Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 40. no. 3. pp. 119–136. ISSN 2079-6641

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MSC 26A33, 92B99

Research Article

Fractional differential model of physical processes with saturation and its application to the description of the dynamics of COVID-19

D. A. Tverdyi¹², R. I. Parovik¹²

¹Vitus Bering Kamchatka State University, 683032, Petropavlovsk-Kamchatsky, st. Pogranichnaya, 4, Russia
²Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, p. Paratunka, Mirnaya st., 7, Russia

In this article, a fractional differential model of physical processes with saturation was used to describe the dynamics of lethal outcomes of COVID-19 infection. The mathematical description of the model is given by the integro-differential Riccati equation with a derivative of a fractional variable order of the Gerasimov-Caputo
type. This description makes it possible to take into account the effects of saturation and memory in the dynamics of the spread of COVID-19 among the population. Here, the saturation effect consists in reaching a plateau in the number of cases and deaths, which indicates the stabilization of the dynamics of the spread of COVID-19. The memory effect is that the symptoms of infection in infected people do not appear immediately, but with some delay. The article examines observational data on new cases of infection and the total number of deaths over a period of 2.5 years (from March to September 2022) in the Russian Federation and the Republic of Uzbekistan. Further, the parameters of the model are refined based on the studied data on the dynamics of COVID-19. With the help of the refined model, a preliminary forecast for the next six months is made with subsequent verification. Good agreement is shown between the model curves and the data curves for the total number of deaths from COVID-19.

Key words: mathematical modeling of dynamic processes, saturation and memory effect, COVID-19, Riccati equation, fractional derivative of variable order, Gerasimov-Caputo derivative.

DOI: 10.26117/2079-6641-2022-40-3-119-136

Original article submitted: 01.12.2022

Revision submitted: 06.12.2022

For citation. Tverdyi D. A., Parovik R. I. Fractional differential model of physical processes with saturation and its application to the description of the dynamics of COVID-19. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 40: 3, 119-137. DOI: 10.26117/2079-6641-2022-40-3-119-136

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (

© Tverdyi D. A., Parovik R. I., 2022

Funding. The study was carried out within the framework of the grant «Development of mathematical models of fractional dynamics for the purpose of studying oscillatory processes and processes with saturation»MD-758.2022.1.1.


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Tverdyi Dmitrii Alexsandrovich – Ph. D. (Phys. & Math.), Researcher laboratory «Natural disasters of Kamchatka — earthquakes and volcanic eruptions»of the Kamchatka State University named after Vitus Bering, Petropavlovsk-Kamchatsky, Russia, Lead programmer laboratory of electromagnetic propogation Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0001-6983-5258.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, chief of the integrative laboratory «Natural disasters of Kamchatka — earthquakes and volcanic eruptions»of the Kamchatka State University named after Vitus Bering, Petropavlovsk-Kamchatsky, Russia, Leading researcher laboratory of modeling physical processes Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0002-1576-1860.