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Optimization of vaccination for COVID-19 in the midst of a pandemic
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Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties
A measure model for the spread of viral infections with mutations
| 1. | School of Mathematical and Statistical Science, Arizona State University, Tempe, AZ, 85281, USA |
| 2. | Department of Mathematical Sciences and Center for Computational and Integrative Biology, Rutgers University, Camden, NJ, 08102, USA |
Genetic variations in the COVID-19 virus are one of the main causes of the COVID-19 pandemic outbreak in 2020 and 2021. In this article, we aim to introduce a new type of model, a system coupled with ordinary differential equations (ODEs) and measure differential equation (MDE), stemming from the classical SIR model for the variants distribution. Specifically, we model the evolution of susceptible $ S $ and removed $ R $ populations by ODEs and the infected $ I $ population by a MDE comprised of a probability vector field (PVF) and a source term. In addition, the ODEs for $ S $ and $ R $ contains terms that are related to the measure $ I $. We establish analytically the well-posedness of the coupled ODE-MDE system by using generalized Wasserstein distance. We give two examples to show that the proposed ODE-MDE model coincides with the classical SIR model in case of constant or time-dependent parameters as special cases.
References:
| [1] |
S. Anita and V. Capasso, Reaction-diffusion systems in epidemiology, 2017. |
| [2] |
N. Bellomo, R. Bingham, M. A. J. Chaplain, G. Dosi, G. Forni, D. A. Knopoff, J. Lowengrub, R. Twarock and M. E. Virgillito, A multi-scale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Models Methods Appl. Sci., 30 (2020), 1591–1651, arXiv: 2006.03915.
doi: 10.1142/S0218202520500323. |
| [3] |
G. A. Bonaschi, J. A. Carrillo, M. Di Francesco and M. A. Peletier,
Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441.
doi: 10.1051/cocv/2014032. |
| [4] |
T. Britton, F. Ball and P. Trapman,
A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2, Science, 369 (2020), 846-849.
doi: 10.1126/science.abc6810. |
| [5] |
Y.-C. Chen, P.-E. Lu and C.-S. Chang, A time-dependent SIR model for COVID-19, IEEE Trans. Network Sci. Eng., 7 (2020), 3279–3294, arXiv: 2003.00122.
doi: 10.1109/TNSE.2020.3024723. |
| [6] |
R. M. Colombo and M. Garavello,
Well posedness and control in a nonlocal SIR model, Appl. Math. Optim., 84 (2021), 737-771.
doi: 10.1007/s00245-020-09660-9. |
| [7] |
R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the COVID-19 pandemic, Journal of Mathematics in Industry, 10 (2020), Paper No. 22, 20 pp.
doi: 10.1186/s13362-020-00090-4. |
| [8] |
G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri,
Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860.
doi: 10.1038/s41591-020-0883-7. |
| [9] |
V. Kala, K. Guo, E. Swantek, A. Tong, M.. Chyba, Y. Mileyko, C. Gray, T. Lee and A. E. Koniges, Pandemics in Hawai'i: 1918 Influenza and COVID-19, The Ninth International Conference on Global Health Challenges GLOBAL HEALTH 2020, IARIA, 2020. |
| [10] |
A. Keimer and L. Pflug, Modeling infectious diseases using integro-differential equations: Optimal control strategies for policy decisions and applications in covid-19, 2020. |
| [11] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics. II.-the problem of endemicity, Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138 (1932), 55-83.
doi: 10.1098/rspa.1932.0171. |
| [12] |
K. Kupferschmidt,
Evolving threat, Science, 373 (2021), 844-849.
doi: 10.1126/science.373.6557.844. |
| [13] |
C. J. E. Metcalf, D. H. Morris and S. W. Park,
Mathematical models to guide pandemic response, Science, 369 (2020), 368-369.
doi: 10.1126/science.abd1668. |
| [14] |
K. R. Moran, G. Fairchild, N. Generous, K. Hickmann, D. Osthus, R. Priedhorsky, J. Hyman and S. Y. Del Valle, Epidemic forecasting is messier than weather forecasting: The role of human behavior and internet data streams in epidemic forecast, The Journal of Infectious Diseases, 214 (2016), S404–S408.
