On mathematical models of COVID-19 pandemic

O. I. Krivorotko ; S. I. Kabanikhin

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1211-1268.

Résumé

The mathematical models for analysis and forecasting of COVID-19 pandemic based on time-series models, differential equations (SIR models based on odinary, partial and stochastic differential equations), agent-based models, mean field games and its combinations are considered. Inverse problems for mathematical models in epidemiology of COVID-19 are formulated in the variational form. The numerical results of modeling and scenarios of COVID-19 propagation in Novosibirsk region are demonstrated and discussed. The epidemiology parameters of COVID-19 propagation in Novosibirsk region (contagiosity, hospitalization and mortality rates, asymptomatic cases) are identified. The combination of differential and agent-based models increases the quality of forecast scenarios.

Détail

Keywords: epidemiology, COVID-19, time-series models, agent-based models, mean field games, inverse problems, forecasting.
Mots-clés : SIR

Bibliographie

[1] M.A. Kondrat'ev, “Forecasting methods and models of disease spread”, Computer Research and Modeling, 5:5 (2013), 863–882 | DOI

[2] V. Zakharov, Y. Balykina, “Balance model of COVID-19 epidemic based on percentage growth rate”, Informatics and Automation, 20:5 (2021), 1034–1064 | DOI

[3] H.S. Burkom, S.P. Murphy, G. Shmueli, “Automated time series forecasting for biosurveillance”, Stat. Med., 26:22 (2007), 4202–4218 | DOI | MR

[4] R.E. Serfling, “Methods for current statistical analysis of excess pneumonia-influenza deaths”, Public Health Rep., 78:6 (1963), 494–506 | DOI

[5] G. Box, G. Jenkins, Time series analysis forecasting and control, Holden-Day, San Francisco etc, 1970 | MR | Zbl

[6] P.P. Dabral, M.Z. Murry, “Modelling and forecasting of rainfall time series using SARIMA”, Environ. Process., 4 (2017), 399–419 | DOI | MR

[7] G.E.P. Box, D.R. Cox, “An analysis of transformations. (With discussion)”, J. R. Stat. Soc., Ser. B, 26 (1964), 211–252 | MR | Zbl

[8] T. Williams, “Adaptive Holt-Winters forecasting”, J. Oper. Res. Soc., 38 (1987), 553–560 | Zbl

[9] Y. In, J. Jung, “Simple averaging of direct and recursive forecasts via partial pooling using machine learning”, Int. J. Forecasting, 38:4 (2022), 1386–1399 | DOI

[10] T. Januschowski, Y. Wang, K. Torkkola, T. Erkkila, H. Hasson, J. Gasthaus, “Forecasting with trees”, Int. J. Forecasting, 38:4 (2022), 1473–1481 | DOI

[11] P. Sebastiani, K.D. Mandl, P. Szolovits, I.S. Kohane, M.F. Ramoni, “A Bayesian dynamic model for influenza surveillance”, Stat. Med., 25:11 (2006), 1803–1816 | DOI | MR

[12] Y. Le Strat, F. Carrat, “Monitoring epidemiologic surveillance data using hidden Markov models”, Stat. Medi., 18:24 (1999), 3463–3478 | 3.0.CO;2-I class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[13] M. Wieczorek, J. Silka, M. Wozniak, “Neural network powered COVID-19 spread forecasting model”, Chaos, Solitons Fractals, 140 (2020), 110203 | DOI | MR

[14] G. Shmueli, S.E. Fienberg, “Current and potential statistical methods for monitoring multiple data streams for biosurveillance”, Statistical Methods in Counterterrorism, eds. Wilson A.G., Wilson G.D., Olwell D.H., Springer, New York, 2006, 109–140 | DOI | MR

[15] E.S. Gardner, “Exponential smoothing: the state of the art”, Int. J. Forecasting, 4:1 (1985), 1–28 | DOI | MR

[16] J.D. Hamilton, Time series analysis, Princeton University Press, Princeton, 1994 | MR | Zbl

[17] F. Petropoulos, D. Apiletti, V. Assimakopoulos et al., “Forecasting: theory and practice”, Int. J. Forecasting, 38:3 (2022), 705–871 | DOI

