On mathematical models of COVID-19 pandemic. (Russian. English summary) Zbl 07896823
Summary: 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.
MSC:
| 92D30 | Epidemiology |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 91A16 | Mean field games (aspects of game theory) |
Keywords:
epidemiology; COVID-19; time-series models; SIR; agent-based models; mean field games; inverse problems; forecastingSoftware:
CovasimReferences:
| [1] | M.A. Kondrat’ev, “Forecasting methods and models of disease spread”, Computer Research and Modeling, 5:5 (2013), 863-882 · doi:10.20537/2076-7633-2013-5-5-863-882 |
| [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:10.15622/20.5.2 |
| [3] | H.S. Burkom, S.P. Murphy, G. Shmueli, “Automated time series forecasting for biosurveillance”, Stat. Med., 26:22 (2007), 4202-4218 · doi:10.1002/sim.2835 |
| [4] | R.E. Serfling, “Methods for current statistical analysis of excess pneumonia-influenza deaths”, Public Health Rep., 78:6 (1963), 494-506 · doi:10.2307/4591848 |
| [5] | G. Box, G. Jenkins, Time series analysis forecasting and control, Holden-Day, San Francisco etc, 1970 · Zbl 0249.62009 |
| [6] | P.P. Dabral, M.Z. Murry, “Modelling and forecasting of rainfall time series using SARIMA”, Environ. Process., 4 (2017), 399-419 · doi:10.1007/s40710-017-0226-y |
| [7] | G.E.P. Box, D.R. Cox, “An analysis of transformations. (With discussion)”, J. R. Stat. Soc., Ser. B, 26 (1964), 211-252 · Zbl 0156.40104 |
| [8] | T. Williams, “Adaptive Holt-Winters forecasting”, J. Oper. Res. Soc., 38 (1987), 553-560 · Zbl 0649.62094 |
| [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:10.1016/j.ijforecast.2021.11.007 |
| [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:10.1016/j.ijforecast.2021.10.004 |
| [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:10.1002/sim.2566 |
| [12] | Y. Le Strat, F. Carrat, “Monitoring epidemiologic surveillance data using hidden Markov models”, Stat. Medi., 18:24 (1999), 3463-3478 · doi:10.1002/(SICI)1097-0258(19991230)18:24<3463::AID-SIM409>3.0.CO;2-I |
| [13] | M. Wieczorek, J. Silka, M. Wozniak, “Neural network powered COVID-19 spread forecasting model”, Chaos, Solitons & Fractals, 140 (2020), 110203 · doi:10.1016/j.chaos.2020.110203 |
| [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:10.1007/0-387-35209-0_8 |
| [15] | E.S. Gardner, “Exponential smoothing: the state of the art”, Int. J. Forecasting, 4:1 (1985), 1-28 · doi:10.1002/for.3980040103 |
| [16] | J.D. Hamilton, Time series analysis, Princeton University Press, Princeton, 1994 · Zbl 0831.62061 |
| [17] | F. Petropoulos, D. Apiletti, V. Assimakopoulos et al., “Forecasting: theory and practice”, Int. J. Forecasting, 38:3 (2022), 705-871 · doi:10.1016/j.ijforecast.2021.11.001 |
| [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 · Zbl 1321.92028 |
| [20] | F. Brauer, “Mathematical epidemiology: Past, present, and future”, Infect. Dis. Model., 2:2 (2017), 113-127 |
| [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:10.1021/ja01453a010 |
| [23] | V. Volterra, “Fluctuations in the abundance of a species considered mathematically”, Nature, 118 (1926), 558-560 · JFM 52.0453.03 · doi:10.1038/118558a0 |
| [24] | A.G. McKendrick, “The application of mathematics to medical problems”, Proc. Edinb. Math. Soc., 44 (1926), 98-130 · JFM 52.0542.04 · doi:10.1017/S0013091500034428 |
| [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 · JFM 53.0517.01 · doi:10.1098/rspa.1927.0118 |
| [26] | E. Pelinovsky, A. Kurkin, O. Kurkina, M. Kokoulina, A. Epifanova, “Logistic equation and COVID-19”, Chaos, Solitons & Fractals, 140 (2020), 110241 · Zbl 1495.92096 · doi:10.1016/j.chaos.2020.110241 |
| [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:10.1016/j.chaos.2020.110058 |
| [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:10.33693/2313-223X-2020-7-1-99-105 |
