Construction and Analysis of a Nonstandard Computational Method for the Solution of SEIR Epidemic Model
Abstract
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This paper is concerned with the numerical methods of susceptible exposed infectious recovered (SEIR) epidemic model of coronavirus disease 2019 (COVID-19). The model is explicated numerically with three numerical schemes, forward Euler, Runge-Kutta of order 4 (RK-4), and the proposed non-standard finite difference (NSFD) technique, respectively. In the epidemic model of infectious diseases, positivity is the main property of a consistent framework, since the negative value of a subpopulation is useless. The NSFD technique ends up being a more important and trustable numerical system than forward Euler and RK-4 techniques since it preserves positivity, stability, and convergence. On the contrary, forward Euler and RK-4 schemes do not hold these characteristics for some choices of step sizes. Numerical simulations confirmed the findings.
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References
Tyrrell DAJ, Bynoe ML. Cultivation of a novel type of common cold virus in organ culture. BMJ. 1965;1(5448):1467–1470. http://dx.doi.org/10.1136/bmj.1.5448.1467
World Health Organization. Coronavirus disease (COVID-19). https://www.who.int/health topics/coronavirus#tab=tab_1
Stadler K, Masignani V, Eickmann M, et al. SARS—beginning to understand a new virus. Nat Rev Microbiol. 2003;1(3):209–218. https://doi.org/10.1038/nrmicro775
Peiris JS, Lai ST, Poon LL, et al. Coronavirus as a possible cause of severe acute respiratory syndrome. Lancet. 2003;361(9366):1319–1325. https://doi.org/10.1016/S0140-6736(03)13077-2
World Health Organization. Novel coronavirus (2019-nCoV). https://www.who.int/docs/default-source/coronaviruse/situation-reports/20200121-sitrep-1-2019-ncov.pdf . Updated January 21, 2021. Accessed September 18,2021.
Abelson R. Brazil reported one of the highest Covid death tolls in the world. The New York Times. https://www.nytimes.com/2021/06/20/health/brazil-deaths-covid.html . Updated June 24, 2021. Accessed September 15, 2021.
Dhandapani PB, Baleanu D, Thippan J, Sivakumar V. On stiff, fuzzy IRD-14 day average transmission model of COVID-19 pandemic disease. AIMS Bioeng. 2020;7(4):208–223.
Ullah S, Khan MA. Modeling the impact of non- pharmaceutical interventions on the dynamics of novel coronavirus with optimal control analysis with a case study. Chaos Solitons Fractal. 2020;139:e110075. https://doi.org/10.1016/j.chaos.2020.110075
Fanelli D, Piazza F. Analysis and forecast of COVID-19 spreading in China, Italy and France. Chaos Solitons Fractal. 2020;134:e109761. https://doi.org/10.1016/j.chaos.2020.109761
Xiao D, Ruan S. Global analysis of an epidemic model with nonmonotone incidence rate. Math Biosci. 2007;208(2):419–429. https://doi.org/10.1016/j.mbs.2006.09.025
Lin Q, Zhao S, Gao D, et al. A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action. Int J Infect Dis. 2020:93;211–216. https://doi.org/10.1016/j.ijid.2020.02.058
Naveed M, Rafiq M, Raza A, et al. Mathematical analysis of novel coronavirus (2019-nCov) delay pandemic model. Comput Mater Continua. 2020;64(3):1401–1414. https://doi.org/10.32604/cmc.2020.011314
Shatanawi W, Raza A, Muhammad Shoaib A, Abodayeh K, Rafiq M, Bibi M. An effective numerical method for the solution of a stochastic coronavirus (2019-nCovid) pandemic model. Comput Mater Continua. 2021;66(2):1121–1137.