[18] D. Bernoulli, Essai d'une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l'inoculation pour la prevenir, Mem. Math. and Phys. de l'Acad. Roy. Sci., Hist. de l'Acad. Roy. Sci., Ann., Paris, 1760

[19] N. Bacaër, A short history of mathematical population dynamics, Springer, London, 2011 | MR | Zbl

[20] F. Brauer, “Mathematical epidemiology: Past, present, and future”, Infect. Dis. Model., 2:2 (2017), 113–127 | MR

[21] R. Ross, The prevention of malaria, John Murray, London, 1911

[22] A.J. Lotka, “Undamped oscillations derived from the law of mass action”, J. Am. Chem. Soc., 42:8 (1920), 1595–1599 | DOI

[23] V. Volterra, “Fluctuations in the abundance of a species considered mathematically”, Nature, 118 (1926), 558–560 | DOI | Zbl

[24] A.G. McKendrick, “The application of mathematics to medical problems”, Proc. Edinb. Math. Soc., 44 (1926), 98–130 | DOI | Zbl

[25] W.O. Kermack, A.G. McKendrick, “A contribution to the mathematical theory of epidemics”, Proc. R. Soc. Lond., Ser. A, 115 (1927), 700–721 | DOI | Zbl

[26] E. Pelinovsky, A. Kurkin, O. Kurkina, M. Kokoulina, A. Epifanova, “Logistic equation and COVID-19”, Chaos, Solitons Fractals, 140 (2020), 110241 | DOI | MR | Zbl

[27] P. Wang, X. Zheng, J. Li, B. Zhu, “Prediction of epidemic trends in COVID-19 with logistic model and machine learning technics”, Chaos, Solitons Fractals, 139 (2020), 110058 | DOI | MR

[28] E.M. Koltsova, E.S. Kurkina, A.M. Vasetsky, “Mathematical modeling of the spread of COVID-19 in Moscow”, Computational Nanotechnology, 7:1 (2020), 99–105 | DOI | MR

[29] Y. Chen, J. Cheng, Y. Jiang, K. Liu, “A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification”, J. Inverse Ill-Posed Probl., 28:2 (2020), 243–250 | DOI | MR | Zbl

[30] M.V. Tamm, “COVID-19 in Moscow: prognoses and scenarios”, FARMAKOEKONOMIKA. Modern Pharmacoeconomic and Pharmacoepidemiology, 13:1 (2020), 43–51 | DOI

[31] E. Unlu, H. Leger, O. Motornyi, A. Rukubayihunga, T. Ishacian, M. Chouiten, Epidemic analysis of COVID-19 Outbreak and Counter-Measures in France, MedRxiv, 2020

[32] O.I. Krivorotko, S.I. Kabanikhin, N.Y. Zyatkov et al, “Mathematical modeling and forecasting of COVID-19 in Moscow and Novosibirsk region”, Numer. Analys. Appl., 13 (2020), 332–348 | DOI | MR

[33] A.I. Borovkov, M.V. Bolsunovskaya, A.M. Gintciak, T.Yu. Kudryavtseva, “Simulation modelling application for balancing epidemic and economic crisis in the region”, Int. J. Technology, 11:8 (2020), 1579–1588 | DOI

[34] H.M. Yang, L.P. Junior Lombardi, F.F.M. Castro, A.C. Yang, “Mathematical modeling of the transmission of SARS-CoV-2-Evaluating the impact of isolation in São Paulo State (Brazil) and lockdown in Spain associated with protective measures on the epidemic of CoViD-19”, PLoS One, 16:6 (2021), e0252271 | DOI | MR

[35] S. Margenov, N. Popivanov, I. Ugrinova, S. Harizanov, T. Hristov, “Mathematical and computer modeling of COVID-19 transmission dynamics in Bulgaria by time-depended inverse SEIR model”, AIP Conference Proceedings, 2333:1 (2021), 090024 | DOI

[36] C.J. Silva, C. Cruz, D.F.M. Torres, et al, “Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in Portugal”, Scientific Reports, 11 (2021), 3451 | DOI

[37] F. Ndaïrou, I. Area, J.J. Nieto, D.F.M. Torres, “Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan”, Chaos Solitons Fractals, 135 (2020), 109846 | DOI | MR | Zbl

[38] I.N. Kiselev, I.R. Akberdin, F.A. Kolpakov, A delay differential equation approach to model the COVID-19 pandemic, MedRxiv, 2021