| [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 · Zbl 1439.92170 · doi:10.1515/jiip-2020-0010 |
| [30] | M.V. Tamm, “COVID-19 in Moscow: prognoses and scenarios”, FARMAKOEKONOMIKA. Modern Pharmacoeconomic and Pharmacoepidemiology, 13:1 (2020), 43-51 · doi:10.17749/2070-4909.2020.13.1.43-51 |
| [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:10.1134/S1995423920040047 |
| [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:10.14716/ijtech.v11i8.4529 |
| [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:10.1371/journal.pone.0252271 |
| [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:10.1063/5.0041868 |
| [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:10.1038/s41598-021-83075-6 |
| [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 · Zbl 1489.92171 · doi:10.1016/j.chaos.2020.109846 |
| [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 · JFM 63.1111.04 · doi:10.1111/j.1469-1809.1937.tb02153.x |
| [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 |
| [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 · Zbl 1469.92127 · doi:10.1007/s00466-020-01888-0 |
| [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:10.3390/physics3010008 |
| [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:10.3390/physics3030033 |
| [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: |
| [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 · Zbl 0872.35053 · doi:10.1137/S0036139995284681 |
| [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 · Zbl 1189.35145 · doi:10.1016/j.nonrwa.2009.04.013 |
| [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:10.1103/PhysRevE.64.066112 |
| [48] | R. Pastor-Satorras, A. Vespignani, “Epidemic spreading in scale-free networks”, Phys. Rev. Lett., 86:14 (2001), 3200-3203 · doi:10.1103/PhysRevLett.86.3200 |
| [49] | M.E. Newman, “Spread of epidemic disease on networks”, Phys. Rev. E Stat. Nonlin. Soft. Matter Phys., 66:1-2 (2002), 016128 · doi:10.1103/PhysRevE.66.016128 |
| [50] | F. Wong, J.J. Collins, “Evidence that coronavirus superspreading is fat-tailed”, Proc. Natl. Acad. Sci. USA, 117:47 (2020), 29416-29418 · Zbl 1485.92168 · doi:10.1073/pnas.2018490117 |
| [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 |
| [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 · Zbl 1244.92051 · doi:10.1016/j.mbs.2011.09.002 |
| [53] | I. Sazonov, M. Kelbert, M.B. Gravenor, “Random migration processes between two stochastic epidemic centers”, Math. Biosci., 274 (2016), 45-57 · Zbl 1365.92132 · doi:10.1016/j.mbs.2016.01.011 |
| [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:10.3390/math9172025 |
| [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:10.30932/1992-3252-2020-18-6-06-29 |
| [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 · Zbl 1466.92199 · doi:10.1515/jiip-2020-0043 |
| [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 · Zbl 1489.82066 · doi:10.1134/S0965542521060075 |
| [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 · Zbl 1206.92038 · doi:10.1007/978-3-540-78911-6_6 |
| [59] | A. Berman, R.J. Plemmons, Nonnegative matrices in the mathematical sciences, Academic Press, New York-San Francisco-London, 1979 · Zbl 0484.15016 |
| [60] | T.C. Schelling, “Dynamic models of segregation”, J. Math. Sociol., 1:2 (1971), 143-186 · Zbl 1355.91061 · doi:10.1080/0022250X.1971.9989794 |
| [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 · Zbl 0834.68086 |
| [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:10.1016/j.cmpb.2020.105832 |
| [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:10.3390/pr9010055 |
| [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 № 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:10.17537/2020.15.338 |
| [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:10.1371/journal.pcbi.1009149 |
| [68] | Covasim documentation: |
| [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:10.1038/s41562-020-0931-9 |
