A.Yu. Perevaryukha1

1 St. Petersburg Federal Research Center of the Russian Academy of Sciences (St. Petersburg, Russia)

Formulation of the problem. Building models of epidemics is a long-developed scientific direction. The practical application of modeling results to predict the development of a modern pandemic in 2020 has shown the fundamental limitations of basic modeling methods. Based on a mathematical analysis of the properties of some deterministic SIRS model, it has not been possible to present the variety of forms that the regional dynamics of the spread of coronavirus has taken. The characteristics of the epidemic process do not remain constant as the competitive evolution of a variable pathogen. The phenomena after the COVID outbreak turned out to be much more complicated than the waves of pandemic strains of influenza that naturally subsided as immunity accumulated, which were previously encountered and based on their study, forecasts of the current situation were built. It is promising to develop a scenario approach to the analysis of epidemic situations, considering the logic of pathogen evolution and adaptation to the features of its distribution in local populations. An urgent task is to analyze the available data from the point of view of oscillation modes to get an idea of the real qualitative diversity of local forms of the wave-like dynamics of the epidemic of coronavirus strains and to identify some special development scenarios described because of well-known bifurcation transformations of the phase portrait.

Target. Determine the features of local forms of oscillatory epidemic processes, classify the observed phenomena of outbreaks and attenuation of waves of coronavirus infections, correlate the dynamics of epidemics with non-linear effects that can be obtained using equations with delay to describe a few scenarios.

Results. Based on a comparison of specific situations, it is shown that the dynamics of epidemics develops in a diverse and variable way at the current stage according to the internal unpredictable logic of the confrontation process, which is determined by the rate of change during the joint evolution of the virus and population immunity. We managed to analyze some of the interesting transformations of COVID waves in equations with delay, using the idea of delimiting the parameters that cause bifurcations. It is substantiated that all non-linear effects and transformations of oscillatory regimes identified from epidemic graphs cannot be obtained in principle within the framework of one model.

Practical significance. A comparative analysis of scenarios confirms the hypothesis that epidemic processes are primarily affected by population immunity, rather than restrictive measures. Vaccination was not able to complete the emergence of new waves of infections, but it obviously affects the properties of local oscillatory morbidity patterns. A generalized predictive model for the development of the coronavirus pandemic in 2020-22. it is impossible to build, even focusing only on aggregated data for a week on all confirmed cases of infection in the world. The experience of Sweden and New Zealand, which adhered to fundamentally different strategies to combat the epidemic, showed that sustainable population immunity that blocks the transmission of the virus for a long time cannot be achieved and administrative barriers are not a panacea. A probable scenario for the end of the pandemic is the separation of up to five stable strains, the circulation of which in the regions is controlled by cross-T-cell immunity.

Perevaryukha A.Yu. Analysis of outbreaks and damped waves of the spread of coronavirus in the regions based on equations with a deviating argument. Nonlinear World. 2023. V. 21. № 1. P. 36-46. DOI: https://doi.org/10.18127/j20700970-202301-05 (In Russian)

