Analytical and Numerical Properties of the Mathematical Model of the Insulin–Glucose Balance in Human Blood
https://doi.org/10.1134/S2304487X19040059
Abstract
A mathematical model of the insulin–glucose balance has been proposed to describe the time dynamics of the insulin and glucose concentrations. The actuality of such models is that the analysis of these concentrations in human blood helps to study serious diseases treatment, e.g., diabetes. The numerical and analytical properties of the model, which is the system of two ordinary differential equations of the first order with the initial conditions, have been presented. The system has been examined for the Painlevé test in two special cases, suggesting that the glucose concentration is outside the allowable range: hypoglycemia and hyperglycemia. Particular analytical solutions have been found taking into account the conditions under which the system passes the Painlevé test. Asymptotic solutions have been obtained for the cases of hypoglycemia and hyperglycemia. In the case of hyperglycemia, the solution has been described by hypergeometric functions of Kummer and Tricomi. Graphs of analytical solutions have been constructed and analyzed. Numerical solutions of the system have been found by the fourth order Runge–Kutta method taking into account various external sources of glucose. Plots of numerical solutions have been obtained, showing fluctuations in the glucose concentration corresponding to the results of medical research in diabetes treatment.
Keywords
model of insulin–glucose balance,
nonlinear differential equations,
Painlevé test,
exact solutions,
numerical solutions,
asymptotic solutions
Supporting agencies: The work was carried out with financial support grant of the Russian Science Foundation “Development of methods for studying nonlinear mathematical models Delay” No. 18-11-00209
About the Authors
K. V. Kan
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
N. A. Kudryashov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute)
Russian Federation
Russian Federation
115409
Moscow
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Review
For citations:
Kan K.V., Kudryashov N.A. Analytical and Numerical Properties of the Mathematical Model of the Insulin–Glucose Balance in Human Blood. Vestnik natsional'nogo issledovatel'skogo yadernogo universiteta "MIFI". 2019;8(4):335-341. (In Russ.) https://doi.org/10.1134/S2304487X19040059
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