doi: 10.1093/infdis/jiw375. |
| [15] |
B. Piccoli,
Measure differential equations, Archive for Rational Mechanics and Analysis, 233 (2019), 1289-1317.
doi: 10.1007/s00205-019-01379-4. |
| [16] |
B. Piccoli and F. Rossi,
Generalized Wasserstein distance and its application to transport equations with source, Archive for Rational Mechanics and Analysis, 211 (2014), 335-358.
doi: 10.1007/s00205-013-0669-x. |
| [17] |
B. Piccoli and F. Rossi,
On properties of the generalized Wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365.
doi: 10.1007/s00205-016-1026-7. |
| [18] |
B. Piccoli and F. Rossi,
Measure dynamics with probability vector fields and sources, Discrete Contin. Dyn. Syst., 39 (2019), 6207-6230.
doi: 10.3934/dcds.2019270. |
| [19] |
N. W. Ruktanonchai, J. R. Floyd, S. Lai, C. W. Ruktanonchai, A. Sadilek, P. Rente-Lourenco, X. Ben, A. Carioli, J. Gwinn and J. E. Steele,
Assessing the impact of coordinated COVID-19 exit strategies across europe, Science, 369 (2020), 1465-1470.
doi: 10.1126/science.abc5096. |
| [20] |
A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, S. V. Scarpino, B. Gutierrez, M. U. G. Kraemer, J. Wu and et al.,
Modelling covid-19, Nature Reviews Physics, 2 (2020), 279-281.
|
| [21] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
| [22] |
J. Zhang, M. Litvinova, Y. Liang, Y. Wang, W. Wang, S. Zhao, Q. Wu, S. Merler, C. Viboud and A. Vespignani,
Changes in contact patterns shape the dynamics of the COVID-19 outbreak in China, Science, 368 (2020), 1481-1486.
doi: 10.1126/science.abb8001. |
| [23] |
J. Zhang, M. Litvinova, W. Wang, Y. Wang, X. Deng, X. Chen, M. Li, W. Zheng, L. Yi and X. Chen,
Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: A descriptive and modelling study, The Lancet Infectious Diseases, 20 (2020), 793-802.
doi: 10.1016/S1473-3099(20)30230-9. |
show all references
References:
| [1] |
S. Anita and V. Capasso, Reaction-diffusion systems in epidemiology, 2017. |
| [2] |
N. Bellomo, R. Bingham, M. A. J. Chaplain, G. Dosi, G. Forni, D. A. Knopoff, J. Lowengrub, R. Twarock and M. E. Virgillito, A multi-scale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Models Methods Appl. Sci., 30 (2020), 1591–1651, arXiv: 2006.03915.
doi: 10.1142/S0218202520500323. |
| [3] |
G. A. Bonaschi, J. A. Carrillo, M. Di Francesco and M. A. Peletier,
Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D, ESAIM Control Optim. Calc. Var., 21 (2015), 414-441.
doi: 10.1051/cocv/2014032. |
| [4] |
T. Britton, F. Ball and P. Trapman,
A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2, Science, 369 (2020), 846-849.
doi: 10.1126/science.abc6810. |
| [5] |
Y.-C. Chen, P.-E. Lu and C.-S. Chang, A time-dependent SIR model for COVID-19, IEEE Trans. Network Sci. Eng., 7 (2020), 3279–3294, arXiv: 2003.00122.
doi: 10.1109/TNSE.2020.3024723. |
| [6] |
R. M. Colombo and M. Garavello,
Well posedness and control in a nonlocal SIR model, Appl. Math. Optim., 84 (2021), 737-771.
doi: 10.1007/s00245-020-09660-9. |
| [7] |
R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the COVID-19 pandemic, Journal of Mathematics in Industry, 10 (2020), Paper No. 22, 20 pp.
doi: 10.1186/s13362-020-00090-4. |
| [8] |
G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri,
Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nature Medicine, 26 (2020), 855-860.
