Analysis of a COVID-19 compartmental model: a mathematical and computational approach

Zita Abreu; Guillaume Cantin; Cristiana J. Silva; Zita Abreu; Guillaume Cantin; Cristiana J. Silva

doi:10.3934/mbe.2021396

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 7979-7998. doi: 10.3934/mbe.2021396

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Analysis of a COVID-19 compartmental model: a mathematical and computational approach

1.
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal
2.
Laboratoire de Mathématiques Appliquées, FR-CNRS-3335, 25, Rue Philippe Lebon, Le Havre Normandie 76063, France

Academic Editor: Zhongwei Shen

Received: 06 July 2021 Accepted: 24 August 2021 Published: 14 September 2021

In this note, we consider a compartmental epidemic mathematical model given by a system of differential equations. We provide a complete toolkit for performing both a symbolic and numerical analysis of the spreading of COVID-19. By using the free and open-source programming language Python and the mathematical software SageMath, we contribute for the reproducibility of the mathematical analysis of the stability of the equilibrium points of epidemic models and their fitting to real data. The mathematical tools and codes can be adapted to a wide range of mathematical epidemic models.
- SAIRP epidemic model,
- COVID-19,
- stability analysis,
- free and open-source software,
- reproducibility of scientific method
Citation: Zita Abreu, Guillaume Cantin, Cristiana J. Silva. Analysis of a COVID-19 compartmental model: a mathematical and computational approach[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 7979-7998. doi: 10.3934/mbe.2021396

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Abstract

In this note, we consider a compartmental epidemic mathematical model given by a system of differential equations. We provide a complete toolkit for performing both a symbolic and numerical analysis of the spreading of COVID-19. By using the free and open-source programming language Python and the mathematical software SageMath, we contribute for the reproducibility of the mathematical analysis of the stability of the equilibrium points of epidemic models and their fitting to real data. The mathematical tools and codes can be adapted to a wide range of mathematical epidemic models.

