The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy

Yujie Sheng; Jing-An Cui; Songbai Guo; Yujie Sheng; Jing-An Cui; Songbai Guo

doi:10.3934/mbe.2023258

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 5966-5992. doi: 10.3934/mbe.2023258

Previous Article Next Article

Research article Special Issues

The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy

School of Science, Beijing University of Civil Engineering and Architecture, Beijing 102616, China

Academic Editor: Xueying Wang

Received: 07 December 2022 Revised: 05 January 2023 Accepted: 05 January 2023 Published: 18 January 2023

The global spread of COVID-19 has not been effectively controlled. It poses a significant threat to public health and global economic development. This paper uses a mathematical model with vaccination and isolation treatment to study the transmission dynamics of COVID-19. In this paper, some basic properties of the model are analyzed. The control reproduction number of the model is calculated and the stability of the disease-free and endemic equilibria is analyzed. The parameters of the model are obtained by fitting the number of cases that were detected as positive for the virus, dead, and recovered between January 20 and June 20, 2021, in Italy. We found that vaccination better controlled the number of symptomatic infections. A sensitivity analysis of the control reproduction number has been performed. Numerical simulations demonstrate that reducing the contact rate of the population and increasing the isolation rate of the population are effective non-pharmaceutical control measures. We found that if the isolation rate of the population is reduced, a short-term decrease in the number of isolated individuals can lead to the disease not being controlled at a later stage. The analysis and simulations in this paper may provide some helpful suggestions for preventing and controlling COVID-19.
- COVID-19,
- epidemic model,
- vaccination,
- isolation,
- numerical simulation
Citation: Yujie Sheng, Jing-An Cui, Songbai Guo. The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 5966-5992. doi: 10.3934/mbe.2023258

Related Papers:

Abstract

The global spread of COVID-19 has not been effectively controlled. It poses a significant threat to public health and global economic development. This paper uses a mathematical model with vaccination and isolation treatment to study the transmission dynamics of COVID-19. In this paper, some basic properties of the model are analyzed. The control reproduction number of the model is calculated and the stability of the disease-free and endemic equilibria is analyzed. The parameters of the model are obtained by fitting the number of cases that were detected as positive for the virus, dead, and recovered between January 20 and June 20, 2021, in Italy. We found that vaccination better controlled the number of symptomatic infections. A sensitivity analysis of the control reproduction number has been performed. Numerical simulations demonstrate that reducing the contact rate of the population and increasing the isolation rate of the population are effective non-pharmaceutical control measures. We found that if the isolation rate of the population is reduced, a short-term decrease in the number of isolated individuals can lead to the disease not being controlled at a later stage. The analysis and simulations in this paper may provide some helpful suggestions for preventing and controlling COVID-19.

