Inverse problem to elaborate and control the spread of COVID-19: A case study from Morocco

Marouane Karim; Abdelfatah Kouidere; Mostafa Rachik; Kamal Shah; Thabet Abdeljawad; Marouane Karim; Abdelfatah Kouidere; Mostafa Rachik; Kamal Shah; Thabet Abdeljawad

doi:10.3934/math.20231194

AIMS Mathematics

2023, Volume 8, Issue 10: 23500-23518. doi: 10.3934/math.20231194

Previous Article Next Article

Research article Special Issues

Inverse problem to elaborate and control the spread of COVID-19: A case study from Morocco

1.
LAMS, Department of Mathematics and Computer Science, Faculty of Sciences Ben M'Sik, Hassan II University of Casablanca, Morocco
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Mathematics, University of Malakand, Chakdara Dir (L), Khyber Pakhtunkhwa 18000, Pakistan
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea
6.
Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 09 June 2023 Revised: 09 July 2023 Accepted: 10 July 2023 Published: 27 July 2023
MSC : 49J15, 49J21, 68P15, 92B05

In this paper, we focus on identifying the transmission rate associated with a COVID-19 mathematical model by using a predefined prevalence function. To do so, we use a Python code to extract the Lagrange interpolation polynomial from real daily data corresponding to an appropriate period in Morocco. The existence of a perfect control scheme is demonstrated. The Pontryagin maximum technique is used to explain these optimal controls. The optimality system is numerically solved using the 4th-order Runge-Kutta approximation.
- COVID-19,
- inverse problem,
- Lagrange interpolation,
- optimal control,
- numerical method
Citation: Marouane Karim, Abdelfatah Kouidere, Mostafa Rachik, Kamal Shah, Thabet Abdeljawad. Inverse problem to elaborate and control the spread of COVID-19: A case study from Morocco[J]. AIMS Mathematics, 2023, 8(10): 23500-23518. doi: 10.3934/math.20231194

Related Papers:

Abstract

In this paper, we focus on identifying the transmission rate associated with a COVID-19 mathematical model by using a predefined prevalence function. To do so, we use a Python code to extract the Lagrange interpolation polynomial from real daily data corresponding to an appropriate period in Morocco. The existence of a perfect control scheme is demonstrated. The Pontryagin maximum technique is used to explain these optimal controls. The optimality system is numerically solved using the 4th-order Runge-Kutta approximation.

