The epidemiological situation development process mathematical modeling taking into account the covid-19 dissemination features .

Authors

  • A. P. Oliinyk
  • Ye. A. Oliinyk
  • L. I. Feshanich

Keywords:

viral infections epidemics, modeling, coefficient systems, differential equations, numerical implementation methods

Abstract

The paper deals with mathematical models of epidemic development and spread. Such models are based on the Cauchy apparatus for nonlinear ordinary differential equations with a wide class of initial conditions. The methodology for selecting the system coefficients is developed and substantiated, their content and influence on the system parameters are described. The method of setting the initial conditions of the problem is proposed and substantiated. Different variants of epidemic models have been analyzed, at the stage of describing the situation with COVID-19 spread, the expediency of choosing the apparatus of systems of nonlinear ordinary differential equations without taking into account argumentation due to the need to develop an express method of simulating the epidemic propagation process is substantiated. The direction of mathematical modeling of epidemiological situations was developed by taking into account the treatment means, influencing the development of epidemics of the economic situation in the country, the presence of other factors positive (the level of communication of the population, its mobility, treatment methods, education of the population, climatic influences, seasonal effects, etc.) and negative factors. distress, political situation in the country). From the point of view of mathematical modeling, the values ​​of the coefficients and their dynamics are analyzed, the limits of their change are revealed for the efficient description and forecasting of the development of the simulated processes.

The numerical implementation of the model uses Runge-Kutta methods, the accuracy of which is selected in the light of the peculiarities of the simulated processes. The calculations confirmed the effectiveness of the proposed model for describing epidemics and pandemics of infections and viruses of various nature, it accurately describes the qualitative behavior of the systems and processes under study, found that all the coefficients of the model accurately describe their impact on the behavior of the model as a whole, reflect trends observed in the detailed study of the spread of COVID-19 in China and Europe Directions for possible further research have been identified.

Downloads

Download data is not yet available.

References

Бабский В.Г., Мышкис А. Д. Математические модели в биологии, связанные с учетом последействия / В. Г. Бабский. М. Мир, 1983. 383 с.

Беляков В. Д., Кравцов Ю. В. , Герасимов Л. Н. Состояние и перспектива математического моделирования в эпидемиологии. Журнал микробиологии, эпидемиологии и иммунобиологии. 1990. № 6. С. 109–113.

Вольтерра В. Математическая теория борьбы за существование. М. : Наука, 1976. 286 с.

Марчук И. Г. Математические модели в иммунологии: вычислительные методы и эксперименты. М. : Наука, 1991. 304 с.

Романюха А. А. Математические модели в иммунологии и эпидемиологии инфекционных заболеваний. М. : БИНОМ. Лаборатория знаний, 2012. 293 с.

Самойленко А. М., Перестюк М. О. , Парасюк І. О. . Диференціальні рівняння. К. : Либідь, 2003. 600 с.

Фельдман Л. П., Петренко А. І., Дмитрієва О. А. Чисельні методи в інформатиці. К. : Видавнича група BHV, 2006. 480 с.

Хусаінов Д. Я., Шатирко А. В. Введення в моделювання динамічних систем. І. І. Харченко. К. : КНУ ім. Тарас Шевченка, 2010. 130 с.

Шахно С. М., Дудикевич А. Т., Левицька С. М. . Практикум з чисельних методів. Львів : ЛНУ імені Івана Франка, 2013. 432 с.

Эдвардс Ч. Г. Пенни Д. Э. Дифференциальные уравнения и краевые задачи: моделирование и вычисление с помощью Mathematica, Maple и MATLAB. М. : ООО "И. Д. Вильямс", 2008. 1104 с.

Bocharov G., Rihan F. A. Numerical modelling in biosciences using delay differential equations. Journal of Computational and Applied Mathematics. 2000. Vol. 125. P. 183–199.

Brauer F., Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology. Spinger, 2012. 508 p.

Хайрер Э., Нерсетт С., Ваннер Г.. Решение обыкновенных дифференциальных уравнений. М.: Мир, 1990. 512 с.

Andriy Oliynyk, Eugene Oliynyk, Olexandr Pyptiuk, Róża Dzierżak, Małgorzata Szatkowska, Svetlana Uvaysova, Ainur Kozbekova. The human body metabolism process mathematical simulation based on Lotka-Volterra model. Proc. SPIE 10445, Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments 2017 104453L (7 August 2017); doi: 10.1117/12.2280972. 7 p.

Олійник А.П., Олійник Є.А. Математичне моделювання процесів обміну речовин в організмі людини та його програмна реалізація. Методи та прилади контролю якості. 2017. №1(38). С. 112-118.