References

[1]	D. Cucinotta, M. Vanelli, WHO declares COVID-19 a pandemic, Acta Biomed., 91 (2020), 157–160. https://doi.org/10.23750/abm.v91i1.9397 doi: 10.23750/abm.v91i1.9397
[2]	S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith, et al., The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. Intern. Med., 172 (2020), 577–582. https://doi.org/10.7326/M20-0504 doi: 10.7326/M20-0504
[3]	A. K. Singh, R. Gupta, A. Misra, Comorbidities in COVID-19: Outcomes in hypertensive cohort and controversies with renin angiotensin system blockers, Diabetes Metab. Syndr. Clin. Res. Rev., 14 (2020), 283–287. https://doi.org/10.1016/j.dsx.2020.03.016 doi: 10.1016/j.dsx.2020.03.016
[4]	Z. Xu, L. Shi, Y. Wang, J. Zhang, L. Huang, C. Zhang, et al., Pathological findings of COVID-19 associated with acute respiratory distress syndrome, Lancet Respir. Med., 8 (2020), 420–422. https://doi.org/10.1016/S2213-2600(20)30076-X doi: 10.1016/S2213-2600(20)30076-X
[5]	W. Guan, Z. Ni, Y. Hu, W. Liang, C. Ou, J. He, et al., Clinical characteristics of coronavirus disease 2019 in China, N. Engl. J. Med., 382 (2020), 1708–1720. https://doi.org/10.1056/NEJMoa2002032 doi: 10.1056/NEJMoa2002032
[6]	World Health Organization, Coronavirus disease 2019 (COVID-19) pandemic, (2020). https://www.who.int/emergencies/diseases/novel-coronavirus-2019 (accessed July 4, 2022).
[7]	Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, et al., Early transmission dynamics in Wuhan, China, of novel coronavirus–infected pneumonia, N. Engl. J. Med., 382 (2020), 1199–1207. https://doi.org/10.1056/NEJMoa2001316 doi: 10.1056/NEJMoa2001316
[8]	C. del Rio, P. N. Malani, COVID-19—new insights on a rapidly changing epidemic, Jama J. Am. Med. Assoc., 323 (2020), 1339–1340. https://doi.org/10.1001/jama.2020.3072 doi: 10.1001/jama.2020.3072
[9]	J. A. Cui, Y. Sun, H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Differ. Equations, 20 (2008), 31–53. https://doi.org/10.1007/s10884-007-9075-0 doi: 10.1007/s10884-007-9075-0
[10]	Y. Li, J. A. Cui, The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun. Nonlinear. Sci. Numer. Simul., 14 (2009), 2353–2365. https://doi.org/10.1016/j.cnsns.2008.06.024 doi: 10.1016/j.cnsns.2008.06.024
[11]	J. Rui, Q. Wang, J. Lv, B. Zhao, Q. Hu, H. Du, et al., The transmission dynamics of middle east respiratory syndrome coronavirus, Travel Med. Infect. Dis., 45 (2022), 102243. https://doi.org/10.1016/j.tmaid.2021.102243 doi: 10.1016/j.tmaid.2021.102243
[12]	J. Li, P. Yuan, J. Heffernan, T. Zheng, N. Ogden, B. Sander, et al., Fangcang shelter hospitals during the COVID-19 epidemic, Wuhan, China, Bull. World Health Organ., 98 (2020), 830–841. https://doi.org/10.2471/BLT.20.258152 doi: 10.2471/BLT.20.258152
[13]	L. Wang, J. Wang, H. Zhao, Y. Shi, K. Wang, P. Wu, et al., Modelling and assessing the effects of medical resources on transmission of novel coronavirus (COVID-19) in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2936–2949. https://doi.org/10.3934/mbe.2020165 doi: 10.3934/mbe.2020165
[14]	B. Yuan, R. Liu, S. Tang, A quantitative method to project the probability of the end of an epidemic: application to the COVID-19 outbreak in Wuhan, 2020, J. Theor. Biol., 545 (2022), 111149. https://doi.org/10.1016/j.jtbi.2022.111149 doi: 10.1016/j.jtbi.2022.111149
[15]	L. Xue, S. Jing, J. C. Miller, W. Sun, H. Li, J. G. Estrada-Franco, et al., A data-driven network model for the emerging COVID-19 epidemics in Wuhan, Toronto and Italy, Math. Biosci., 326 (2020), 108391. https://doi.org/10.1016/j.mbs.2020.108391 doi: 10.1016/j.mbs.2020.108391
[16]	C. Yang, J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708–2724. https://doi.org/10.3934/mbe.2020148 doi: 10.3934/mbe.2020148
[17]	Z. Li, T. Zhang, Analysis of a COVID-19 epidemic model with seasonality, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01105-4 doi: 10.1007/s11538-022-01105-4
[18]	X. Wang, S. Wang, J. Wang, L. Rong, A multiscale model of COVID-19 dynamics, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01058-8 doi: 10.1007/s11538-022-01058-8