| [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:10.1073/pnas.2011802117 |
| [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:10.1016/S1473-3099(20)30457-6 |
| [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:10.1038/s41591-020-1001-6 |
| [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:10.1016/S2214-109X(20)30074-7 |
| [74] | B.F. Nielsen, K. Sneppen, COVID-19 superspreading suggests mitigation by social network modulation, medRxiv, |
| [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:10.17537/2021.16.169 |
| [76] | Household Size, |
| [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:10.7326/M20-0504 |
| [78] | R. Wölfel, V.M. Corman, W. Guggemos et al, “Virological assessment of hospitalized patients with COVID-2019”, Nature, 581 (2020), 465-469 · doi:10.1038/s41586-020-2196-x |
| [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:10.1016/S1473-3099(20)30243-7 |
| [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:10.1001/jama.2020.1585 |
| [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:10.1073/pnas.2014564118 |
| [82] | J. Hatzius, D. Struyven, I. Rosenbery, Face masks and GDP, |
| [83] | J.M. Smith, G.R. Price, “The logic of animal conflict”, Nature, 246:5427 (1973), 15-18 · Zbl 1369.92134 · doi:10.1038/246015a0 |
| [84] | B. Jovanovic, R.W. Rosenthal, “Anonymous sequential games”, J. Math. Econ., 17:1 (1988), 77-87 · Zbl 0643.90107 · doi:10.1016/0304-4068(88)90029-8 |
| [85] | J.-M. Lasry, P.-L. Lions, “Mean field games”, Jpn. J. Math., 2:1 (2007), 229-260 · Zbl 1156.91321 · doi:10.1007/s11537-007-0657-8 |
| [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 1287.91019 |
| [87] | V.N. Kolokoltsov, J. Li., W. Yang, Mean field games and nonlinear Markov processes, 2012, arXiv: |
| [88] | D. Andersson, B. Djehiche, “A maximum principle for SDEs of meanfield type”, Appl. Math. Optim., 63:3 (2011), 341-356 · Zbl 1215.49034 · doi:10.1007/s00245-010-9123-8 |
| [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 · Zbl 1176.60042 · doi:10.1214/08-AOP442 |
| [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 · Zbl 1371.92122 · doi:10.1016/j.mbs.2015.03.002 |
| [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 · Zbl 1458.92075 · doi:10.1137/20M1342690 |
| [92] | H. Tembine, “COVID-19: data-driven mean-field-type game perspective”, Games, 11:4 (2020), 51 · Zbl 1457.91062 · doi:10.3390/g11040051 |
| [93] | W. Lee, S. Liu, W. Li, S. Osher, “Mean field control problems for vaccine distribution”, Res. Math. Sci., 9:3 (2022), 51 · Zbl 1498.92114 · doi:10.1007/s40687-022-00350-2 |
| [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 · Zbl 0077.13605 |
| [96] | M. Fischer, “On the connection between symmetric N-player games and mean field games”, Ann. Appl. Probab., 27:2 (2017), 757-810 · Zbl 1375.91009 · doi:10.1214/16-AAP1215 |
| [97] | A. Bensoussan, J. Frehse, Ph. Yam, Mean field games and mean field type control theory, Springer, New York, 2013 · Zbl 1287.93002 |
| [98] | D. Gomes, J. Mohr, R.R. Souza, “Continuous time finite state mean field games”, Appl. Math. Optim., 68:1 (2013), 99-143 · Zbl 1283.91016 · doi:10.1007/s00245-013-9202-8 |
| [99] | V.V. Kolokoltsov, J.J. Li, W. Yang, Mean field games and nonlinear Markov processes, 2011, arXiv: |
| [100] | V.V. Kolokoltsov, W. Yang, Sensitivity analysis for HJB equations with an application to a coupled backward-forward system, 2013, arXiv: |
| [101] | R. Carmona, F. Delarue, “Probabilistic analysis of mean-field games”, SIAM J. Control Optim., 51:4 (2013), 2705-2734 · Zbl 1275.93065 · doi:10.1137/120883499 |
| [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:10.1016/S0304-3800(01)00463-X |
| [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:10.1016/j.ecolmodel.2009.01.014 |
| [104] | Y. Achdou, I. Capuzzo-Dolcetta, “Mean field games: numerical methods”, SIAM J. Numer. Anal., 48:3 (2010), 1136-1162 · Zbl 1217.91019 · doi:10.1137/090758477 |
