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doi: 10.3934/cpaa.2021093
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Data-driven optimal control of a seir model for COVID-19

1. 

Department of Mathematics, Iowa State University, Ames, IA 50011, USA

2. 

Department of Mathematics, Iowa State University, Ames, IA 50011, USA

* Corresponding author

Received  December 2020 Revised  April 2021 Early access June 2021

Fund Project: This research was supported by the National Science Foundation under Grant DMS1812666

We present a data-driven optimal control approach which integrates the reported partial data with the epidemic dynamics for COVID-19. We use a basic Susceptible-Exposed-Infectious-Recovered (SEIR) model, the model parameters are time-varying and learned from the data. This approach serves to forecast the evolution of the outbreak over a relatively short time period and provide scheduled controls of the epidemic. We provide efficient numerical algorithms based on a generalized Pontryagin's Maximum Principle associated with the optimal control theory. Numerical experiments demonstrate the effective performance of the proposed model and its numerical approximations.

Citation: Hailiang Liu, Xuping Tian. Data-driven optimal control of a seir model for COVID-19. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021093
References:
[1]

V. V. Aleksandrov, On the accumulation of perturbations in the linear systems with two coordinates, Vestnik MGU, 3. Google Scholar

[2]

L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology, vol. 1945, Springer, Berlin, 2008, doi: 10.1007/978-3-540-78911-6_3.  Google Scholar

[3]

C. AnastassopoulouL. RussoA. Tsakris and C. Siettos, Data-based analysis, modelling and forecasting of the COVID-19 outbreak, PLOS ONE, 15 (2020), 1-21.   Google Scholar

[4]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367.   Google Scholar

[5]

S. Arik, C. L. Li, J. Yoon, R. Sinha, A. Epshteyn, L. T. Le, V. Menon, S. Singh, L. Zhang, M. Nikoltchev, Y. K. Sonthalia, H. Nakhost, E. Kanal and T. Pfister, Interpretable sequence learning for covid-19 forecasting, arXiv: 2008.00646. Google Scholar

[6]

C. T. Bauch and D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. USA, 101 (2004), 13391-13394.  doi: 10.1073/pnas.0403823101.  Google Scholar

[7]

H. Behncke, Optimal control of deterministic epidemics, Optim. Contr. Appl. Met., 21 (2000), 269-285.  doi: 10.1002/oca.678.  Google Scholar

[8]

H. Berestycki, J. M. Roquejoffre and L. Rossi, Propagation of epidemics along lines with fast diffusion, arXiv: 2005.01859. Google Scholar

[9]

A. BertozziE. FrancoG. MohlerM. Short and D. Sledge, The challenges of modeling and forecasting the spread of COVID-19, P. Natl. Acad. Sci., 117 (2020), 16732-16738.   Google Scholar

[10]

M. C. J. Bootsma and N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in u.s. cities, P. Natl. Acad. Sci., 104 (2007), 7588-7593.   Google Scholar

[11]

A. Bressan and B. Piccoli, Introduction to the mathematical theory of control, in AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[12]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[13]

V. Capasso, Reaction-diffusion models for the spread of a class of infectious diseases, in Proceedings of the Second European Symposium on Mathematics in Industry (Oberwolfach, 1987), vol. 3 of European Consort. Math. Indust., Teubner, Stuttgart, 1988, doi: 10.1007/978-94-009-2979-1_11.  Google Scholar

[14]

F. Castiglione and B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control, J. Theor. Biol., 247 (2007), 723-732.  doi: 10.1016/j.jtbi.2007.04.003.  Google Scholar

[15]

S. L. ChangM. PiraveenanP. Pattison and M. Prokopenko, Game theoretic modelling of infectious disease dynamics and intervention methods: a review, J. Biol. Dyn., 14 (2020), 57-89.  doi: 10.1080/17513758.2020.1720322.  Google Scholar

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R. T. Q. Chen, Y. Rubanova, J. Bettencourt and D. K. Duvenaud, Neural ordinary differential equations, in Conference on Neural Information Processing Systems (NIPS), 2018. Google Scholar

[17]

F. L. Chernous and A. A. Lyubushin, Method of successive approximations for solution of optimal control problems, Optimal Control Appl. Methods, 3 (1982), 101-114.  doi: 10.1002/oca.4660030201.  Google Scholar

