques (covering methods, nature-like algorithms, multilevel gradient methods) and local techniques (gradient methods and the Nelder–Mead method)

Inverse problems for Maxwell's equations

The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology