The alarming conditions arising in this world due to the surveillance of COVID-19 have been a threat to the human beings. Mathematics has been widely employed to obtain a comprehensive understanding of the transmission dynamics and regulation of such a disease. This article explores the dynamical and asymmetrical behaviors of the recent pandemic COVID-19 transmission using a mathematical model. The local and global stabilities of equilibrium points of the system are examined based on the basic reproduction number R₀. Jacobian matrices and the Routh-Hurwitz criterion are used to confirm the local stability of the equilibrium state. The findings of the simulations reveal that the transmission of COVID-19 can be symmetrically controlled by limiting the interaction rate of sick people and growing the quarantine of exposed people. Further, the model is solved using the Galerkin technique which demonstrated the impact of significant parameters entangled in the model on the population dynamics of state variables. Furthermore, the confirmed results are obtained through Galerkin scheme by solving the model using the well-known classical Runge-Kutta method. Finally, comparing the findings, it is found that the outcomes of both the schemes coincide in symmetric patterns, which in turn, demonstrated the viability of the suggested approaches.

mathematical model, computational and stability analysis, COVID-19, Galerkin technique.

Received: February 4, 2023; Accepted: March 14, 2023; Published: April 11, 2023

How to cite this article: Mdi Begum Jeelani, Stability and computational analysis of COVID-19 using a higher order Galerkin time discretization scheme, Advances and Applications in Statistics 86(2) (2023), 167-206. http://dx.doi.org/10.17654/0972361723022

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] R. M. Anderson and R. M. May, Helminth infections of humans: mathematical models, population dynamics, and control, Advances in Parasitology, Elsevier, Vol. 24, 1985, pp. 1-101.
[2] N. T. Bailey, B. Sendov and R. Tsanev, Mathematical models in biology and medicine, IFIP-TC4 Working Conference on Mathematical Models in Biology and Medicine (1972: Varna, Bulgaria), North-Holland Pub. Co., 1974.
[3] D. Dourado-Neto, D. Teruel, K. Reichardt, D. Nielsen, J. Frizzone and O. Bacchi, Principles of crop modeling and simulation: I. Uses of mathematical models in agricultural science, Sci. Agricola. 55 (1998), 46-50.
[4] L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, 2005.
[5] Attaullah, R. Jan and Ş. Yüzbaşı, Dynamical behaviour of HIV infection with the influence of variable source term through Galerkin method, Chaos Solitons Fractals 152 (2021), 111429.
[6] Attaullah and M. Sohaib, Mathematical modeling and numerical simulation of HIV infection model, Results in Applied Mathematics 7 (2020), 100118.
[7] H. A. Rothana and S. N. Byrareddy, The epidemiology and pathogenesis of coronavirus disease (COVID-19) outbreak, J. Autoimmun. 109 (2020), 102433.
[8] H. Lu, Drug treatment options for the 2019-new coronavirus (2019-nCoV), Biosci. Trends 14 (2020), 69-71. https://doi.org/10.5582/bst.2020.01020.
[9] M. Bassetti, A. Vena and D. R. Giacobbe, The novel Chinese coronavirus (2019-nCoV) infections: challenges for fighting the storm, Eur. J. Clin. Invest. 50(3) (2020), e13209.
[10] D. Wrapp, N. Wang, K. S. Corbett, J. A. Goldsmith, C. L. Hsieh, O. Abiona, B. S. Graham and J. S. McLellan, Cryo-EM structure of the 2019-nCoV spike in the prefusion conformation, Science 367(6483) (2020), 1260-1263.
[11] J. Cao, W. J. Tu, W. Chang, L. Yu, Y. K. Liu, X. Hu and Q. Liu, Clinical features and short-term outcomes of 102 patients with coronavirus disease 2019 in Wuhan, China, Clinical Infectious Diseases 71(15) (2020), 748-755.
[12] N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, X. Zhao, B. Huang, W. Shi, R. Lu and P. Niu, A novel coronavirus from patients with pneumonia in China, 2019, New England J. Med. 382 (2020), 727-733.