[18]

M. Cranmer, S. Greydanus, S. Hoyer, P. Battaglia, D. Spergel and S. Ho, Lagrangian neural networks, arXiv: 2003.04630. Google Scholar

[19]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

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O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases, in Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2000  Google Scholar

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W. E, A proposal on machine learning via dynamical systems, Math. Sci., 5 (2017), 1-11.  doi: 10.1007/s40304-017-0103-z.  Google Scholar

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G. GiordanoF. BlanchiniR. BrunoP. ColaneriA. FilippoA. Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in italy, Nature Med., 26 (2020), 1-6.   Google Scholar

[23]

D. Greenhalgh, Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. Comput. Model., 25 (1997), 85-107.  doi: 10.1016/S0895-7177(97)00009-5.  Google Scholar

[24]

D. Greenhalgh and R. Das, Modeling epidemics with variable contact rates, Theor. Population Biol., 47 (1995), 129-179.   Google Scholar

[25]

S. Greydanus, M. Dzamba and J. Yosinski, Hamiltonian neural networks, arXiv: 1906.01563. Google Scholar

[26]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1966.  Google Scholar

[27]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[28]

S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Comput., 9 (1997), 1735-1780.   Google Scholar

[29]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.  Google Scholar

[30]

E. HunterB. Mac Namee and J. Kelleher, An open-data-driven agent-based model to simulate infectious disease outbreaks, PLOS ONE, 13 (2018), 1-35.   Google Scholar

[31]

M. M. Tiberiu Harko Francisco S.N. Lobo, Exact analytical solutions of the susceptible-infected-recovered (SIR) epidemic model and of the SIR model with equal death and birth rates, Appl. Math. Comput., 236 (2014), 184-194.  doi: 10.1016/j.amc.2014.03.030.  Google Scholar

[32]

J. JangH. Kwon and J. Lee, Optimal control problem of an SIR reaction-diffusion model with inequality constraints, Math. Comput. Simul., 171 (2020), 136-151.  doi: 10.1016/j.matcom.2019.08.002.  Google Scholar

[33]

H. Jo, H. Son, H. J. Hwang and S. Y. Jung, Analysis of COVID-19 spread in {South Korea} using the SIR model with time-dependent parameters and deep learning, medRxiv. Google Scholar

[34]

M. Keeling and K. Eames, Networks and epidemic models, J. Roy. Soc. Interface, 2 (2005), 295-307.   Google Scholar

[35]

D. G. Kendall, Mathematical models of the spread of infection, Math. Comput. Sci. Biol. Med., 171 (1965), 213-225.   Google Scholar

[36]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, P. Roy. Soc. Lond. A, 115(772) (1927), 700-721.   Google Scholar

[37]

A. Kleczkowski and B. T. Grenfell, Mean-field-type equations for spread of epidemics: The 'small world' model, Physica A, 274 (1999), 355-360.   Google Scholar

[38]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83.  doi: 10.1007/s11538-008-9352-z.  Google Scholar

[39]

I. A. Krylov and F. L. Černous' ko, The method of successive approximations for solving optimal control problems, Ž. Vyčisl. Mat i Mat. Fiz., 2 (1962), 1132-1139.  Google Scholar

[40]

W. Lee, S. Liu, H. Tembine, W. Li and S. Osher, Controlling propagation of epidemics via mean-field games, arXiv: 2006.01249. Google Scholar

[41]

M. Y. LiJ. R. GraefL. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[42]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164.  doi: 10.1016/0025-5564(95)92756-5.  Google Scholar

[43]

Q. LinS. ZhaoD. GaoY. LouS. YangS. MusaM. WangW. WangL. Yang and D. He, A conceptual model for the outbreak of coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action, Int. J. Infect. Dis., 93 (2020), 211-216.   Google Scholar

[44]

H. Liu and P. Markowich, Selection dynamics for deep neural networks, J. Differ. Equ., 269 (2020), 11540-11574.  doi: 10.1016/j.jde.2020.08.041.  Google Scholar

[45]

W. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.  Google Scholar

[46]

M. Lutter, C. Ritter and J. Peters, Deep lagrangian networks: Using physics as model prior for deep learning, arXiv: 1907.04490. Google Scholar

[47]

L. Magri and N. A. K. Doan, First-principles machine learning modelling of COVID-19, arXiv: 2004.09478. Google Scholar

[48]

