Stability analysis and optimal control of COVID-19 with quarantine and media awareness

Jiajia Zhang; Yuanhua Qiao; Yan Zhang; Jiajia Zhang; Yuanhua Qiao; Yan Zhang

doi:10.3934/mbe.2022230

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 4911-4932. doi: 10.3934/mbe.2022230

Previous Article Next Article

Research article Special Issues

Stability analysis and optimal control of COVID-19 with quarantine and media awareness

Faculty of Science, Beijing University of Technology, Beijing 100124, China

Academic Editor: Wandi Ding

Received: 08 December 2021 Revised: 27 February 2022 Accepted: 07 March 2022 Published: 14 March 2022

In this paper, an improved COVID-19 model is given to investigate the influence of treatment and media awareness, and a non-linear saturated treatment function is introduced in the model to lay stress on the limited medical conditions. Equilibrium points and their stability are explored. Basic reproduction number is calculated, and the global stability of the equilibrium point is studied under the given conditions. An object function is introduced to explore the optimal control strategy concerning treatment and media awareness. The existence, characterization and uniqueness of optimal solution are studied. Several numerical simulations are given to verify the analysis results. Finally, discussion on treatment and media awareness is given for prevention and treatment of COVID-19.
- COVID-19,
- epidemic model,
- globally stable,
- sensitivity analysis,
- optimal control
Citation: Jiajia Zhang, Yuanhua Qiao, Yan Zhang. Stability analysis and optimal control of COVID-19 with quarantine and media awareness[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 4911-4932. doi: 10.3934/mbe.2022230

Related Papers:

Abstract

In this paper, an improved COVID-19 model is given to investigate the influence of treatment and media awareness, and a non-linear saturated treatment function is introduced in the model to lay stress on the limited medical conditions. Equilibrium points and their stability are explored. Basic reproduction number is calculated, and the global stability of the equilibrium point is studied under the given conditions. An object function is introduced to explore the optimal control strategy concerning treatment and media awareness. The existence, characterization and uniqueness of optimal solution are studied. Several numerical simulations are given to verify the analysis results. Finally, discussion on treatment and media awareness is given for prevention and treatment of COVID-19.