[39] R.A. Fisher, “The wave of advance of advantageous genes”, Ann. Eugenics, London, 7 (1937), 355–369 | DOI | Zbl

[40] A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, “Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem”, Bull. Moscow Univ. Math. Mech., 1:6 (1937), 1–25 | Zbl

[41] A. Viguerie, A. Veneziani, G. Lorenzo et al, “Diffusion-reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study”, Comput. Mech., 66:5 (2020), 1131–1152 | DOI | MR | Zbl

[42] V.V. Aristov, A.V. Stroganov, A.D. Yastrebov, “Simulation of spatial spread of the COVID-19 pandemic on the basis of the kinetic-advection model”, Physics, 3:1 (2021), 85–102 | DOI

[43] G. Bärwolff, “A local and time resolution of the COVID-19 propagation – a two-dimensional approach for Germany including diffusion phenomena to describe the spatial spread of the COVID-19 pandemic”, Physics, 3:3 (2021), 536–548 | DOI | MR

[44] Z. Lau, I.M. Griffiths, A. English, K. Kaouri, Predicting the spatially varying infection risk in indoor spaces using an efficient airborne transmission model, 2021, arXiv: 2012.12267v3

[45] V. Capasso, R.E. Wilson, “Analysis of a reaction-diffusion system modeling man-environment-man epidemics”, SIAM J. Appl. Math., 57:2 (1997), 327–346 | DOI | MR | Zbl

[46] A.S. Bratus', V.P. Posvyanskii, A.S. Novozhilov, “Existence and stability of stationary solutions to spatially extended autocatalytic and hypercyclic systems under global regulation and with nonlinear growth rates”, Nonlinear Anal., Real World Appl., 11:3 (2010), 1897–1917 | DOI | MR | Zbl

[47] R.M. May, A.L. Lloyd, “Infection dynamics on scale-free networks”, Phys. Rev. E Stat. Nonlin. Soft. Matter Phys., 64:6-2 (2001), 066112 | DOI

[48] R. Pastor-Satorras, A. Vespignani, “Epidemic spreading in scale-free networks”, Phys. Rev. Lett., 86:14 (2001), 3200–3203 | DOI

[49] M.E. Newman, “Spread of epidemic disease on networks”, Phys. Rev. E Stat. Nonlin. Soft. Matter Phys., 66:1-2 (2002), 016128 | DOI | MR

[50] F. Wong, J.J. Collins, “Evidence that coronavirus superspreading is fat-tailed”, Proc. Natl. Acad. Sci. USA, 117:47 (2020), 29416–29418 | DOI

[51] L.J.S. Allen, “An introduction to stochastic epidemic models”, Mathematical epidemiology, Lecture Notes in Mathematics, eds. Bauer Fred et al., Springer, Berlin, 1945 | MR | Zbl

[52] I. Sazonov, M. Kelbert, M.B. Gravenor, “A two-stage model for the SIR outbreak: Accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage”, Math. Biosci., 234:2 (2011), 108–117 | DOI | MR | Zbl

[53] I. Sazonov, M. Kelbert, M.B. Gravenor, “Random migration processes between two stochastic epidemic centers”, Math. Biosci., 274 (2016), 45–57 | DOI | MR | Zbl

[54] I. Sazonov, D. Grebennikov, A. Meyerhans, G. Bocharov, “Markov chain-based stochastic modelling of HIV-1 life cycle in a CD4 T cell”, Mathematics, 9:17 (2021), 2025 | DOI

[55] A.S. Bratus, A.S. Ocheretyanaya, “Mathematical model of spread of the epidemic inside a railway compartment coach”, World of Transport and Transportation, 18:6 (2020), 6–29 | DOI

[56] G.Z. Lotova, G.A. Mikhailov, “Numerically statistical investigation of the partly super-exponential growth rate in the COVID-19 pandemic (throughout the world)”, J. Inverse Ill-Posed Probl., 28:6 (2020), 877–879 | DOI | MR | Zbl

[57] G.Z. Lotova, G.A. Mikhailov, “Numerical-statistical and analytical study of asymptotics for the average multiplication particle flow in a random medium”, Comput. Math. Math. Phys., 61:8 (2021), 1330–1338 | DOI | MR | Zbl