Shahid N, Baleanu D, Ahmed N, et al. Optimality of solution with numerical investigation for coronavirus epidemic model. Comput Mater Continua. 2021;67(2):1713–1728. https://doi.org/10.32604/cmc.2021.014191
Ahmed N, Baleanu D, Rafiq M, et al., Dynamical behavior and sensitivity analysis of a delayed coronavirus epidemic model. Comput Mater Continua. 2020;65(1):225–241. https://doi.org/10.32604/cmc.2020.011534
Aba-Oud MA, Ali A, Alrabaiah H, et al. A fractional order mathematical model for COVID-19 dynamics with quarantine, isolation, and environmental viral load. Adv Differ Equ. 2021;106:1–19. https://doi.org/10.1186/s13662-021-03265-4
Gao W, Veeresha P, Prakasha DG, et al. Novel dynamic structures of 2019-nCOV with nonlocal operator via powerful computational technique. Biology. 2020;9(5):e107. https://doi.org/10.3390/biology9050107
Khan MA, Atangana A. Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alexandria Eng J. 2020;59(4):2379–2389. https://doi.org/10.1016/j.aej.2020.02.033
Atangana A, Retaraz SIG. A novel Covid-19 model with fractional differential operators with singular and non-singular kernels: Analysis and numerical scheme based on Newton polynomial. Alexandria Eng J. 2021;60(4):3781–3806. https://doi.org/10.1016/j.aej.2021.02.016
Khan MA, Atangana A, Fatmawati EA. The dynamics of COVID-19 with quarantined and isolation. Adv Differ Equ. 2020;2020(1):1–22. https://doi.org/10.1186/s13662-020-02882-9
Khan MA, Ullah S, Kumar S. A robust study on 2019-nCOV outbreaks through non-singular derivative. Eur Phys J Plus. 2021;136:e168. https://doi.org/10.1140/epjp/s13360-021-01159-8
Rafiq M, Ali J, Riaz MB, et al. Numerical analysis of a bi-modal covid-19 SITR model. Alexandria Eng J. 2022;61(1):227–235. https://doi.org/10.1016/j.aej.2021.04.102
Mickens RE. Advances in applications of non-standard finite difference schemes. Singapore, World Scientific Publishing Company; 2019.
Cresson J, Pierret F. Nonstandard finite difference schemes preserving dynamical properties. J Comput Appl Math. 2016;303:15–30. https://doi.org/10.1016/j.cam.2016.02.007
Anguelov R, Dumont Y, Lubuma J, Mureithi E. Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model. Math Popul Stud. 2013;(20):101–122. https://doi.org/10.1080/08898480.2013.777240
Riaz S, Rafiq M, Ahmad O. Non standard finite difference method for quadratic riccati differential equation. Punjab Unive J Math. 2015;47(2):49–55.
Rafiq M, Raza A, Anayat A. Numerical modeling of virus transmission in a computer network. Paper presented at: 14th International Bhurban Conference on Applied Sciences & Technology (IBCAST) 219; January 10–14, 2017; Islamabad, Pakistan. https://doi.org/10.1109/IBCAST.2017.7868087
Rafiq M, Ahmad W, Abbas M. et al. A reliable and competitive mathematical analysis of Ebola epidemic model. Adv Differ Equ. 2020;2020:e540. https://doi.org/10.1186/s13662-020-02994-2
Rafiq M, Ali J, Riaz MB, Jan Awrejcewicz, Numerical analysis of a bi-modal covid-19 SITR model. Alexandria Eng J. 2022;61(1):227–235, http://doi/doi.org/10.1016/j.aej.2021.04.102
Ahmed N, Tahira S, Rafiq M, et al. Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model. Open Math. 2019;17(1):313–330. https://doi.org/10.1515/math-2019-0027
Dayan F, Ahmed N, Rafiq M, et al. Construction and numerical analysis of a fuzzy non-standard computational method for the solution of an SEIQR model of COVID-19 dynamics. AIMS Math. 2022;7(5):8449–8470. https://doi.org/10.3934/math.2022471
Dayan F, Rafiq M, Ahmed N, et al. Design and numerical analysis of fuzzy nonstandard computational methods for the solution of rumor based fuzzy epidemic model. Physica A. 2022;600:e127542. https://doi.org/10.1016/j.physa.2022.127542
Dayan F, Ahmed N, Rafiq M, et al. A dynamically consistent approximation for an epidemic model with fuzzy parameters. Expe Syst Appl. 2022;210:e118066. https://doi.org/10.1016/j.eswa.2022.118066
Dayan F, Rafiq M, Ahmed N, Raza A, Ahmad MO. A dynamical study of a fuzzy epidemic model of Mosquito-Borne Disease. Comput Biol Med. 20221;148:e105673. https://doi.org/10.1016/j.compbiomed.2022.105673
Dayan F, Baleanu D, Ahmed N, et al., Numerical investigation of malaria disease dynamics in fuzzy environment. Comput Mater Continua. 2023;74(2):2345–2361.
Raza A, Awrejcewicz J, Rafiq M, Ahmed N, Mohsin M. Stochastic analysis of nonlinear cancer disease model through virotherapy and computational methods. Mathematics. 2022;10(3):e368. https://doi.org/10.3390/math10030368
Alsallami SA, Raza A, Elmahi M, et al. Computational approximations for Real-World application of epidemic model. Intell Autom Soft Comput. 2022;33(3):1923–1939.
Chitnis N, Hyman JM, Cushing JM. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin Math Biol. 2008;70:1272-1296. https://doi.org/10.1007/s11538-008-9299-0
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