Олийник Е. А., Гой Т. П. Математическое моделирование эпидемиологических процессов с помощью дифференциальных уравнений с запаздывающим аргументом. Материалы Междунар. молодежного симпозиума «Современные проблемы математики. Методы, модели, приложения», 17–20 нояб. 2015 г. Воронеж : ФГБОУ ВПО «ВГЛТА», 2015. С. 34–37.

Babskiy V.G., Myishkis A. D. Matematicheskie modeli v biologii, svyazannyie s uchetom posledeystviya. M. Mir, 1983. 383 р. [in Russian]

Belyakov V. D., Kravtsov Yu. V. , Gerasimov L. N. Sostoyanie i perspektiva matematicheskogo modelirovaniya v epidemiologii. Zhurnal mikrobiologii, epidemiologii i immunobiologii. 1990. No 6. P. 109–113. [in Russian]

Volterra V. Matematicheskaya teoriya borbyi za suschestvovanie. M. : Nauka, 1976. 286 p. [in Russian]

Marchuk I. G. Matematicheskie modeli v immunologii: vyichislitelnyie metodyi i eksperimentyi. M. : Nauka, 1991. 304 p. [in Russian]

Romanyuha A. A. Matematicheskie modeli v immunologii i epidemiologii infektsionnyih zabolevaniy. M. : BINOM. Laboratoriya znaniy, 2012. 293 p. [in Russian]

Samoylenko A. M., Perestyuk M. O. , Parasyuk I. O. . DiferentsIalni rivnyannya. K. : LibId, 2003. 600 p. [in Ukrainian]

Feldman L. P., Petrenko A. I., DmitrIEva O. A. Chiselni metody v informatitsi. K. : Vidavnicha grupa BHV, 2006. 480 p. [in Ukrainian]

HusaInov D. Ya., Shatirko A. V. Vvedennya v modelyuvannya dinamIchnih sistem. I. I. Harchenko. K. : KNU Im. Taras Shevchenka, 2010. 130 p. [in Russian]

Shahno S. M., Dudikevich A. T., Levitska S. M. . Praktikum z chiselnih metodiv. LvIv : LNU Imeni Ivana Franka, 2013. 432 p. [in Ukrainian]

Edvards Ch. G. Penni D. E. Differentsialnyie uravneniya i kraevyie zadachi: modelirovanie i vyichislenie s pomoschyu Mathematica, Maple i MATLAB. M. : OOO "I. D. Vilyams", 2008. 1104 p. [in Russian]

Bocharov G., Rihan F. A. Numerical modelling in biosciences using delay differential equations. Journal of Computational and Applied Mathematics. 2000. Vol. 125. P. 183–199.

Brauer F., Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology. Spinger, 2012. 508 p.

Hayrer E., Nersett S., Vanner G.. Reshenie obyiknovennyih differentsialnyih uravneniy. M.: Mir, 1990. 512 p. [in Russian]

Andriy Oliynyk, Eugene Oliynyk, Olexandr Pyptiuk, Róża Dzierżak, Małgorzata Szatkowska, Svetlana Uvaysova, Ainur Kozbekova. The human body metabolism process mathematical simulation based on Lotka-Volterra model. Proc. SPIE 10445, Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments 2017 104453L (7 August 2017); doi: 10.1117/12.2280972. 7 p.

OlIynik A.P., OlIynik E.A. Matematichne modelyuvannya protsesIv obmInu rechovin v organIzmI lyudini ta yogo programna realIzatsIya. Metodi ta priladi kontrolyu yakostI. 2017. No 1(38). P. 112-118. [in Ukrainian]

Oliynik E. A., Goy T. P. Matematicheskoye modelirovaniye epidemiologicheskikh protsessov s pomoshchyu differentsialnykh uravneniy s zapazdyvayushchim argumentom. Materialy Mezhdunar. molodezhnogo simpoziuma «Sovremennyye problemy matematiki. Metody. modeli. prilozheniya». 17–20 noyab. 2015 g. Voronezh : FGBOU VPO «VGLTA». 2015. P. 34–37. [in Russian]

Published

2020-08-31

How to Cite

Олійник, . А. П., Олійник, . . Є. А., & Фешанич, . Л. І. . (2020). The epidemiological situation development process mathematical modeling taking into account the covid-19 dissemination features . METHODS AND DEVICES OF QUALITY CONTROL, (1(44). Retrieved from http://mpky.nung.edu.ua/index.php/mpky/article/view/507

Issue

Section

MATHEMATICAL MODELLING FOR THE UNDESTROYED CONTROL PROBLEMS