DIFFERENTIAL EPIDEMIC MODELS AND SCENARIOS FOR RESTRICTIVE MEASURES

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Abstract

We consider algorithms for calculating the spread of epidemics and analyzing the consequences of introducing or removing restrictive measures based on the SIR model and the Hamilton–Jacobi–Bellman equation. After studying the identifiability and sensitivity of the SIR models, the correctness in the neighborhood of the exact solution and the convergence of the numerical algorithms for solving forward and inverse problems, the optimal control problem is formulated. Numerical simulation results show that feedback control can help determine vaccination policies. The use of PINN neural networks reduced the computation time by a factor of 5, which seems important for promptly changing restrictive measures.

About the authors

S. I Kabanikhin

S.L. Sobolev Institute of Mathematics SB RAS

Novosibirsk, Russia

O. I Krivorotko

S.L. Sobolev Institute of Mathematics SB RAS

Email: krivorotko.olya@mail.ru
Novosibirsk, Russia

A. V Neverov

S.L. Sobolev Institute of Mathematics SB RAS

Novosibirsk, Russia

References

  1. Abbasi Z., Zamani I., Hossein A.A.M., Shafieirad M., Ibeas A. Optimal Control Design of Impulsive SQEIAR Epidemic Models with Application to COVID-19 // Chaos Solitons Fractals. 2020. V. 139. P. 110054.
  2. Joshi H.R., Lenhart S., Li M.Y., Wang L. Optimal control methods applied to disease models // AMS Volume on Mathematical Studies on Human Disease Dynamics: Emerging Paradigms and Challenges, AMS Contemporary Mathematics Series. 2006. V. 410. P. 187–207.
  3. Lenhart S., Workman J.T. Optimal Control Applied to Biological Models. London: CRC Press, Taylor & Francis Group, 2007.
  4. Antonopoulos C.G., Akrami M.H., Basios V., Latifi A. A generic model for pandemics in networks of communities and the role of vaccination // Chaos 2022. V. 32. № 6. P. 063127.
  5. Bunimovich C.H., Gusev A.A., Derbov V.L., Krassovitskiy P.M., Penkov F.M., Chulumbaatar G. Редуцированная модель SIR пандемии COVID-19 // Ж. вычисл. матем. и матем. физ. 2021. Т. 61. № 3. С. 400–412.
  6. Penkov F.M., Derbov V.L., Chulumbaatar G., Gusev A.A., Vinitsky S.I., Gozdz M., Krassovitskiy P.M. Approximate Solutions of the RSIR Model of COVID-19 Pandemic // IFIP Advances in Information and Communication Technology. 2021. V. 616. P. 53.
  7. Lazzigzer I. An analytic approximate solution of the SIR model // Appl. Math. 2021. V. 12. P. 58–73.
  8. Sokolov A.V., Sokolova L.A. COVID-19 dynamic model: balanced identification of general biological and country specific features // Procedia Comput Sci. 2020. V. 178. P. 301–310.
  9. Sokolov A.V., Sokolova L.A., Monitoring and forecasting the development of the COVID-19 epidemic in Moscow: choosing models based on balanced identification technology, COVID19-PREPRINTS.MICROBE.RU, 2021.
  10. Wang S., Gao F., Teo K. L. An upwind finite-difference method for the approximation of viscosity solutions to Hamilton-Jacobi-Bellman equations // IMA Journal of Mathematical Control and Information. 2000. V. 17. № 2. P. 167–178.
  11. Zhuliang C., Peter A. F. A Semi-Lagrangian Approach for Natural Gas Storage Valuation and Optimal Operation // SIAM Journal on Scientific Computing. 2008. V. 30. № 1. P. 339–368.
  12. Kermack W.O., McKendrick A.G. A contribution to the mathematical theory of epidemics // Proc. R. Soc. Lond. A. 1927. V. 115. P. 700–721.
  13. Pelinovsky E., Kurkin A., Kurkina O., Kokoulina M., Epifanova A. Logistic equation and COVID-19 // Chaos, Solitons & Fractals. 2020. V. 140. P. 110241.
  14. Wang P., Zheng X., Li J., Zhu B. Prediction of epidemic trends in COVID-19 with logistic model and machine learning technics // Chaos, Solitons & Fractals. 2020. V. 139. P. 110058.
  15. Koltsova E.M., Kurkina E.S., Yasetsky A.M. Mathematical modeling of the spread of COVID-19 in Moscow // Computational nanotechnology. 2020. V. 7. № 1. P. 99–105.
  16. Chen Y., Cheng J., Jiang Y., Liu K. A time delay dynamical model for outbreak of 2019-nCoV and the parameter identification // J. Inverse Ill-Posed Probl. 2020. V. 28. № 2. P. 243–250.
  17. Tamm M.V. COVID-19 in Moscow: prognoses and scenarios // FARMAKOEKONOMIKA. Modern Pharmacoeconomic and Pharmacoepidemiology. 2020. V. 13. № 1. P. 43–51.