References

[1]	W. O. Kermack, A. G. Mckendrick, Contributions to the mathematical theory of epidemics–II. The problem of endemicity, Bull. Math. Biol., 53 (1991), 57–87. https://doi.org/10.1007/BF02464424 doi: 10.1007/BF02464424
[2]	S. Gao, D. Xie, L. Chen, Pulse vaccination strategy in a delayed SIR epidemic model with vertical transmission, Discrete Cont. Dyn. B, 7 (2007), 77–86. https://doi.org/10.3934/dcdsb.2007.7.77 doi: 10.3934/dcdsb.2007.7.77
[3]	W. Liu, S. A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biology, 23 (1986), 187–204. https://doi.org/10.1007/BF00276956 doi: 10.1007/BF00276956
[4]	A. Lahrouz, L. Omari, Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probabil. Lett., 83 (2013), 960–968. https://doi.org/10.1016/j.spl.2012.12.021 doi: 10.1016/j.spl.2012.12.021
[5]	L. F. Chen, M. W. V. Weg, D. A. Hofmann, H. S. Reisinger, The Hawthorne effect in infection prevention and epidemiology, Infect. Cont. Hosp. Ep., 36 (2015), 1444–1450. https://doi.org/10.1017/ice.2015.216 doi: 10.1017/ice.2015.216
[6]	Y. Guo, T. Li, Modeling the competitive transmission of the Omicron strain and Delta strain of COVID-19, J. Math. Anal. Appl., 526 (2023), 127283. https://doi.org/10.1016/j.jmaa.2023.127283 doi: 10.1016/j.jmaa.2023.127283
[7]	T. Li, Y. Guo, Modeling and optimal control of mutated COVID-19 (Delta strain) with imperfect vaccination, Chaos Soliton. Fract., 156 (2022), 111825. https://doi.org/10.1016/j.chaos.2022.111825 doi: 10.1016/j.chaos.2022.111825
[8]	Y. Guo, T. Li, Fractional-order modeling and optimal control of a new online game addiction model based on real data, Commun. Nonlinear Sci., 121 (2023), 107221. https://doi.org/10.1016/j.cnsns.2023.107221 doi: 10.1016/j.cnsns.2023.107221
[9]	Y. Guo, T. Li, Modeling and dynamic analysis of novel coronavirus pneumonia (COVID-19) in China, J. Appl. Math. Comput., 68 (2022), 2641–2666. https://doi.org/10.1007/s12190-021-01611-z doi: 10.1007/s12190-021-01611-z
[10]	M. Karim, S. B. Rhila, H. Boutayeb, M. Rachik, COVID-19 spatiotemporal SIR model: Regional optimal control approach, Commun. Math. Biol. Neurosci., 2022 (2022), 115. https://doi.org/10.28919/cmbn/7734 doi: 10.28919/cmbn/7734
[11]	H. Khan, A. Khan, F. Jarad, A. Shah, Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system, Chaos Soliton. Fract., 131 (2020), 109477. https://doi.org/10.1016/j.chaos.2019.109477 doi: 10.1016/j.chaos.2019.109477
[12]	S. Hussain, O. Tunç, G. U. Rahman, H. Khan, E. Nadia, Mathematical analysis of stochastic epidemic model of MERS-corona and application of ergodic theory, Math. Comput. Simulat., 207 (2023), 130–150. https://doi.org/10.1016/j.matcom.2022.12.023 doi: 10.1016/j.matcom.2022.12.023
[13]	H. Khan, J. Alzabut, A. Shan, Z. Y. He, S. Etemad, S. Rezapour, et al., On fractal-fractional waterborne disease model: A study on theoretical and numerical aspects of solutions via simulations, Fractals, 31 (2023), 2340055. https://doi.org/10.1142/S0218348X23400558 doi: 10.1142/S0218348X23400558
[14]	B. Wacker, J. Schlüter, Time-discrete parameter identification algorithms for two deterministic epidemiological models applied to the spread of COVID-19, submitted for publication.
[15]	K. P. Hadeler, Parameter identification in epidemic models, Math. Biosci., 229 (2011), 185–189. https://doi.org/10.1016/j.mbs.2010.12.004 doi: 10.1016/j.mbs.2010.12.004
[16]	M. Pollicott, H. Wang, H. Weiss, Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem, J. Biol. Dynam., 6 (2012), 509–523. https://doi.org/10.1080/17513758.2011.645510 doi: 10.1080/17513758.2011.645510
[17]	A. Harding, New Covid variant: South Africa's pride and punishment, BBC News, 2021, Available from: https://www.bbc.com/news/world-africa-59432579.
[18]	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 Contin. Dyn. Syst. Ser. S, 6 (2018), 1179–1199. https://doi.org/10.3934/dcdss.2018067 doi: 10.3934/dcdss.2018067
[19]	T. Zhang, Z. Teng, On a nonautonomous SEIRS model in epidemiology, Bull. Math. Biol., 69 (2007), 2537–2559. https://doi.org/10.1007/s11538-007-9231-z doi: 10.1007/s11538-007-9231-z
[20]	A. Kouidere, B. Khajji, A. E. Bhih, O. Balatif, M. Rachik, A mathematical modeling with optimal control strategy of transmission of COVID-19 pandemic virus, Commun. Math. Biol. Neurosci., 2020 (2020), 24. https://doi.org/10.28919/cmbn/4599 doi: 10.28919/cmbn/4599
[21]	Z. Q. Xia, J. Zhang, Y. K. Xue, G. Q. Sun, Z. Jin, Modeling the transmission of Middle East respirator syndrome corona virus in the Republic of Korea, Plos One, 10 (2015), e0144778. https://doi.org/10.1371/journal.pone.0144778 doi: 10.1371/journal.pone.0144778
[22]	Maroc Github Topics, Available from: https://github.com/topics/maroc.
[23]	R. A. Addi, A. Benksim, M. Amine, M. Cherkaoui, COVID-19 outbreak and perspective in Morocco, Electron. J. Gen. Med., 17 (2020), em204. https://doi.org/10.29333/ejgm/7857 doi: 10.29333/ejgm/7857
[24]	Royaume du Maroc Ministere de la Santr et de la Protection Sociale, Available from: https://www.sante.gov.ma.
[25]	M. Alkama, A. Larrache, M. Rachik, I. Elmouki, Optimal duration and dosage of BCG intravesical immunotherapy: A free final time optimal control approach, Math. Method. Appl. Sci., 41 (2018), 2209–2219. https://doi.org/10.1002/mma.4745 doi: 10.1002/mma.4745
[26]	W. Fleming, R. Rishel, Deterministic and stochastic optimal control, New York: Springer, 2012. https://doi.org/10.1007/978-1-4612-6380-7
[27]	E. Roxin, Differential equations: classical to controlled, Am. Math. Mon., 92 (1985), 223–225. https://doi.org/10.1080/00029890.1985.11971586 doi: 10.1080/00029890.1985.11971586
[28]	W. E. Boyce, R. C. Diprima, D. B. Meade, Elementary differential equations and boundary value problems, New York: John Wiley and Sons, 2017.
[29]	L. S. Pontryagin, Mathematical theory of optimal processes, London: Routledge, 1987. https://doi.org/10.1201/9780203749319
[30]	M. Elhia, L. Boujallal, M. Alkama, O. Balatif, M. Rachik, Set-valued control approach applied to a COVID-19 model with screening and saturated treatment function, Complexity, 2020 (2020), 9501028. https://doi.org/10.1155/2020/9501028 doi: 10.1155/2020/9501028
[31]	M. Layelmam, Y. A. Laaziz, S. Benchelha, Y. Diyer, S. Rarhibou, Forecasting COVID-19 in Morocco, J. Clin. Exp. Invest., 11 (2020), em00748. https://doi.org/10.5799/jcei/8264 doi: 10.5799/jcei/8264

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)