[58] P. van den Driessche, J. Watmough, “Further notes on the basic reproduction number”, Mathematical epidemiology, Lecture Notes in Mathematics, 1945, eds. Brauer F., van den Driessche P., Wu J., Springer, Berlin–Heidelberg, 2008, 159–178 | DOI | MR | Zbl

[59] A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York–San Francisco–London, 1979 | MR | Zbl

[60] T.C. Schelling, “Dynamic models of segregation”, J. Math. Sociol., 1:2 (1971), 143–186 | DOI | Zbl

[61] M. Mitchell, P.T. Hraber, J.P. Crutchfield, “Revisiting the edge of chaos: Evolving cellular automata to perform computations”, Complex Syst., 7:2 (1993), 89–130 | MR | Zbl

[62] P.H.T. Schimit, “A model based on cellular automata to estimate the social isolation impact on COVID-19 spreading in Brazil”, Comput. Methods Prog. Biomed., 200 (2021), 105832 | DOI

[63] J. Dai, C. Zhai, J. Ai, J. Ma, J. Wang, W. Sun, “Modeling the Spread of Epidemics Based on Cellular Automata”, Processes, 9:1 (2021), 55 | DOI | MR

[64] P. Patlolla, V. Gunupudi, A.R. Mikler, R.T. Jacob, “Agent-based simulation tools in computational epidemiology”, International Conference on Innovative Internet Community Systems, LNCS, 3473, eds. Böhme T., Larios Rosillo V.M., Unger H., Unger H., Springer, Berlin–Heidelberg, 2004, 212–223

[65] V.A. Adarchenko, S.A. Baban, A.A. Bragin, K.F. Grebenkin et al., Modeling the development of the coronavirus epidemic using differential and statistical models, Preprint No 264, RFNC-VNIITF, 2020

[66] A.I. Vlad, T.E. Sannikova, A.A. Romanyukha, “Transmission of acute respiratory infections in a city: agent-based approach”, Mat. Biolog. Bioinform., 15:2 (2020), 338–356 | DOI | MR

[67] C.C. Kerr, R.M. Stuart, D. Mistry, R.G. Abeysuriya, K. Rosenfeld, G.R. Hart, et al, “Covasim: An agent-based model of COVID-19 dynamics and interventions”, PLoS Comput. Biol., 17:7 (2021), e1009149 | DOI

[68] Covasim documentation: https://docs.idmod.org/projects/covasim/en/latest/index.html

[69] A. Aleta, D. Martin-Corral, Y. Pastore, A. Piontti, M. Ajelli, M. Litvinova, et al, “Modelling the impact of testing, contact tracing and household quarantine on second waves of COVID-19”, Nat. Hum. Behav., 4:9 (2020), 964–971 | DOI

[70] M.S.Y. Lau, B. Grenfell, M. Thomas, M. Bryan, K. Nelson, B. Lopman, “Characterizing superspreading events and age-specific infectiousness of SARS-CoV-2 transmission in Georgia, USA”, Proc. Natl. Acad. Sci. USA, 117:36 (2020), 22430–22435 | DOI

[71] A.J. Kucharski, P. Klepac, A.J.K. Conlan, S.M. Kissler, M.L. Tang, H. Fry, et al, “Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: a mathematical modelling study”, Lancet Infect. Dis., 20:10 (2020), 1151–1160 | DOI

[72] N. Hoertel, M. Blachier, C. Blanco, et al, “A stochastic agent-based model of the SARS-CoV-2 epidemic in France”, Nat. Med., 26:9 (2020), 1417–1421 | DOI

[73] J. Hellewell, S. Abbott, A. Gimma, N.I. Bosse, C.I. Jarvis, T.W. Russell, et al, “Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts”, Lancet. Glob. Health, 8:4 (2020), e488–e496 | DOI

[74] B.F. Nielsen, K. Sneppen, COVID-19 superspreading suggests mitigation by social network modulation, medRxiv, 2020.09.15.20195008

[75] K.K. Loginov, N.V. Pertsev, “Direct statistical modeling of spread of epidemic based on a stage-dependent stochastic model”, Mat. Biolog. Bioinform., 16:2 (2021), 169–200 | DOI

[76] Household Size, , UN, 2019 https://population.un.org/Household/#/countries/840