J. Mena-Lorcat and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.  doi: 10.1007/BF00173264.  Google Scholar

[49]

R. Parshani, S. Carmi and S. Havlin, Epidemic threshold for the susceptible-infectious-susceptible model on random networks, Phys. Rev. Lett., 104 (2010), 258701. Google Scholar

[50] L. PontryaginV. BoltyanskiiR. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, CRC Press, 1962.   Google Scholar
[51]

M. RaissiP. Perdikaris and G. E. Karniadakis, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686-707.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[52]

T. C. Reluga and A. P. Galvani, A general approach for population games with application to vaccination, Math. Biosci., 230 (2011), 67-78.  doi: 10.1016/j.mbs.2011.01.003.  Google Scholar

[53]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[54]

W. H. Schmidt, Numerical methods for optimal control problems with ODE or integral equations, in Large-Scale Scientific Computing, vol. 3743 of Lecture Notes in Comput. Sci., Springer, Berlin, 2006, doi: 10.1007/11666806_28.  Google Scholar

[55]

R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291.  doi: 10.1016/j.camwa.2010.08.020.  Google Scholar

[56]

H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111 (1992), 99-130.  doi: 10.1016/0025-5564(92)90081-7.  Google Scholar

[57]

D. J. Watts, Small worlds, in Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 1999,  Google Scholar

[58]

S. H. WhiteA. M. del Rey and G. R. Sánchez, Modeling epidemics using cellular automata, Appl. Math. Comput., 186 (2007), 193-202.  doi: 10.1016/j.amc.2006.06.126.  Google Scholar

show all references

References:
[1]

V. V. Aleksandrov, On the accumulation of perturbations in the linear systems with two coordinates, Vestnik MGU, 3. Google Scholar

[2]

L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology, vol. 1945, Springer, Berlin, 2008, doi: 10.1007/978-3-540-78911-6_3.  Google Scholar

[3]

C. AnastassopoulouL. RussoA. Tsakris and C. Siettos, Data-based analysis, modelling and forecasting of the COVID-19 outbreak, PLOS ONE, 15 (2020), 1-21.   Google Scholar

[4]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367.   Google Scholar

[5]

S. Arik, C. L. Li, J. Yoon, R. Sinha, A. Epshteyn, L. T. Le, V. Menon, S. Singh, L. Zhang, M. Nikoltchev, Y. K. Sonthalia, H. Nakhost, E. Kanal and T. Pfister, Interpretable sequence learning for covid-19 forecasting, arXiv: 2008.00646. Google Scholar

[6]

C. T. Bauch and D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. USA, 101 (2004), 13391-13394.  doi: 10.1073/pnas.0403823101.  Google Scholar

[7]

H. Behncke, Optimal control of deterministic epidemics, Optim. Contr. Appl. Met., 21 (2000), 269-285.  doi: 10.1002/oca.678.  Google Scholar

[8]

H. Berestycki, J. M. Roquejoffre and L. Rossi, Propagation of epidemics along lines with fast diffusion, arXiv: 2005.01859. Google Scholar

[9]

A. BertozziE. FrancoG. MohlerM. Short and D. Sledge, The challenges of modeling and forecasting the spread of COVID-19, P. Natl. Acad. Sci., 117 (2020), 16732-16738.   Google Scholar

[10]

M. C. J. Bootsma and N. M. Ferguson, The effect of public health measures on the 1918 influenza pandemic in u.s. cities, P. Natl. Acad. Sci., 104 (2007), 7588-7593.   Google Scholar

[11]

A. Bressan and B. Piccoli, Introduction to the mathematical theory of control, in AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2007.  Google Scholar

[12]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[13]

V. Capasso, Reaction-diffusion models for the spread of a class of infectious diseases, in Proceedings of the Second European Symposium on Mathematics in Industry (Oberwolfach, 1987), vol. 3 of European Consort. Math. Indust., Teubner, Stuttgart, 1988, doi: 10.1007/978-94-009-2979-1_11.  Google Scholar

[14]

F. Castiglione and B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control, J. Theor. Biol., 247 (2007), 723-732.  doi: 10.1016/j.jtbi.2007.04.003.  Google Scholar

[15]