  18. Unlu E., Leger H., Motornyi O., Rukubayihunga A., Ishaeian T., Chouien M. Epidemic analysis of COVID-19 Outbreak and Counter-Measures in France // MedRxiv 2020. doi: 10.1101/2020.04.27.20079962
  19. Кабанихин С.И., Криворотько О.И. Математическое моделирование эпидемии Уханьского коронавируса COVID-2019 и обратные задачи // Ж. вычисл. матем. и матем. физ. 2020. Т. 50. № 11. С. 1950–1961.
  20. Krivorotko O.I., Kabanikhin S.I., Zyatkov N.Y. et al. Mathematical modeling and forecasting of COVID-19 in Moscow and Novosibirsk region // Numer. Analys. Appl. 2020. V. 13. P. 332–348.
  21. Borovkov A.I., Bolsunovskaya M.V., Gintciak A.M., Kadryavtseva T.Yu. Simulation modelling application for balancing epidemic and economic crisis in the region // International Journal of Technology. 2020. V. 11. № 8. P. 1579–1588.
  22. Ndairou F., Area I., Nieto J.J., Torres D.F.M. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan // Chaos Solitons & Fractals. 2020. V. 135. P. 109846.
  23. 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. 2021. V. 2333. P. 090024.
  24. Kiselev I.N., Akberdin I.R., Kolpakov F.A. A delay differential equation approach to model the COVID-19 pandemic // MedRxiv. 2021. doi: 10.1101/2021.09.01.21263002
  25. Krivorotko O.I., Zyatkov N.Y. Data-driven regularization of inverse problem for SEIR-HCD model of COVID-19 propagation in Novosibirsk region // Eurasian J. Math. Comput. Appl. 2022. V. 10. № 1. P. 51–68.
  26. Криворотько О.И., Кабанихин С.И. О математическом моделировании COVID-19 // Сибирские электронные математические известия. 2023. Т. 20. № 2. С. 1211–1268.
  27. Криворотько О.И., Кабанихин С.И., Петракова В.С. Идентифицируемость математических моделей эпидемиологии: туберкулез, ВИЧ, COVID-19 // Математическая биология и биоинформатика. 2023. Т. 18. № 1. С. 177–214.
  28. Криворотько О.И., Кабанихин С.И., Сосновская М.Н., Андорова Д.В. Анализ чувствительности и идентифицируемости математических моделей распространения эпидемии COVID-19 // Вавиловский журнал генетики и селекции. 2021. Т. 25. № 1. С. 82–91.
  29. Криворотько О.И., Затьков Н.Ю., Кабанихин С.И. Моделирование эпидемий: нейросеть на основе данных и SIR-модели // Ж. вычисл. матем. и матем. физ. 2023. Т. 63. № 10. С. 1733–1746.
  30. Соломко А. Моделирование распространения Covid-19 в Москве с использованием имитационной модели SEIR-HCD // Информационно-телекоммуникационные технологии и математическое моделирование высокотехнологичных систем. 2021. С. 484–491.
  31. Hwang Yoon-gu, Kwon Hee-Dae, Lee Jeehyun. Feedback control problem of an SIR epidemic model based on the Hamilton-Jacobi-Bellman equation // Mathematical Biosciences and Engng. 2020. V. 17. № 3. P. 2284–2301.
  32. Courant R., Isaacson E., Rees M. On the solution of nonlinear hyperbolic differential equations by finite differences // Comm. Pure Appl. Math. 1952. V. 5. P. 243–255.
  33. Bellman R. The theory of dynamic programming // Bull. Am. Math. Soc. 1954. V. 60. P. 503–515.
  34. Yong J., Zhou X.Y. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York: Springer, 1999.
  35. Bardi M., Capuzzo-Dolcetta I. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. New York: Springer, 2008.
  36. Lee W., Liu S., Tembine H., Li W., Osher S. Controlling propagation of epidemics via mean-field control // SIAM J. Appl. Math. 2021. V. 81. № 1. P. 190–207.
  37. Petrakova V. Inverse Coefficient Problem for Epidemiological Mean-Field Formulation // Mathematics. 2024. V. 12. № 22. P. 3581.
  38. Grigorieva E.V., Khailov E.N. Optimal Vaccination, Treatment, and Preventive Campaigns in Regard to the SIR Epidemic Model // Math. Model. Nat. Phenom. 2014. V. 9. № 4. P. 105–121.
  39. Петракова В.С., Криворотько О.И. Несколько подходов к моделированию динамики доходов населения в условиях эпидемии // Успехи кибернетики. 2023. Т. 4. № 1. С. 24–32.
  40. Petrakova V., Krivorotko O. Mean field game for modeling of COVID-19 spread // Journal of Mathematical Analysis and Application. 2022. V. 514. P. 126271.
  41. Petrakova V., Krivorotko O. Mean Field Optimal Control Problem for Predicting the Spread of Viral Infections // 2023 19th International Asian School-Seminar on Optimization Problems of Complex Systems (OPCS), Novosibirsk, Moscow, Russian Federation. 2023. P. 79–84.
  42. Petrakova V., Krivorotko O., Neverov A. Review of the mean field models for predicting the spread of viral infections // 2023 IEEE CSGB. 2023. P. 45–50.

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