[77] S.A. Lauer, K.H. Grantz, Q. Bi et al, “The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application”, Ann. Intern. Med., 172:9 (2020), 577–582 | DOI

[78] R. Wölfel, V.M. Corman, W. Guggemos et al, “Virological assessment of hospitalized patients with COVID-2019”, Nature, 581 (2020), 465–469 | DOI

[79] R. Verity, L.C. Okell, I. Dorigatti et al, “Estimates of the severity of coronavirus disease 2019: a model-based analysis”, The Lancet Infect. Dis., 20:6 (2020), 669–677 | DOI | MR

[80] D. Wang, B. Hu, C. Hu et al, “Clinical characteristics of 138 hospitalized patients with 2019 novel coronavirus-infected pneumonia in Wuhan, China”, JAMA, 323:11 (2020), 1061–1069 | DOI | MR

[81] J. Howard, A. Huang, Z. Li et al, “An evidence review of face masks against COVID-19”, Proc. Natl. Acad. Sci. USA, 118:4 (2021), e2014564118 | DOI | MR

[82] J. Hatzius, D. Struyven, I. Rosenbery, Face masks and GDP, https://www.goldmansachs.com/insights/pages/face-masks-and-gdp.html

[83] J.M. Smith, G.R. Price, “The logic of animal conflict”, Nature, 246:5427 (1973), 15–18 | DOI | Zbl

[84] B. Jovanovic, R.W. Rosenthal, “Anonymous sequential games”, J. Math. Econ., 17:1 (1988), 77–87 | DOI | MR | Zbl

[85] J.-M. Lasry, P.-L. Lions, “Mean field games”, Jpn. J. Math., 2:1 (2007), 229–260 | DOI | MR | Zbl

[86] V.N. Kolokoltsov, M.S. Troeva, Wei Yang, “Mean field games based on the stable-like processes”, Mat. Teor. Igr Pril., 5:4 (2013), 33–65 | Zbl

[87] V.N. Kolokoltsov, J. Li., W. Yang, Mean field games and nonlinear Markov processes, 2012, arXiv: 1112.3744 | MR

[88] D. Andersson, B. Djehiche, “A maximum principle for SDEs of meanfield type”, Appl. Math. Optim., 63:3 (2011), 341–356 | DOI | MR | Zbl

[89] R. Buckdahn, B. Djehiche, J. Li, S. Peng, “Mean-field backward stochastic differential equations: a limit approach”, Ann. Prob., 37:4 (2009), 1524–1565 | DOI | MR | Zbl

[90] L. Laguzet, G. Turinici, “Global optimal vaccination in the SIR model: properties of the value function and application to cost-effectiveness analysis”, Math. Biosci., 263 (2015), 180–197 | DOI | MR | Zbl

[91] W. Lee, S. Liu, H. Tembine, W. Li, S. Osher, “Controlling propagation of epidemics via mean-field control”, SIAM J. Appl. Math., 81:1 (2021), 190–207 | DOI | MR | Zbl

[92] H. Tembine, “COVID-19: data-driven mean-field-type game perspective”, Games, 11:4 (2020), 51 | DOI | MR | Zbl

[93] W. Lee, S. Liu, W. Li, S. Osher, “Mean field control problems for vaccine distribution”, Res. Math. Sci., 9:3 (2022), 51 | DOI | MR | Zbl

[94] N.N.Bogolybov, N.M.Krylov, “Ob uravneniyah Fokkera-Planka, kotorye vyvodyatsya v teorii vozmushchenij metodom, osnovannym na spektral'nyh svojstvah vozmushchyonnogo gamil'toniana”, Zapiski kafedry matematicheskoj fiziki Instituta nelinejnoj mekhaniki AN USSR, 1939:4 (1939), 5–80

[95] R.E. Bellman, Dynamic programming, Princeton University Press, Princeton, New Jersey, 1957 | MR | Zbl

[96] M. Fischer, “On the connection between symmetric N-player games and mean field games”, Ann. Appl. Probab., 27:2 (2017), 757–810 | DOI | MR | Zbl

[97] A. Bensoussan, J. Frehse, Ph. Yam, Mean field games and mean field type control theory, Springer, New York, 2013 | MR | Zbl

[98] D. Gomes, J. Mohr, R.R. Souza, “Continuous time finite state mean field games”, Appl. Math. Optim., 68:1 (2013), 99–143 | DOI | MR | Zbl