1. Perevarjuha A.Ju. Haoticheskie rezhimy v modeljah teorii formirovanija popolnenija populjacij. Nelinejnyj mir. 2009. T. 7. № 12. S. 925-932 (In Russian).
2. Perevarjuha A.Ju. Interpretacija povedenija modelej dinamiki bioresursov i momental'naja haotizacija v novoj modeli. Nelinejnyj mir. 2012. T. 10. № 4. S. 255-262 (In Russian).
3. Mihajlov V.V., Perevarjuha A.Ju. Modelirovanie processa stremitel'noj jevtrofikacii krupnogo ozera i ego vlijanija na blagopoluchie avtohtonnoj ihtiofauny. Nelinejnyj mir. 2018. T. 16. № 4. S. 45-53 (In Russian).
4. Perevarjuha A.Ju. Parametricheskiye intervaly interpretiruyemosti v iteratsionnykh modelyakh real'nykh ekologicheskikh protsessov. Nelinejnyj mir. 2022. Т. 20. № 1. S. 55-64 (In Russian).
5. Vasil'yeva I.I., Druzhinina O.V., Masina O.N. Postroyeniye i issledovaniye populyatsionnykh dinamicheskikh modeley tipa "dva konkurenta - dva areala migratsii". Nelinejnyj mir. 2022. T. 20. № 4. S. 60–68 (In Russian).
6. Krivorot'ko O. I., Kabanikhin S. I., Zyat'kov N. YU., Prikhod'ko A. YU., Prokhoshin N. M., Shishlenin M. A. Matematicheskoye modelirovaniye i prognozirovaniye COVID-19 v Moskve i Novosibirskoy oblasti. Sib. zh. vychisl. matem., 2020. T. 23. № 4. S. 395-414 (In Russian).
7. Shabunin A.V. Gibridnaya SIRS-model' rasprostraneniya infektsiy. Izvestiya vysshikh uchebnykh zavedeniy. Prikladnaya nelineynaya dinamika. 2022. T. 30. № 6. S. 717–731 (In Russian).
8. Abotaleb M.S., Makarovskikh T.A. Development of algorithms for choosing the best time series models and neural networks to predict COVID-19 cases. Bulletin of the South Ural State University. 2021. Т. 21. № 3. S. 26-35 (In Russian).
9. Zvyagintsev A.I. O nelineynoy differentsial'noy sisteme, modeliruyushchey dinamiku pandemii COVID-19. Mezhdunarodnyy nauchno-issledovatel'skiy zhurnal. 2022. № 7-1 (121). S. 115-121 (In Russian).
10. Gol'dberg V.M. Razvitiye tsepnoy modeli dinamiki rasprostraneniya pandemii COVID. Goreniye i vzryv. 2021. T. 14. № 3. S. 3-10 (In Russian).
11. Nikitina A.V. Study of the spread of viral diseases based on modifications of the SIR model. Computational Mathematics and Information Technologies. 2020. № 1. P. 19-30.
12. Hutchinson G. E. Circular causal systems in ecology. Ann. N. Y. Acad. Sci. 1948. Vol. 50, №. 4. S. 22-246 (In Russian).
13. Silva S.J., Pardee K., Pena L., Kohl A. Recent insights into SARS-COV-2 omicron variant. Reviews in Medical Virology. 2022. V. 33. Article number: e2373. doi: 10.1002/rmv.2373.
14. Oshitani H. COVID lessons from Japan: the right messaging empowers citizens. Nature 2022. 605, Article number: 589. https://doi.org/10.1038/d41586-022-01385-9.
15. Borisova T.Ju., Solov'eva I.V. Problemnye aspekty modelirovanija populjacionnyh processov i kriterii ih soglasovanija. Matematicheskie mashiny i sistemy. 2017. № 1. S. 71-81 (In Russian).
16. Dubrovskaja V.A., Trofimova I.V. Model' dinamiki strukturirovannyh subpopuljacij osetrovyh ryb Kaspija s uchetom otklonenij v tempah razvitija molodi. Zhurnal Belorusskogo gosudarstvennogo universiteta. Biologija. 2017. № 3. S. 76-86 (In Russian).
17. Nikitina A.V. Jekologo-gidrofizicheskoe obosnovanie vlijanija kojefficienta vertikal'nogo turbulentnogo obmena na soderzhanie rastvorennogo kisloroda v pridonnom sloe melkovodnogo vodoema. Uspehi sovremennogo estestvoznanija. 2018. № 1.
    S. 115-119 (In Russian).
18. Solov'eva T.N. Dinamicheskaja model' degradacii zapasov osetrovyh ryb so slozhnoj vnutripopuljacionnoj strukturoj. Informacionno-upravljajushhie sistemy. 2016. № 4. S. 60-67 (In Russian).
19. Mihajlov V.V., Reshetnikov Ju.S. Model' dinamiki populjacii ryb s raschetom tempov rosta osobej i scenariev gidrologicheskoj obstanovki. Informacionno-upravljajushhie sistemy. 2018. № 4. S. 31-38 (In Russian).
20. Perevaryukha A.Y. A continuous model of three scenarios of the infection process with delayed immune response factors. Biophysics. 2021. Т. 66. № 2. S. 327-348.
21. Belov A.B., Kulikov P.V. Reshennyye i problemnyye voprosy epidemiologii grippa cherez sto let posle pandemii "ispanki". Epidemiologiya i vaktsinoprofilaktika. 2019. T. 18. № 5. S. 109-120 (In Russian).