S. L. ChangM. PiraveenanP. Pattison and M. Prokopenko, Game theoretic modelling of infectious disease dynamics and intervention methods: a review, J. Biol. Dyn., 14 (2020), 57-89.  doi: 10.1080/17513758.2020.1720322.  Google Scholar

[16]

R. T. Q. Chen, Y. Rubanova, J. Bettencourt and D. K. Duvenaud, Neural ordinary differential equations, in Conference on Neural Information Processing Systems (NIPS), 2018. Google Scholar

[17]

F. L. Chernous and A. A. Lyubushin, Method of successive approximations for solution of optimal control problems, Optimal Control Appl. Methods, 3 (1982), 101-114.  doi: 10.1002/oca.4660030201.  Google Scholar

[18]

M. Cranmer, S. Greydanus, S. Hoyer, P. Battaglia, D. Spergel and S. Ho, Lagrangian neural networks, arXiv: 2003.04630. Google Scholar

[19]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[20]

O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases, in Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2000  Google Scholar

[21]

W. E, A proposal on machine learning via dynamical systems, Math. Sci., 5 (2017), 1-11.  doi: 10.1007/s40304-017-0103-z.  Google Scholar

[22]

G. GiordanoF. BlanchiniR. BrunoP. ColaneriA. FilippoA. Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in italy, Nature Med., 26 (2020), 1-6.   Google Scholar

[23]

D. Greenhalgh, Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. Comput. Model., 25 (1997), 85-107.  doi: 10.1016/S0895-7177(97)00009-5.  Google Scholar

[24]

D. Greenhalgh and R. Das, Modeling epidemics with variable contact rates, Theor. Population Biol., 47 (1995), 129-179.   Google Scholar

[25]

S. Greydanus, M. Dzamba and J. Yosinski, Hamiltonian neural networks, arXiv: 1906.01563. Google Scholar

[26]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1966.  Google Scholar

[27]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[28]

S. Hochreiter and J. Schmidhuber, Long short-term memory, Neural Comput., 9 (1997), 1735-1780.   Google Scholar

[29]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.  Google Scholar

[30]

E. HunterB. Mac Namee and J. Kelleher, An open-data-driven agent-based model to simulate infectious disease outbreaks, PLOS ONE, 13 (2018), 1-35.   Google Scholar

[31]

M. M. Tiberiu Harko Francisco S.N. Lobo, Exact analytical solutions of the susceptible-infected-recovered (SIR) epidemic model and of the SIR model with equal death and birth rates, Appl. Math. Comput., 236 (2014), 184-194.  doi: 10.1016/j.amc.2014.03.030.  Google Scholar

[32]

J. JangH. Kwon and J. Lee, Optimal control problem of an SIR reaction-diffusion model with inequality constraints, Math. Comput. Simul., 171 (2020), 136-151.  doi: 10.1016/j.matcom.2019.08.002.  Google Scholar

[33]

H. Jo, H. Son, H. J. Hwang and S. Y. Jung, Analysis of COVID-19 spread in {South Korea} using the SIR model with time-dependent parameters and deep learning, medRxiv. Google Scholar

[34]

M. Keeling and K. Eames, Networks and epidemic models, J. Roy. Soc. Interface, 2 (2005), 295-307.   Google Scholar

[35]

D. G. Kendall, Mathematical models of the spread of infection, Math. Comput. Sci. Biol. Med., 171 (1965), 213-225.   Google Scholar

[36]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, P. Roy. Soc. Lond. A, 115(772) (1927), 700-721.   Google Scholar

[37]

A. Kleczkowski and B. T. Grenfell, Mean-field-type equations for spread of epidemics: The 'small world' model, Physica A, 274 (1999), 355-360.   Google Scholar

[38]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83.  doi: 10.1007/s11538-008-9352-z.  Google Scholar

[39]

I. A. Krylov and F. L. Černous' ko, The method of successive approximations for solving optimal control problems, Ž. Vyčisl. Mat i Mat. Fiz., 2 (1962), 1132-1139.  Google Scholar

[40]

W. Lee, S. Liu, H. Tembine, W. Li and S. Osher, Controlling propagation of epidemics via mean-field games, arXiv: 2006.01249. Google Scholar

[41]

M. Y. LiJ. R. GraefL. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[42]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164.  doi: 10.1016/0025-5564(95)92756-5.  Google Scholar

[43]