[99] V.V. Kolokoltsov, J.J. Li, W. Yang, Mean field games and nonlinear Markov processes, 2011, arXiv: 1112.3744v2 | MR

[100] V.V. Kolokoltsov, W. Yang, Sensitivity analysis for HJB equations with an application to a coupled backward-forward system, 2013, arXiv: 1303.6234

[101] R. Carmona, F. Delarue, “Probabilistic analysis of mean-field games”, SIAM J. Control Optim., 51:4 (2013), 2705–2734 | DOI | MR | Zbl

[102] L. Berec, “Techniques of spatially explicit individual-based models: Construction, simulation, and mean-field analysis”, Ecol. Modell., 150:1-2 (2002), 55–81 | DOI

[103] P.H.T. Schimit, L.H.A Monteiro, “On the basic reproduction number and the topological properties of the contact network: An epidemiological study in mainly locally connected cellular automata”, Ecol. Modell., 220:7 (2009), 1034–1042 | DOI | MR

[104] Y. Achdou, I. Capuzzo-Dolcetta, “Mean field games: numerical methods”, SIAM J. Numer. Anal., 48:3 (2010), 1136–1162 | DOI | MR | Zbl

[105] Y. Achdou, “Finite difference methods for mean field games”, Hamilton-Jacobi equations: approximations, numerical analysis and applications, Lecture Notes in Mathematics, 2074, eds. Loreti Paola et al., Springer, Berlin, 2013, 1–47 | DOI | MR | Zbl

[106] V.V. Shaydurov, S. Zhang, V.S. Kornienko, “A finite-difference solution of mean field problem with the fractional derivative for subdiffusion”, AIP Conference Proceeding, 2302 (2020), 110002 | DOI

[107] B. Kaltenbacher, A. Neubauer, O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems, De Gruyter, Berlin, 2008 | MR | Zbl

[108] S. Kabanikhin, “Definitions and examples of inverse and ill-posed problems”, J. Inverse Ill-Posed Probl., 16:4 (2008), 317–357 | DOI | MR | Zbl

[109] A.N. Tikhonov, A.V. Goncharskij, V.V. Stepanov, A.G. Yagola, Regularizing algorithms and a priori information, Nauka Fizmatlit, M., 1983 | MR | Zbl

[110] A.N. Tikhonov, A.S. Leonov, A.G. Yagola, Nonlinear incorrect problems, Nauka Fizmatlit, M., 1995 | Zbl

[111] H. Miao, X. Xia, A.S. Perelson, H. Wu, “On identifiability of nonlinear ODE models and applications in viral dynamics”, SIAM Rev., 53:1 (2011), 3–39 | DOI | MR | Zbl

[112] B.M. Adams, H.T. Banks et al., “HIV dynamics: modeling, data analysis, and optimal treatment protocols”, J. Comput. Appl. Math., 184:1 (2005), 10–49 | DOI | MR | Zbl

[113] I. Andrianakis, I.R. Vernon, N. McCreesh, T.J. McKinley, J.E. Oakley, R.N. Nsubuga, M. Goldstein, R.G. White, “Bayesian history matching of complex infectiousdisease models using emulation: a tutorial and a case study on HIV in Uganda”, PLOS Computational Biology, 11:1 (2015), e1003968 | DOI

[114] O.I. Krivorotko, D.V. Andornaya, S.I. Kabanikhin, “Sensitivity analysis and practical identifiability of some mathematical models in biology”, J. Appl. Ind. Math., 14:1 (2020), 115–130 | DOI | MR | Zbl

[115] A.N. Tikhonov, “On the stability of inverse problems”, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., 39 (1943), 176–179 | MR | Zbl

[116] M.M. Lavrent'ev, “On the issue of improving accuracy solving a system of linear equations”, Dokl. Akad. Nauk SSSR, n. Ser., 92:5 (1953), 885–886 | MR | Zbl

[117] S.K. Godunov, A.G. Antonov, O.P. Kirilyuk, V.I. Kostin, Guaranteed accuracy in numerical linear algebra, Mathematics and its Applications (Dordrecht), 252, Kluwer Academic Publishers, Dordrecht, 1993 | MR | Zbl

