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Mathematics Subject Classification: 49N90, 58E25, 92D25, 92D30
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Revista de Matemática Teoría y Aplicaciones
versión impresa ISSN 1409-2433
Rev. Mat vol.28 no.1 San José ene./jul. 2021
http://dx.doi.org/10.15517/rmta.v28i1.42077
Artículo
Optimizing the quarantine cost for suppression of the COVID-19 epidemic in México
Optimización del costo de la cuarentena para la supresión de la epidemia del COVID-19 en México
1Universidad Michoacana de San Nicolás de Hidalgo; Institute of Physics and Mathematics; Morelia; México; abdon.choque@umich.mx
2Lomonosov Moscow State University; Faculty of Computational Mathematics and Cybernetics; Moscow; Russia; khailov@cs.msu.su
3Texas Woman’s University; Department of Mathematics and Computer Sciences; Denton; United States; egrigorieva@twu.edu
This paper is one of the few attempts to use the optimal control theory to find optimal quarantine strategies for eradication of the spread of the COVID-19 infection in the Mexican human population. This is achieved by introducing into the SEIR model a bounded control function of time that reflects these quarantine measures. The objective function to be minimized is the weighted sum of the total infection level in the population and the total cost of the quarantine. An optimal control problem reflecting the search for an effective quarantine strategy is stated and solved analytically and numerically. The properties of the corresponding optimal control are established analytically by applying the Pontryagin maximum principle. The optimal solution is obtained numerically by solving the two-point boundary value problem for the maximum principle using MATLAB software. A detailed discussion of the results and the corresponding practical conclusions are presented.
Keywords: coronavirus; quarantine cost; Pontryagin maximum principal; optimal control.
En este trabajo empleamos la teoría de control óptimo para encontrar una cuarentena óptima y estrategias para la erradicación de la propagación de la infección por COVID-19 en la población humana mexicana. En un modelo SEIR, introducimos un control acotado que es una función respecto del tiempo, la cual refleja las medidas de la cuarentena. La función objetivo a minimizar es la suma ponderada del nivel total de infección en la población y el costo total de la cuarentena. Planteamos un problema de control óptimo que representa la búsqueda de una estrategia eficaz de una cuarentena. Resolvemos este problema analíticamente y numéricamente. Establecemos analíticamente las propiedades del control óptimo correspondiente aplicando el principio del máximo de Pontryagin. La solución óptima se obtiene resolviendo un problema de valor de frontera de dos puntos asociado al principio del máximo. Usamos el software MATLAB. Presentamos una discusión detallada de los resultados y las correspondientes conclusiones prácticas.
Palabras clave: coronavirus; costo de una cuarentena; principio del máximo de Pontryagin; control óptimo.
References
I,H, Aslan; M, Demir; M,M, Wise; S, Lenhart. Modeling COVID-19: forecasting and analyzing the dynamics of the outbreak in Hubei and Turkey, MedRxiv (2020), 1-17. Doi: 10.1101/2020.04.11.20061952 [ Links ]
F, Brauer; C, Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology, Springer, New York-Heidelberg-Dordrecht-London, 2012. Doi: 10.1007/978-1-4614-1686-9 [ Links ]
Z, Cao; Q, Zhang; X, Lu; D, Pfeiffer; Z, Jia; H, Song; D,D, Zeng. Estimating the effective reproduction number of the 2019-nCoV in China, MedRxiv, (2020), 1-8. Doi: 10.1101/2020.01.27.20018952 [ Links ]
T,-M, Chen; J, Rui; Q,-P, Wang; Z,-Y, Zhao; J,-A, Cui; L, Yin. A mathematical model for simulating the phase-based transmissibility of a novel coronavirus, Infect. Dis. Poverty. 9(2020), no. 24, 1-8. Doi: 10.1186/s40249-020-00640-3 [ Links ]
W,H, Fleming; R,W, Rishel. Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin, 1975. Doi: 10.1007/ 978-1-4612-6380-7 [ Links ]
H, Gaff; E, Schaefer. Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng. 6(2009), no. 3, 469-492. Doi: 10.3934/mbe.2009.6.469 [ Links ]
Gobierno de la Ciudad de México. El modelo epidemiológico del Gobierno de la Ciudad de México, 2020. https://modelo.covid19.cdmx. gob.mx/modelo-epidemico [ Links ]
Gobierno de México. Proyecciones de la población de México y de las entidades federativas, 2016-2050, CONAPO, 2019. https://www.gob. mx/cms/uploads/attachment/file/487395/09_CMX.pdf [ Links ]
Gobierno de México. Indicadores demográficos de México de 1950 a 2050, CONAPO, http://www.conapo.gob.mx/work/models/ CONAPO/Mapa_Ind_Dem18/index_2.html [ Links ]
Gobierno de México. Aviso epidemiológico - Casos de infección respiratoria asociados a Coronavirus (COVID-19), Dirección General de Epidemiología, 2020. https://www.gob.mx/salud/es [ Links ]
E,V, Grigorieva; E,N, Khailov. Optimal intervention strategies for a SEIR control model of Ebola epidemics, Mathematics 3(2015), no. 4, 961-983. Doi 10.3390/math3040961 [ Links ]