[13] L. Li, C. Sun and J. Jia, Optimal control of a delayed SIRC epidemic model with saturated incidence rate, Optim. Control Appl. Met. 40(2) (2018), 367-374.
[14] B. Acay, M. Inc, A. Khan and A. Yusuf, Fractional methicillin-resistant Staphylococcus aureus infection model under Caputo operator, J. Appl. Math. Comput. 67(1-2) (2021), 755-783. https://doi.org/10.1007/s12190-021-01502-3.
[15] J. Lin, R. Xu and X. Tian, Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence, Appl. Math. Model. 63 (2018), 688-708.
[16] A. Yusuf, B. Acay, U. T. Mustapha, M. Inc and D. Baleanu, Mathematical modeling of pine wilt disease with Caputo fractional operator, Chaos Solitons Fractals 143 (2021), 110569.
[17] I. Ahmed, E. F. D. Goufo, A. Yusuf, P. Kumam, P. Chaipanya and K. Nonlaopon, An epidemic prediction from analysis of a combined HIV-COVID-19 co-infection model via ABC fractional operator, Alexand Eng. J. 60 (2021), 2979-2995.
[18] S. Kirtphaiboon, U. Humphries, A. Khan and A. Yusuf, Model of rice blast disease under tropical climate conditions, Chaos Solitons Fractals 143 (2021), 110530.
[19] S. S. Musa, S. Qureshi, S. Zhao, A. Yusuf, U. T. Mustapha and D. He, Mathematical modeling of COVID-19 epidemic with effect of awareness programs, Infect Disease Model 6 (2021), 448-460.
[20] B. F. Maier and D. Brockmann, Effective containment explains subexponential growth in recent confirmed COVID-19 cases in China, Science 8 (2020). https://doi.org/10.1126/science.abb4557
[21] F. Ndaïrou, I. Area, J. J. Nieto and D. F. M. Torres, Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan, Chaos Solitons Fractals 135 (2020), 109846. https://doi.org/10.1016/j.chaos.2020.109846.
[22] C. Yang and J. Wang, A mathematical model for the novel coronavirus epidemic in Wuhan, China, Math. Biosci. Eng. 17(3) (2020), 2708-2724.
[23] Y. Fang, Y. Nie and M. Penny, Transmission dynamics of the covid-19 outbreak and effectiveness of government interventions: a data-driven analysis, J. Med. Virol. 2 (2020), 6-21.
[24] X. Rong, L. Yang, H. Chu and M. Fan, Effect of delay in diagnosis on transmission of COVID-19, Math. Biosci. Eng. 17(3) (2020), 2725-2740.
[25] K. Mizumoto and G. Chowell, Estimating risk for death from coronavirus disease, China, January-February 2020, Emerg. Infect. Diseases 26(6) (2020), 1251-1256.
[26] J. Hellewell, S. Abbott, A. Gimma, N. I. Bosse, C. I. Jarvis, T. W. Russell, J. D. Munday, A. J. Kucharski, W. J. Edmunds, F. Sun and S. Flasche, Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The Lancet Global Health 8(4) (2020), 488-496.
[27] M. J. Attaullah, S. Alyobi, M. F. Yassen and W. Weera, A higher order Galerkin time discretization scheme for the novel mathematical model of COVID-19, AIMS Mathematics 8(2) (2023), 3763-3790.
[28] P. Pandey, J. F. Gómez-Aguilar, M. K. Kaabar, Z. Siri and A. M. Abd Allah, Mathematical modeling of COVID-19 pandemic in India using Caputo-Fabrizio fractional derivative, Computers in Biology and Medicine 145 (2022), 105518.
[29] I. Ahmed, G. U. Modu, A. Yusuf, P. Kumam and I. Yusuf, A mathematical model of Coronavirus disease (COVID-19) containing asymptomatic and symptomatic classes, Results in Physics 21 (2021), 103776.