References

[1]	Z. Ahmad, M. Arif, F. Ali, I. Khan, K. S. Nisar, A report on COVID-19 epidemic in Pakistan using SEIR fractional model, Sci. Rep., 10 (2020), 1–14. doi: 10.1038/s41598-019-56847-4
[2]	S. Ahmad, A. Ullah, Q. M. Al-Mdallal, H. Khan, K. Shah, A. Khan, Fractional order mathematical modeling of COVID-19 transmission, Chaos Solitons Fractals, 139 (2020), 110256. doi: 10.1016/j.chaos.2020.110256
[3]	M. Amouch, N. Karim, Modeling the dynamic of COVID-19 with different types of transmissions, Chaos Solitons Fractals, 150 (2021), 111188. doi: 10.1016/j.chaos.2021.111188
[4]	I. A. Baba, A. Yusuf, K. S. Nisar, A. Abdel-Aty, T. A. Nofal, Mathematical model to assess the imposition of lockdown during COVID-19 pandemic, Results Phy., 20 (2021), 103716. doi: 10.1016/j.rinp.2020.103716
[5]	N. Bacaër, McKendrick and Kermack on epidemic modelling (1926–1927), in A Short History of Mathematical Population Dynamics, Springer, (2011), 89–96.
[6]	S. Bugalia, V. P. Bajiya, J. P. Tripathi, M. T. Li, G. Q. Sun, Mathematical modeling of COVID-19 transmission: the roles of intervention strategies and lockdown, Math. Biosci. Eng., 17 (2020), 5961–5986. doi: 10.3934/mbe.2020318
[7]	S. A. Cheema, T. Kifayat, A. R. Rahman, U. Khan, A. Zaib, et al., Is social distancing, and quarantine effective in restricting covid-19 outbreak? Statistical evidences from Wuhan, China, Comput. Mater. Con., 66 (2021), 1977–1985.
[8]	J. Danane, K. Allali, Z. Hammouch, K. S. Nisar, Mathematical analysis and simulation of a stochastic COVID-19 L évy jump model with isolation strategy, Results Phys., 23 (2021), 103994. doi: 10.1016/j.rinp.2021.103994
[9]	Z. B. Dieudonné, Mathematical model for the mitigation of the economic effects of the Covid-19 in the Democratic Republic of the Congo, Plos One, 16 (2021), e0250775. doi: 10.1371/journal.pone.0250775
[10]	G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo, et al., Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy, Nat. Med., 26 (2020), 855–860. doi: 10.1038/s41591-020-0883-7
[11]	G. Hussain, T. Khan, A. Khan, M. Inc, G. Zaman, K. S. Nisar, A. Akgul, Modeling the dynamics of novel coronavirus (COVID-19) via stochastic epidemic model, Alex. Eng. J., 60 (2021), 4121–4130. doi: 10.1016/j.aej.2021.02.036
[12]	S. Khajanchi, K. Sarkar, J. Mondal, K. S. Nisar, S. F. Abdelwahab, Mathematical modeling of the COVID-19 pandemic with intervention strategies, Results Phys., 25 (2021), 104285. doi: 10.1016/j.rinp.2021.104285
[13]	A. Khan, H. M. Alshehri, T. Abdeljawad, Q. M. Al-Mdallal, H. Khan, Stability analysis of fractional nabla difference COVID-19 model, Results Phys., 22 (2021), 103888. doi: 10.1016/j.rinp.2021.103888
[14]	A. P. Lemos-Paião, C. J. Silva, D. F. Torres, A new compartmental epidemiological model for COVID-19 with a case study of portugal, Ecol. Complex., 44 (2020), 100885. doi: 10.1016/j.ecocom.2020.100885
[15]	K. Logeswari, C. Ravichandran, K. S. Nisar, Mathematical model for spreading of COVID-19 virus with the Mittag –Leffler kernel, Numer. Meth. Part. Differ. Equations, (2020), 1–16.
[16]	L. López, X. Rodo, A modified SEIR model to predict the COVID-19 outbreak in Spain and Italy: Simulating control scenarios and multi-scale epidemics, Results Phys., 21 (2021), 103746. doi: 10.1016/j.rinp.2020.103746
[17]	J. Y. Mugisha, J. Ssebuliba, J. N. Nakakawa, C. R. Kikawa, A. Ssematimba, Mathematical modeling of COVID-19 transmission dynamics in Uganda: Implications of complacency and early easing of lockdown, Plos One, 16 (2021), e0247456. doi: 10.1371/journal.pone.0247456
[18]	F. Ndairou, I. Area, J. J. Nieto, C. J. Silva, D. F. M. Torres, Fractional model of COVID-19 applied to Galicia, Spain and Portugal, Chaos Solitons Fractals, 144 (2021), 110652. doi: 10.1016/j.chaos.2021.110652
[19]	K. S. Nisar, S. Ahmad, A. Ullah, K. Shah, H. Alrabaiah, M. Arfan, Mathematical analysis of SIRD model of COVID-19 with Caputo fractional derivative based on real data, Results Phys., 21 (2021), 103772. doi: 10.1016/j.rinp.2020.103772
[20]	Python Software Foundation, Python Language Reference, (2019). Available from: http://www.python.org.
[21]	A. Radulescu, C. Williams, K. Cavanagh, Management strategies in a SEIR-type model of COVID 19 community spread, Sci. Rep., 10 (2020), 1–16. doi: 10.1038/s41598-019-56847-4
[22]	The Sage Developers, The Sage Mathematics Software System, (2020). Available from: https://www.sagemath.org.
[23]	A. S. Shaikh, I. N. Shaikh, K. S. Nisar, A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control, Adv. Differ. Equ., (2020), 1–19.
[24]	S. Sharma, V. Volpert, M. Banerjee, Extended SEIQR type model for COVID-19 epidemic and data analysis, Math. Biosic. Eng., 17 (2020), 7562–7604.
[25]	C. J. Silva, G. Cantin, C. Cruz, R. Fonseca-Pinto, R. Fonseca, E. S. Santos, et al., Complex network model for COVID-19: human behavior, pseudo-periodic solutions and multiple epidemic waves, J. Math. Anal. Appl., Forthcoming 2006.
[26]	C. J. Silva, C. Cruz, D. F. M. Torres, A. P. Muñuzuri, A. Carballosa, I. Area, et al., Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in Portugal, Sci. Rep., 11 (2021), 1–15. doi: 10.1038/s41598-020-79139-8
[27]	T. N. Sindhu, A. Shafiq, Q. M. Al-Mdallal, On the analysis of number of deaths due to Covid-19 outbreak data using a new class of distributions, Results Phys., 21 (2021), 103747. doi: 10.1016/j.rinp.2020.103747
[28]	T. N. Sindhu, A. Shafiq, Q. M. Al-Mdallal, Exponentiated transformation of Gumbel Type-II distribution for modeling COVID-19 data, Alex. Eng. J., 60 (2021), 671–689. doi: 10.1016/j.aej.2020.09.060
[29]	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. doi: 10.1016/S0025-5564(02)00108-6
[30]	C. Y. Yang, J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng., 17 (2020), 2708–2724. doi: 10.3934/mbe.2020148
[31]	Direção Geral da Saúde – COVID-19, Ponto de Situação Atual em Portugal, (2021). Available from: https://covid19.min-saude.pt/ponto-de-situacao-atual-em-portugal.
[32]	GitHub, Dados Relativos a Pandemia COVID-19 em Portugal, (2021). Available from: https://github.com/dssg-pt/covid19pt-data.
[33]	SciPy.org, scipy.optimize.curve fit, (2021). Available from: https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html.

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