MATHEMATICAL MODEL FOR MEDIUM-TERM COVID-19 FORECASTS IN KAZAKHSTAN
DOI:
https://doi.org/10.26577/JMMCS.2021.v111.i3.08Keywords:
inverse problems, identification, differential evolutionAbstract
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.
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[3] Hongyu Miao, Xiaohua Xia, Alan S. Perelson, Hulin Wu, "On Identifiability of nonlinear ODE models and applications in viral dynamics" , SIAM Rev Soc Ind Appl Math 53-1 (2011): 3-39.
[4] Bellu G., Saccomani M.P., Audoly S. and D’Angio’ L., "Differential Algebra for Identifiability of SYstems software (DAISY)" , URL: http://www.dei.unipd.it/ pia/ (2008).
[5] Kaltenbacher B., "All-at-once versus reduced iterative methods for time dependent inverse problems" , Inverse Problems 33 (2017): 31.
[6] Herowitz J., "Ill-posed inverse problems in economics" , Annual Review of Economics 6 (2014): 21-51.
[7] Dunker F. and Hohage Th., "On parameter identification in stochastic differential equations by penalized maximum likelihood" , Inverse Problem 30 (2014): 20.
[8] Hohage Th., Werner F., "Inverse problems with Poisson data: statistical regularization theory, applications and algorithms" , Inverse Problem 32 (2016): 56.
[9] Marco A. Iglesias, Kui Lin, Shuai Lu, Andrew M. Stuart., "Filter Based Methods For Statistical Linear Inverse Problems" , ARXIV. E-print arXiv:1512.01955. (2015).
[10] Soner H.M., Stochastic Optimal Control in Finance (Istanbul: Koc Univercity, 2004).
[11] Krivorotko O.I., Kabanikhin S.I., Sosnovskaya M. and Andornaya D., "Sensitivity and identifiability analysis of COVID-19 pandemic models" , Vavilov Journal of Genetics and Breeding 25-1 (2021): 82-91.
[12] Krivorotko O.I., Kabanikhin S.I., Zyat’kov N. and others, "Mathematical Modeling and Forecasting of COVID-19 in Moscow and Novosibirsk Region" , Numerical Analysis and Applications 13-4 (2020): 332-348.
[13] Margenov S., Popivanov N., Ugrinova I., Harizanov S., Hristov T., "Mathematical and computer modeling of COVID-19 transmission dynamics in Bulgaria by time-depended inverse SEIR model" , AIP Conference Proceedings 2333 (2021).
[14] Cooper I., Mondal A., Antonopoulos C., "A SIR model assumption for the spread of COVID-19 in different communities" , Chaos, Solitons and Fractals 139 (2020): 1-15.
[15] Lima L., "Modeling based in the stochastic dynamics for the time evolution of the COVID-19" , Preprint (2020): 1-4.
[16] Paticchio A., Scarlatti T., Mattheakis M., Protopapas P., Brambilla M., "Semi-supervised Neural Networks solve an inverse problem for modeling Covid-19 spread" , Preprint (2020): 1-6.
[17] Lasry J.-M., Lions P.-L., "Mean field games" , Jpn. J. Math. 2-1 (2007): 229-260.
[18] Lasry J-M., Lions P-L., Gueant O., "Application of mean field games to growth theory" , Technical report, INRIA a CCSD electronic archive server based on P.A.O.L. (2008): URL: http://hal.inria.fr/oai/oai.php.
[19] Krivorot’ko O.I., Kabanihin S.I., Zyat’kov N.YU., Prihod’ko A.YU., Prohoshin N.M., SHishlenin M.A.,
"Matematicheskoe modelirovanie i prognozirovanie COVID-19 v Moskve i Novosibirskoj oblasti" , URL:
https://arxiv.org/pdf/2006.12619.pdfLi (2020) [in Russian].
[20] Wang B. Y., Peng R., Zhou C., Zhan Y., Liu Z., et al., "Mathematical Modeling and Epidemic Prediction of COVID-19 and Its Significance to Epidemic Prevention and Control Measures " , Ann. Infect. Dis. Epidemiol. 5, no.1 (2020): 10-52.
[21] Sameni R., "Mathematical Modeling of Epidemic Diseases: A Case Study of the COVID-19 Coronavirus." , URL: arXiv:2003.11371 (2020).
[22] Price K.V., Storn R. and Lampinen J.A., Differential Evolution (Nat. Comput. Ser., Springer, Berlin. 2004).
[23] Storn R., Differential evolution research: Trends and open questions, in: Advances in Differential Evolution (Stud.Comput. Intell. 143. Springer, Berlin. 2004): 1-31.
[24] Storn R., Price K., Differential evolution: A simple and efficient adaptive scheme for global optimization over continuous spaces (Report no. TR 012, International Computer Science Institute, Berkeley. 1995).
[25] Storn R., Price K., "Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces" , J. Global Optim. 11, no. 4 (1997): 341-359.
[26] Qing A., Diferential Evolution: Fundamentals and Applications in Electrical Engineering JohnWiley & Sons, NewYork.2009).
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How to Cite
Kabanikhin, S. I., Bektemesov, M. A., & Bektemessov, J. M. (2021). MATHEMATICAL MODEL FOR MEDIUM-TERM COVID-19 FORECASTS IN KAZAKHSTAN. Journal of Mathematics, Mechanics and Computer Science, 111(3), 95–106. https://doi.org/10.26577/JMMCS.2021.v111.i3.08
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Applied Mathematics