| [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 · Zbl 1271.65120 · doi:10.1007/978-3-642-36433-4_1 |
| [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:10.1063/5.0033522 |
| [107] | B. Kaltenbacher, A. Neubauer, O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems, De Gruyter, Berlin, 2008 · Zbl 1145.65037 |
| [108] | S. Kabanikhin, “Definitions and examples of inverse and ill-posed problems”, J. Inverse Ill-Posed Probl., 16:4 (2008), 317-357 · Zbl 1170.35100 · doi:10.1515/JIIP.2008.019 |
| [109] | A.N. Tikhonov, A.V. Goncharskij, V.V. Stepanov, A.G. Yagola, Regularizing algorithms and a priori information, Nauka Fizmatlit, M., 1983 · Zbl 0529.65027 |
| [110] | A.N. Tikhonov, A.S. Leonov, A.G. Yagola, Nonlinear incorrect problems, Nauka Fizmatlit, M., 1995 · Zbl 0856.65066 |
| [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 · Zbl 1215.34015 · doi:10.1137/090757009 |
| [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 · Zbl 1075.92030 · doi:10.1016/j.cam.2005.02.004 |
| [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:10.1371/journal.pcbi.1003968 |
| [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 · Zbl 1505.92201 · doi:10.1134/S1990478920010123 |
| [115] | A.N. Tikhonov, “On the stability of inverse problems”, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., 39 (1943), 176-179 · Zbl 0061.23308 |
| [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 · Zbl 0051.34604 |
| [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 · Zbl 0793.65014 |
| [118] | S.I. Kabanikhin, O.I. Krivorotko, “Singular value decomposition in an inverse source problem”, Numer. Analysis Appl., 5:2 (2012), 168-174 · Zbl 1299.35314 · doi:10.1134/S1995423912020115 |
| [119] | M.M. Lavrent’ev, Some improperly posed problems of mathematical physics, Springer-Verlag, Berlin-Heidelberg-New York, 1967 · Zbl 0149.41902 |
| [120] | A.N.Tikhonov, “Solution of incorrectly formulated problems and the regularization method”, Sov. Math., Dokl., 5 (1963), 1035-1038 · Zbl 0141.11001 |
| [121] | V.K.Ivanov, “On ill-posed problems”, Mat. Sb., N. Ser., 61(103):2 (1963), 211-223 · Zbl 0208.16203 |
| [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 · Zbl 1369.92002 · doi:10.1515/jiip-2015-0072 |
| [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 · Zbl 1460.35394 · doi:10.1515/jiip-2020-0083 |
| [124] | A.A. Zhiglyavskij, A.G. Zhilinskas, Methods for the search of a global extremum, Nauka, M., 1991 · Zbl 0803.90115 |
| [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 · Zbl 0827.15029 |
| [126] | I.V. Oseledets, “Tensor-train decomposition”, SIAM J. Sci. Comput., 33:5 (2011), 2295-2317 · Zbl 1232.15018 · doi:10.1137/090752286 |
| [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 · Zbl 1383.92047 · doi:10.1515/jiip-2016-0083 |
| [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) |
| [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 |
| [131] | Software package for modelling COVID-19 in Novosibirsk region, |
| [132] | OPTUNA: hyperparameter optimization framework, |
| [133] | R. Casagrandi, L. Bolzoni, S. Levin, V. Andreasen, “The SIRC modeland influenza A”, Math. Biosci., 200:2 (2006), 152-169 · Zbl 1089.92043 · doi:10.1016/j.mbs.2005.12.029 |
| [134] | V. Petrakova, O. Krivorotko, “Mean field game for modeling of COVID-19 spread”, J. Math. Anal. Appl., 514:1 (2022), 126271 · Zbl 1492.91044 · doi:10.1016/j.jmaa.2022.126271 |
| [135] | Federal State Statistics Service, Novosibirsk region, |
| [136] | Telegram channel with the results of modeling the spread of COVID-19 in the Novosibirsk region, |
| [137] | Self-isolation index from Yandex, |
| [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, |
| [140] | Software package for reconstruction of epidemiologic parameters of agent model of Covasim package, |
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