E,V, Grigorieva; E,N, Khailov. Estimating the number of switchings of the optimal interventions strategies for SEIR control models of Ebola epidemics, Pure and Applied Functional Analysis 1(2016), no. 4, 541-572. [ Links ]
E,V, Grigorieva; P,B, Deignan; E,N, Khailov. Optimal control problem for a SEIR type model of Ebola epidemics, Revista de Matemática: Teoría y Aplicaciones 24(2017), no. 1, 79-96. Doi: 10.15517/RMTA.V24I1. 27771 [ Links ]
E,V, Grigorieva; E,N, Khailov. Optimal preventive strategies for SEIR type model of the 2014 Ebola epidemics, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms. 24(2017), 155-182. [ Links ]
E,V, Grigorieva; E,N, Khailov. Determination of the optimal controls for an Ebola epidemic model, Discrete Cont. Dyn. Syst. Ser. S. 2018, vol. 11 (6), 1071-1101. Doi: 10.3934/dcdss.2018062 [ Links ]
J, Jia; J, Ding; S, Liu; G, Liao; J, Li; B, Duan; G, Wang; R, Zhang. Modeling the control of COVID-19: impact of policy interventions and meteorological factors, Electron J. Differ. Eq. 23(2020), 1-24. https://ejde.math.txstate.edu/Volumes/2020/23/ jia.pdf [ Links ]
Y, Jing; L, Minghui; L, Gang; Z,K, Lu. Monitoring transmissibility and mortality of COVID-19 in Europe, Int. J. Infect. Dis. (2020), 1-16. Doi: 10.1016/j.ijid.2020.03.050 [ Links ]
E, Jung; S, Lenhart; Z, Feng. Optimal control of treatments in a twostrain tuberculosis model, Discrete Cont. Dyn. Syst. Ser. B. 2(2002), no.4, 473-482. Doi: 10.3934/dcdsb.2002.2.473 [ Links ]
U, Ledzewicz; H, Schättler. On optimal singular controls for a general SIR-model with vaccination and treatment, Discrete Cont. Dyn. Syst. 2011, supplement, 981-990. [ Links ]
E,B, Lee; L, Marcus. Foundations of Optimal Control Theory, John Wiley & Sons, New York NY, USA, 1967. Doi: 10.2307/2343766 [ Links ]
Y, Liu; A,A, Gayle; A, Wilder-Smith; J, Rocklöv. The reproductive number of COVID-19 is higher compared to SARS coronavirus, J. Travel Med. 2020, 1-4. Doi: 10.1093/jtm/taaa021. [ Links ]
Z, Liu; P, Magal; O, Seydi; G, Webb. Understanding unreported cases in the COVID-19 epidemic outbreak in Wuhan, China, and the importance of major public health interventions, Biology 9(2020), no. 50, 1-12. Doi: 10.3390/biology9030050 [ Links ]
J,P, Mateus; P, Rebelo; S, Rosa; C,M, Silva; D,F,M, Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment, Discrete Cont. Dyn. Syst. Ser. S. 11(2018), no. 6, 1179-1199. Doi:10.3934/ dcdss.2018067 [ Links ]
M, Park; A,R, Cook; J,T, Lim; Y, Sun; B,L, Dickens. A systematic review of COVID-19 epidemiology based on current evidence, J. Clin. Med. 2020, 9, 967, 1-13. Doi: 10.3390/jcm9040967 [ Links ]
L,S, Pontryagin; V,G, Boltyanskii; R,V, Gamkrelidze; and E,F, Mishchenko. Mathematical Theory of Optimal Processes, John Wiley & Sons, New York NY, USA, 1962. [ Links ]
F, Saldaña; H, Flores-Arguedas; J,A, Camacho-Gutiérrez; I, Barrandas. Modeling the transmission dynamics and the impact of the control interventions for the COVID-19 epidemic outbreak, Math. Biosci. Eng., 2020. https://www.researchgate.net/publication/ 340902930_Modeling_the_transmission_dynamics_and_ the_impact_of_the_control_interventions_for_the_ COVID-19_epidemic_outbreak [ Links ]
H, Schättler; U, Ledzewicz. Optimal Control for Mathematical Models of Cancer Therapies: An Application of Geometric Methods, Springer, New York-Heidelberg-Dordrecht-London, 2015. [ Links ]
C,J, Silva; D,F,M, Torres. Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci. 244 (2013), 153 -164. Doi: 10.1016/j.mbs.2013.05.005 [ Links ]
R, Smith? Modelling Disease Ecology with Mathematics, AIMS, Springfield MO, USA, 2008. [ Links ]
B, Tang; X, Wang; Q, Li; N,L, Bragazzi; S, Tang; Y, Xiao; J, Wu. Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med. 9(2020), no. 462, 1-13. Doi: 10.3390/jcm9020462 [ Links ]
P, van den Driessche; J, Watmough. Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180(2002), 29-48. Doi: 10.1016/ s0025-5564(02)00108-6 [ Links ]
World Health Organization. Coronavirus disease (COVID-2019) situation reports. https://www.who.int/emergencies/diseases/ novel-coronavirus-2019/situation-reports [ Links ]
J,T, Wu; K, Leung; G,M, Leung. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, Lancet, 395(2020), no. 10225 , 689- 697. Doi: 10.1016/S0140-6736(20)30260-9 [ Links ]
S, Zhao; Q, Lin; J, Ran; S,S, Musa; G, Yang; W, Wang;...; M,H, Wang. Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak, Int. J. Infect. Dis. 92(2020), 214-217. Doi: 10.1016/j.ijid.2020.01.050 [ Links ]
Z, Zhuang; S, Zhao; Q, Lin; P, Cao; Y, Lou; L, Yang; D, He. Preliminary estimation of the novel coronavirus disease (COVID-19) cases in Iran: A modelling analysis based on overseas cases and air travel data, Int. J. Infect. Dis. 94(2020), 29-31. Doi: /10.1016/j.ijid.2020.01. 050 [ Links ]
Recibido: 07 de Agosto de 2020; Revisado: 12 de Noviembre de 2020; Aprobado: 12 de Noviembre de 2020
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