[19]	L. Xue, S. Jing, W. Sun, M. Liu, Z. Peng, H. Zhu, Evaluating the impact of the travel ban within mainland China on the epidemic of the COVID-19, Int. J. Infect. Dis., 107 (2021), 278–283. https://doi.org/10.1016/j.ijid.2021.03.088 doi: 10.1016/j.ijid.2021.03.088
[20]	S. Wang, Y. Pan, Q. Wang, H. Miao, A. N. Brown, L. Rong, Modeling the viral dynamics of SARS-CoV-2 infection, Math. Biosci., 328 (2020), 108438. https://doi.org/10.1016/j.mbs.2020.108438 doi: 10.1016/j.mbs.2020.108438
[21]	H. Wan, J. A. Cui, G. J. Yang, Risk estimation and prediction of the transmission of coronavirus disease-2019 (COVID-19) in the mainland of China excluding Hubei province, Infect. Dis. Poverty, 9 (2020). https://doi.org/10.1186/s40249-020-00683-6 doi: 10.1186/s40249-020-00683-6
[22]	K. S. Al-Basyouni, A. Q. Khan, Discrete-time COVID-19 epidemic model with chaos, stability and bifurcation, Results. Phys., 43 (2022), 106038. https://doi.org/10.1016/j.rinp.2022.106038 doi: 10.1016/j.rinp.2022.106038
[23]	A. Abbes, A. Ouannas, N. Shawagfeh, G. Grassi, The effect of the Caputo fractional difference operator on a new discrete COVID-19 model, Results Phys., 39 (2022), 105797. https://doi.org/10.1016/j.rinp.2022.105797 doi: 10.1016/j.rinp.2022.105797
[24]	S. He, J. Yang, M. He, D. Yan, S. Tang, L. Rong, The risk of future waves of COVID-19: modeling and data analysis, Math. Biosci. Eng., 18 (2021), 5409–5426. https://doi.org/10.3934/mbe.2021274 doi: 10.3934/mbe.2021274
[25]	P. Y. Liu, S. He, L. B. Rong, S. Y. Tang, The effect of control measures on COVID-19 transmission in Italy: comparison with Guangdong province in China, Infect. Dis. Poverty, 9 (2020). https://doi.org/10.1186/s40249-020-00730-2 doi: 10.1186/s40249-020-00730-2
[26]	H. Song, Z. Jia, Z. Jin, S. Liu, Estimation of COVID-19 outbreak size in Harbin, China, Nonlinear. Dyn., 106 (2021), 1229–1237. https://doi.org/10.1007/s11071-021-06406-2 doi: 10.1007/s11071-021-06406-2
[27]	X. Ma, X. F. Luo, L. Li, Y. Li, G. Q. Sun, The influence of mask use on the spread of COVID-19 during pandemic in New York City, Results Phys., 34 (2022), 105224. https://doi.org/10.1016/j.rinp.2022.105224 doi: 10.1016/j.rinp.2022.105224
[28]	J. K. K. Asamoah, E. Okyere, A. Abidemi, S. E. Moore, G. Q. Sun, Z. Jin, et al., Optimal control and comprehensive cost-effectiveness analysis for COVID-19, Results Phys., 33 (2022), 105177. https://doi.org/10.1016/j.rinp.2022.105177 doi: 10.1016/j.rinp.2022.105177
[29]	L. Masandawa, S. S. Mirau, I. S. Mbalawata, J. N. Paul, K. Kreppel, O. M. Msamba, Modeling nosocomial infection of COVID-19 transmission dynamics, Results Phys., 37 (2022), 105503. https://doi.org/10.1016/j.rinp.2022.105503 doi: 10.1016/j.rinp.2022.105503
[30]	C. Legarreta, S. Alonso-Quesada, M. De la Sen, Analysis and parametrical estimation with real COVID-19 data of a new extended SEIR epidemic model with quarantined individuals, Discrete Dyn. Nat. Soc., 2022 (2022), 1–29. https://doi.org/10.1155/2022/5151674 doi: 10.1155/2022/5151674
[31]	M. De la Sen, A. Ibeas, On an SE (Is) (Ih) AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic, Adv. Differ. Equations, 2021 (2021). https://doi.org/10.1186/s13662-021-03248-5 doi: 10.1186/s13662-021-03248-5
[32]	M. Rangasamy, N. Alessa, P. B. Dhandapani, K. Loganathan, Dynamics of a novel IVRD pandemic model of a large population over a long time with efficient numerical methods, Symmetry, 14 (2022), 1919. https://doi.org/10.3390/sym14091919 doi: 10.3390/sym14091919
[33]	A. K. Paul, M. A. Kuddus, Mathematical analysis of a COVID-19 model with double dose vaccination in Bangladesh, Results Phys., 35 (2022), 105392. https://doi.org/10.1016/j.rinp.2022.105392 doi: 10.1016/j.rinp.2022.105392
[34]	U. A. P. de León, E. Avila-Vales, K. Huang, Modeling COVID-19 dynamic using a two-strain model with vaccination, Chaos Solitons Fractals, 157 (2022), 111927. https://doi.org/10.1016/j.chaos.2022.111927 doi: 10.1016/j.chaos.2022.111927
[35]	X. Wang, H. Wu, S. Tang, Assessing age-specific vaccination strategies and post-vaccination reopening policies for COVID-19 control using SEIR modeling approach, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-01064-w doi: 10.1007/s11538-022-01064-w