Q. LinS. ZhaoD. GaoY. LouS. YangS. MusaM. WangW. WangL. Yang and D. He, A conceptual model for the outbreak of coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action, Int. J. Infect. Dis., 93 (2020), 211-216.   Google Scholar

[44]

H. Liu and P. Markowich, Selection dynamics for deep neural networks, J. Differ. Equ., 269 (2020), 11540-11574.  doi: 10.1016/j.jde.2020.08.041.  Google Scholar

[45]

W. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.  Google Scholar

[46]

M. Lutter, C. Ritter and J. Peters, Deep lagrangian networks: Using physics as model prior for deep learning, arXiv: 1907.04490. Google Scholar

[47]

L. Magri and N. A. K. Doan, First-principles machine learning modelling of COVID-19, arXiv: 2004.09478. Google Scholar

[48]

J. Mena-Lorcat and H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693-716.  doi: 10.1007/BF00173264.  Google Scholar

[49]

R. Parshani, S. Carmi and S. Havlin, Epidemic threshold for the susceptible-infectious-susceptible model on random networks, Phys. Rev. Lett., 104 (2010), 258701. Google Scholar

[50] L. PontryaginV. BoltyanskiiR. Gamkrelidze and E. Mishchenko, The Mathematical Theory of Optimal Processes, CRC Press, 1962.   Google Scholar
[51]

M. RaissiP. Perdikaris and G. E. Karniadakis, Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686-707.  doi: 10.1016/j.jcp.2018.10.045.  Google Scholar

[52]

T. C. Reluga and A. P. Galvani, A general approach for population games with application to vaccination, Math. Biosci., 230 (2011), 67-78.  doi: 10.1016/j.mbs.2011.01.003.  Google Scholar

[53]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.  Google Scholar

[54]

W. H. Schmidt, Numerical methods for optimal control problems with ODE or integral equations, in Large-Scale Scientific Computing, vol. 3743 of Lecture Notes in Comput. Sci., Springer, Berlin, 2006, doi: 10.1007/11666806_28.  Google Scholar

[55]

R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291.  doi: 10.1016/j.camwa.2010.08.020.  Google Scholar

[56]

H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111 (1992), 99-130.  doi: 10.1016/0025-5564(92)90081-7.  Google Scholar

[57]

D. J. Watts, Small worlds, in Princeton Studies in Complexity, Princeton University Press, Princeton, NJ, 1999,  Google Scholar

[58]

S. H. WhiteA. M. del Rey and G. R. Sánchez, Modeling epidemics using cellular automata, Appl. Math. Comput., 186 (2007), 193-202.  doi: 10.1016/j.amc.2006.06.126.  Google Scholar

Figure 1.  (a) Reported and fitted cumulative infection and death cases in the US (b) Estimated SEIR parameters and the basic reproduction number. $ \beta $ ($ \mu $) corresponds to the left (right) vertical axis, $ \epsilon = 0.2 $ and $ \gamma = 0.1 $ are almost constant. The dashed line in $ R_0 $ is a zoomed-in version on the tail of the solid line
Figure 2.  Scheduled control for the US in $ 270-300 $ days by SEIR model
Figure 3.  (a) Reported and fitted cumulative infection and death cases in the UK (b) Estimated SEIR parameters and the basic reproduction number. $ \beta $ ($ \mu $) corresponds to the left (right) vertical axis, $ \epsilon = 0.2 $ and $ \gamma = 0.1 $ are almost constant. The dashed line in $ R_0 $ is a zoomed-in version on the tail of the solid line
Figure 4.  (a) Reported and fitted cumulative infection and death cases in France (b) Estimated SEIR parameters and the basic reproduction number. $ \beta $ ($ \mu $) corresponds to the left (right) vertical axis, $ \epsilon = 0.2 $ and $ \gamma = 0.1 $ are almost constant. The dashed line in $ R_0 $ is a zoomed-in version on the tail of the solid line
Figure 5.  (a) Reported and fitted cumulative infection and death cases in China (b) Estimated SEIR parameters and the basic reproduction number. $ \beta $ ($ \mu $) corresponds to the left (right) vertical axis, $ \epsilon = 0.2 $ and $ \gamma = 0.2 $ are almost constant. The dashed line in $ R_0 $ is a zoomed-in version on the tail of the solid line
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