References

[1]	The Government of Wuhan Homepage. Available from: http://english.wh.gov.cn/.
[2]	A. E. Gorbalenya, S. C. Baker, R. S. Baric, R. J. de Groot, C. Drosten, A. A. Gulyaeva, et al., The species severe acute respiratory syndrome-related coronavirus: Classifying 2019-nCoV and naming it SARS-CoV-2, Nat. Microbiol., 5 (2020), 536–544. https://doi.org/10.1038/s41564-020-0695-z doi: 10.1038/s41564-020-0695-z
[3]	L. Zhou, M. Fan, Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012), 312–324. https://doi.org/10.1016/j.nonrwa.2011.07.036 doi: 10.1016/j.nonrwa.2011.07.036
[4]	N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of Malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272–1296. https://doi.org/10.1007/s11538-008-9299-0 doi: 10.1007/s11538-008-9299-0
[5]	T. K. Kar, P. K. Mondal, Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear Anal. Real World Appl., 12 (2011), 2058–2068. https://doi.org/10.1016/j.nonrwa.2010.12.021 doi: 10.1016/j.nonrwa.2010.12.021
[6]	C. Shan, H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differ. Equations, 257 (2014), 1662–1688. https://doi.org/10.1016/j.jde.2014.05.030 doi: 10.1016/j.jde.2014.05.030
[7]	J. R. Artalejo, A. Economou, M. J. Lopez-Herrero, The stochastic SEIR model before extinction: computational approaches, Appl. Math. Comput., 265 (2015), 1026–1043. https://doi.org/10.1016/j.amc.2015.05.141 doi: 10.1016/j.amc.2015.05.141
[8]	P. Diaz, P. Constantine, K. Kalmbach, E. Jones, S. Pankavich, A modified SEIR model for the spread of Ebola in Western Africa and metrics for resource allocation, Appl. Math. Comput., 324 (2018), 141–155. https://doi.org/10.1016/j.amc.2017.11.039 doi: 10.1016/j.amc.2017.11.039
[9]	R. Almeida, Analysis of a fractional SEIR model with treatment, Appl. Math. Lett., 84 (2018), 56–62. https://doi.org/10.1016/j.aml.2018.04.015 doi: 10.1016/j.aml.2018.04.015
[10]	S. Jana, P. Haldar, T. K. Kar, Mathematical analysis of an epidemic model with isolation and optimal controls, Int. J. Comput. Math., 94 (2017), 1318–1336. https://doi.org/10.1080/00207160.2016.1190009 doi: 10.1080/00207160.2016.1190009
[11]	D. Gao, N. Huang, Optimal control analysis of a tuberculosis model, Appl. Math. Model., 58 (2018), 47–64. https://doi.org/10.1016/j.apm.2017.12.027 doi: 10.1016/j.apm.2017.12.027
[12]	D. K. Das, S. Khajanchi, T. K. Kar, The impact of the media awareness and optimal strategy on the prevalence of tuberculosis, Appl. Math. Comput., 366 (2020), 124732. https://doi.org/10.1016/j.amc.2019.124732 doi: 10.1016/j.amc.2019.124732
[13]	T. K. Kar, S. K. Nandi, S. Jana, M. Mandal, Stability and bifurcation analysis of an epidemic model with the effect of media, Chaos Solitons Fractals, 120 (2019), 188–199. https://doi.org/10.1016/j.chaos.2019.01.025 doi: 10.1016/j.chaos.2019.01.025
[14]	T. Chen, J. Rui, Q. Wang, Z. Zhao, J. Cui, L. Yin, A mathematical model for simulating the phase-based transmissibility of a novel coronvirus, Infect. Dis. Poverty, 9 (2020), 24. https://doi.org/10.1186/s40249-020-00640-3 doi: 10.1186/s40249-020-00640-3
[15]	M. A. Khan, A. Atangana, Modeling the dynamics of novel coronavirus (2019-nCoV) with fractional derivative, Alexandria Eng. J., 59 (2020), 2379–2389. https://doi.org/10.1016/j.aej.2020.02.033 doi: 10.1016/j.aej.2020.02.033
[16]	B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao, et al., Estimation of the transmission risk of 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. https://doi.org/10.3390/jcm9020462 doi: 10.3390/jcm9020462
[17]	B. Tang, N. L. Bragazzi, Q. Li, S. Tangd, Y. Xiao, J. Wu, An updated estimation of the risk of transmission of the novel coronavirus (2019-nCoV), Infect. Dis. Model., 5 (2020), 248–255. https://doi.org/10.1016/j.idm.2020.02.001 doi: 10.1016/j.idm.2020.02.001
[18]	P. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[19]	L. S. Pontryagin, V. G. Boltyanski, R. V. Gamkrelidze, E. F. Misenko, The Mathematical Theory of Optimal Process, Wiley, New Jersey, 1962.
[20]	H. R. Joshi, Optimal control of an HIV immunology model, Optim. Control Appl. Methods, 23 (2002), 199–213. https://doi.org/10.1002/oca.710 doi: 10.1002/oca.710
[21]	W. H. Fleming, R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975. https://doi.org/10.1007/978-1-4612-6380-7
[22]	D. L. Lukes, Differential Equations Classical to Controlled, Academia Press, New York, 1982.
[23]	H. Zhang, Z. Yang, K. A. Pawelek, S. Liu, Optimal control strategies for a two-group epidemic model with vaccination-resource constraints, Appl. Math. Comput., 371 (2020), 124956. https://doi.org/10.1016/j.amc.2019.124956 doi: 10.1016/j.amc.2019.124956

Reader Comments

Your name:*

Email:*
© 2022 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)