doi: 10.1038/s41591-020-0883-7. |
| [9] |
V. Kala, K. Guo, E. Swantek, A. Tong, M.. Chyba, Y. Mileyko, C. Gray, T. Lee and A. E. Koniges, Pandemics in Hawai'i: 1918 Influenza and COVID-19, The Ninth International Conference on Global Health Challenges GLOBAL HEALTH 2020, IARIA, 2020. |
| [10] |
A. Keimer and L. Pflug, Modeling infectious diseases using integro-differential equations: Optimal control strategies for policy decisions and applications in covid-19, 2020. |
| [11] |
W. O. Kermack and A. G. McKendrick,
Contributions to the mathematical theory of epidemics. II.-the problem of endemicity, Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138 (1932), 55-83.
doi: 10.1098/rspa.1932.0171. |
| [12] |
K. Kupferschmidt,
Evolving threat, Science, 373 (2021), 844-849.
doi: 10.1126/science.373.6557.844. |
| [13] |
C. J. E. Metcalf, D. H. Morris and S. W. Park,
Mathematical models to guide pandemic response, Science, 369 (2020), 368-369.
doi: 10.1126/science.abd1668. |
| [14] |
K. R. Moran, G. Fairchild, N. Generous, K. Hickmann, D. Osthus, R. Priedhorsky, J. Hyman and S. Y. Del Valle, Epidemic forecasting is messier than weather forecasting: The role of human behavior and internet data streams in epidemic forecast, The Journal of Infectious Diseases, 214 (2016), S404–S408.
doi: 10.1093/infdis/jiw375. |
| [15] |
B. Piccoli,
Measure differential equations, Archive for Rational Mechanics and Analysis, 233 (2019), 1289-1317.
doi: 10.1007/s00205-019-01379-4. |
| [16] |
B. Piccoli and F. Rossi,
Generalized Wasserstein distance and its application to transport equations with source, Archive for Rational Mechanics and Analysis, 211 (2014), 335-358.
doi: 10.1007/s00205-013-0669-x. |
| [17] |
B. Piccoli and F. Rossi,
On properties of the generalized Wasserstein distance, Archive for Rational Mechanics and Analysis, 222 (2016), 1339-1365.
doi: 10.1007/s00205-016-1026-7. |
| [18] |
B. Piccoli and F. Rossi,
Measure dynamics with probability vector fields and sources, Discrete Contin. Dyn. Syst., 39 (2019), 6207-6230.
doi: 10.3934/dcds.2019270. |
| [19] |
N. W. Ruktanonchai, J. R. Floyd, S. Lai, C. W. Ruktanonchai, A. Sadilek, P. Rente-Lourenco, X. Ben, A. Carioli, J. Gwinn and J. E. Steele,
Assessing the impact of coordinated COVID-19 exit strategies across europe, Science, 369 (2020), 1465-1470.
doi: 10.1126/science.abc5096. |
| [20] |
A. Vespignani, H. Tian, C. Dye, J. O. Lloyd-Smith, R. M. Eggo, M. Shrestha, S. V. Scarpino, B. Gutierrez, M. U. G. Kraemer, J. Wu and et al.,
Modelling covid-19, Nature Reviews Physics, 2 (2020), 279-281.
|
| [21] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058. |
| [22] |
J. Zhang, M. Litvinova, Y. Liang, Y. Wang, W. Wang, S. Zhao, Q. Wu, S. Merler, C. Viboud and A. Vespignani,
Changes in contact patterns shape the dynamics of the COVID-19 outbreak in China, Science, 368 (2020), 1481-1486.
doi: 10.1126/science.abb8001. |
| [23] |
J. Zhang, M. Litvinova, W. Wang, Y. Wang, X. Deng, X. Chen, M. Li, W. Zheng, L. Yi and X. Chen,
Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: A descriptive and modelling study, The Lancet Infectious Diseases, 20 (2020), 793-802.
doi: 10.1016/S1473-3099(20)30230-9. |
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