[118] S.I. Kabanikhin, O.I. Krivorotko, “Singular value decomposition in an inverse source problem”, Numer. Analysis Appl., 5:2 (2012), 168–174 | DOI | Zbl

[119] M.M. Lavrent'ev, Some improperly posed problems of mathematical physics, Springer-Verlag, Berlin-Heidelberg-New York, 1967 | Zbl

[120] A.N.Tikhonov, “Solution of incorrectly formulated problems and the regularization method”, Sov. Math., Dokl., 5 (1963), 1035–1038 | MR | Zbl

[121] V.K.Ivanov, “On ill-posed problems”, Mat. Sb., N. Ser., 61(103):2 (1963), 211–223 | MR | Zbl

[122] S.I. Kabanikhin, O.I. Krivorotko, “Identification of biological models described by systems of nonlinear differential equations”, J. Inverse Ill-Posed Probl., 23:5 (2015), 519–527 | DOI | MR | Zbl

[123] O. Krivorotko, S. Kabanikhin, Sh. Zhang, V. Kashtanova, “Global and local optimization in identification of parabolic systems”, J. Inverse Ill-Posed Probl., 28:6 (2020), 899–913 | DOI | MR | Zbl

[124] A.A. Zhiglyavskij, A.G. Zhilinskas, Methods for the search of a global extremum, Nauka, M., 1991 | MR | Zbl

[125] E.E. Tyrtyshnikov, “New theorems on the distribution of eigenvalues and singular numbers of multilevel Toeplitz matrices”, Russ. Acad. Sci., Dokl., Math., 48:3 (1994), 534–528 | MR | Zbl

[126] I.V. Oseledets, “Tensor-train decomposition”, SIAM J. Sci. Comput., 33:5 (2011), 2295–2317 | DOI | MR | Zbl

[127] D.A. Zheltkov, I.V. Oferkin, E.V. Katkova, A.V. Sulimov, V.B. Sulimov, E.E. Tyrtyshnikov, “TTDock: a docking method based on tensor train decompositions”, Num. Meth. Prog., 14:3 (2013), 279–291

[128] V.V. Zheltkova, D.A. Zheltkov, Z. Grossman, G.A. Bocharov, E.E. Tyrtyshnikov, “Tensor based approach to the numerical treatment of the parameter estimation problems in mathematical immunology”, J. Inverse Ill-Posed Probl., 26:1 (2018), 51–66 | DOI | MR | Zbl

[129] O.I. Krivorotko, M.I. Sosnovskaia, I. Vashchenko, “Agent-based mathematical model of COVID-19 spread in Novosibirsk region: identifiability, optimization and forecasting”, J. Inverse Ill-Posed Probl., 2021 (to appear) | MR | Zbl

[130] O.I. Krivorotko, S.I. Kabanikhin, M.I. Sosnovskaya, D.V. Andornaya, “Sensitivity and identifiability analysis of COVID-19 pandemic models”, Vavilovskii Zhurnal Genet. Selektsii, 25:1 (2021), 82–91 | MR

[131] Software package for modelling COVID-19 in Novosibirsk region, http://covid19-modeling.ru/

[132] OPTUNA: hyperparameter optimization framework, https://optuna.org/

[133] R. Casagrandi, L. Bolzoni, S. Levin, V. Andreasen, “The SIRC modeland influenza A”, Math. Biosci., 200:2 (2006), 152–169 | DOI | MR | Zbl

[134] V. Petrakova, O. Krivorotko, “Mean field game for modeling of COVID-19 spread”, J. Math. Anal. Appl., 514:1 (2022), 126271 | DOI | MR | Zbl

[135] Federal State Statistics Service, Novosibirsk region, https://novosibstat.gks.ru/folder/31729

[136] Telegram channel with the results of modeling the spread of COVID-19 in the Novosibirsk region, https://t.me/COVID2019_official

[137] Self-isolation index from Yandex, https://yandex.ru/company/researches/2020/podomam

[138] Laboratory data from the Invitro medical center in Novosibirsk on the daily dynamics of the percentage of individuals with antibodies to COVID-19

[139] Software package Covasim, , GitHub https://github.com/InstituteforDiseaseModeling/covasim

[140] Software package for reconstruction of epidemiologic parameters of agent model of Covasim package, , GitHub https://github.com/msosnovskaya/autocalibration-covasim-pub