[30] A. Zeb, E. Alzahrani, V. S. Erturk and G. Zaman, Mathematical model for coronavirus disease 2019 (COVID-19) containing isolation class, BioMed Research International (2020), 1-7.
[31] J. K. Ghosh, S. K. Biswas, S. Sarkar and U. Ghosh, Mathematical modelling of COVID-19: A case study of Italy, Math. Comput. Simulation 194 (2022), 1-18.
[32] I. H. K. Premarathna, H. M. Srivastava, Z. A. M. S. Juman, A. AlArjani, M. S. Uddin and S. S. Sana, Mathematical modeling approach to predict COVID-19 infected people in Sri Lanka, AIMS Mathematics (2022), 4672-4699.
[33] A. S. Shaikh, I. N. Shaikh and K. S. Nisar, A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control, Advances in Difference Equations 2020(1) (2020), 1-19.
[34] N. H. Tuan, V. V. Tri and D. Baleanu, Analysis of the fractional corona virus pandemic via deterministic modeling, Math. Methods Appl. Sci. 44(1) (2021), 1086-1102.
[35] X. Rong, H. Chu, L. Yang, S. Tan, C. Yang, P. Yuan, Y. Tan, L. Zhou, Y. Liu, Q. Zhen and S. Wang, Recursive Zero-COVID model and quantitation of control efforts of the Omicron epidemic in Jilin province, Infectious Disease Modelling 8(1) (2023), 11-26.
[36] L. Kalachev, E. L. Landguth and J. Graham, Revisiting classical SIR modelling in light of the COVID-19 pandemic, Infectious Disease Modelling 8(1) (2023), 72 83.
[37] B. Yu, Q. Li, J. Chen and D. He, The impact of COVID-19 vaccination campaign in Hong Kong SAR China and Singapore, Infectious Disease Modelling (2022), 101-106.
[38] V. P. Bajiya, S. Bugalia and J. P. Tripathi, Mathematical modeling of COVID-19: Impact of non-pharmaceutical interventions in India, Chaos: J. Nonlinear Sci. 30(11) (2020), 113143. http://dx.doi.org/10.1063/5.0021353.
[39] S. Ullah and M. A. Khan, Modeling the impact of non-pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study, Chaos Solitons Fractals 139 (2020), 110075.
http://dx.doi.org/10.1016/j.chaos.2020.110075.
[40] S. Olaniyi, O. S. Obabiyi, K. O. Okosun, A. T. Oladipo and S. O. Adewale, Mathematical modelling and optimal cost-effective control of COVID-19 transmission dynamics, Eur. Phys. J. Plus. 135(11) (2020), 1-20.
http://dx.doi.org/10.1140/epjp/s13360-20-00954-z.
[41] W. Ming, J. V. Huang and C. J. P. Zhang, Breaking down of the healthcare system: mathematical modelling for controlling the novel coronavirus (2019-nCoV) outbreak in Wuhan, China, medRxiv and bioRxiv, 2020.
[42] I. Nesteruk, Statistics-based predictions of coronavirus epidemic spreading in Mainland China, Innovative Biosystems and Bioengineering 4(1) (2020), 13-18.
[43] M. Batista, Estimation of the final size of the coronavirus epidemic by SIR model, Research Gate, 2020.
[44] V. A. Okhuese, Mathematical predictions for coronavirus as a global pandemic, Research Gate, 2020.
[45] S. Hussain, F. Schieweck and S. Turek, Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation, J. Numer. Math. 19(1) (2011), 41 61.
[46] S. Hussain, F. Schieweck and S. Turek, A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary stokes equations, Open Numer. Methods J. 4(1) (2012), 35-45.
[47] O. Diekmann, J. A. Heesterbeek and M. G. Roberts, The construction of next generation matrices for compartmental epidemic models, J. R. Soc. Interface 7(47) (2010), 873-885.
[48] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29-48. http://dx.doi.org/10.1016/S0025-5564(02)00108-6.
[49] L. J. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, 2007.
[50] C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng. 1(2) (2004), 361.