[36]	F. Zhang, Z. Jin, Effect of travel restrictions, contact tracing and vaccination on control of emerging infectious diseases: transmission of COVID-19 as a case study, Math. Biosci. Eng., 19 (2022), 3177–3201. https://doi.org/10.3934/mbe.2022147 doi: 10.3934/mbe.2022147
[37]	J. H. Buckner, G. Chowell, M. R. Springborn, Dynamic prioritization of COVID-19 vaccines when social distancing is limited for essential workers, Appl. Biol. Sci., 118 (2021). https://doi.org/10.1073/pnas.2025786118 doi: 10.1073/pnas.2025786118
[38]	S. Moore, E. M. Hill, M. J. Tildesley, L. Dyson, M. J. Keeling, Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study, Lancet Infect. Dis., 21 (2021), 793–802. https://doi.org/10.1016/S1473-3099(21)00143-2 doi: 10.1016/S1473-3099(21)00143-2
[39]	Z. Ma, Y. Zhou, C. Li, Qualitative and stability methods of ordinary differential equations (in Chinese), 2nd ed, Science Press, Beijing, 2015.
[40]	P. van den 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
[41]	S. Guo, W. Ma, Remarks on a variant of Lyapunov-LaSalle theorem, Math. Biosci. Eng., 16 (2019), 1056–1066. https://doi.org/10.3934/mbe.2019050 doi: 10.3934/mbe.2019050
[42]	S. Guo, Y. Xue, X. Li, Z. Zheng, Dynamics of COVID-19 models with asymptomatic infections and quarantine measures, arXiv preprint, (2022). https://doi.org/10.21203/rs.3.rs-2291574/v1 doi: 10.21203/rs.3.rs-2291574/v1
[43]	Italian Ministry of Health, COVID-19 Vaccines Report, (2022). Available from: https://www.governo.it/it/cscovid19/report-vaccini/(accessed January 5, 2023).
[44]	United States Food and Drug Administration, FDA Briefing Document Pfizer-BioNTech COVID-19 Vaccine, (2020). Available from: https://www.fda.gov/media/144245/download.
[45]	Presidency of the Council of Ministers, DECREE-LAW No. 172 of December 18, 2020, (2020). Available from: https://www.normattiva.it/uri-res/N2Ls?urn:nir:stato:decreto.legge:2020-12-18;172!vig= (accessed October 9, 2022).
[46]	Governo Italiano, Council of Ministers Press Release No. 97, (2021). Available from: https://www.sitiarcheologici.palazzochigi.it/www.governo.it/febbraio%202021/node/16180.html (accessed October 8, 2022).
[47]	Italian Ministry of Health, OJ General Series No. 75, 27-03-2021, (2021). Available from: https://www.gazzettaufficiale.it/eli/id/2021/03/27/21A01967/sg (accessed October 9, 2022).
[48]	Italy Civil Protection Department, Italian COVID-19 data, (2022). Available from: https://github.com/pcm-dpc/COVID-19 (accessed January 5, 2023).
[49]	World Bank, Average life expectancy in Italy, (2020). Available from: https://data.worldbank.org/indicator/SP.DYN.LE00.IN?locations=IT (accessed January 5, 2023).
[50]	World Bank, Italy Birth rate, crude (per 1,000 people), (2020). Available from: https://data.worldbank.org/indicator/SP.DYN.CBRT.IN?locations=IT (accessed January 5, 2023).
[51]	Our World in Data, Italian COVID-19 vaccine dataset, (2022).
[52]	World Bank, Italian population data. Available from: https://data.worldbank.org/indicator/SP.POP.TOTL?locations=IT (accessed January 5, 2023).
[53]	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. https://doi.org/10.1007/s11538-008-9299-0 doi: 10.1007/s11538-008-9299-0
[54]	H. Tian, Y. Liu, Y. Li, C. H. Wu, B. Chen, M. U. G. Kraemer, et al., An investigation of transmission control measures during the first 50 days of the COVID-19 epidemic in China, Science, 368 (2020), 638–642. https://doi.org/10.1126/science.abb6105 doi: 10.1126/science.abb6105
[55]	M. Duan, Z. Jin, The heterogeneous mixing model of COVID-19 with interventions, J. Theor. Biol., 553 (2022), 111258. https://doi.org/10.1016/j.jtbi.2022.111258 doi: 10.1016/j.jtbi.2022.111258
[56]	J. A. Cui, Y. Wu, S. Guo, Effect of non-homogeneous mixing and asymptomatic individuals on final epidemic size and basic reproduction number in a meta-population model, Bull. Math. Biol., 84 (2022). https://doi.org/10.1007/s11538-022-00996-7 doi: 10.1007/s11538-022-00996-7

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)