Covid 19 için SIR modelinin Chebyshev Polinom Çözümleri

Year 2023, Volume: 9 Issue: 4, 39 - 47, 31.12.2023

Demet Özdek

Abstract

Bu çalışma, Türkiye’deki COVID-19 salgınına yönelik matematiksel modelin başlangıç değer probleminin sayısal olarak çözümü ile ilgilidir. Bu model, nonlineer denklem sisteminden oluşan SIR modelidir. Bu denklemleri çözmek için Chebyshev polinomlarına dayanan bir kollokasyon yöntemi kullanılmıştır. Chebyshev polinomları ortonormal polinomlardır ve yöntemin bir avantajı olarak, ortonormallik metodun hesaplama maliyetini düşürmektedir. Bir diğer avantaj ise mevcut yöntem, alan ayrıştırması gerektirmemektedir. Dolayısıyla yöntemin uygulanması kolaydır. Metodun ana düşüncesi modeli nonlineer cebirsel denklemlere dönüştürmektir. Bunun için sistemin yaklaşık çözümü ve birinci türevi, matris formlarında katsayıları bilinmeyen Chebyshev polinomlarının kesik serisi olarak yazılmaktadır ve sonra kollokasyon noktalarından yararlanarak, SIR modeli nonlineer denklemler sistemine dönüştürülmektedir. Elde edilen sistem Chebshev polinomlarının bilinmeyen katsayıları için Matlab kullanılarak çözülür ve böylece yaklaşık çözüm elde edilir. Metodun doğruluğunu kontrol etmek için çözümün artık mutlak hatası incelenmiştir. Sonuçlar mevcut metodun etkili ve doğru olduğunu göstermektedir.

Keywords

SIR modeli, kollokasyon metodu, COVID-19 modellemesi, hata analizi, matematiksel modelleme

Ethical Statement

Makalede etik açıdan herhangi bir sorun bulunmamaktadır.

Supporting Institution

İzmir Ekonomi Üniversitesi

Chebyshev Polynomial Solution For The SIR Model Of COVID 19

Abstract

. In this study, we deal with solving numerically initial value problem of a mathematical model of COVID-19 pandemic in Turkey. This model is a SIR model consisting of a nonlinear system of differential equations. In order to solve these equations, a collocation approach based on the Chebyshev polynomials is used. Chebyshev polynomials are orthonormal polynomials and the orthonormality reduces the computation cost of the method as an advantage. Another advantage is that the present method does not require any discretization of the domain. So the method is easy to implement. The main idea of the method is to convert the model to a system of nonlinear algebraic equations. For this we write the approximate solution of the system and its first derivative as the truncated series of Chebyshev polynomials with unknown coefficients in matrix forms and then utilizing the collocation points, the SIR model is converted to a system of the nonlinear equations. The obtained system is solved for the unknown coefficients of the assumed Chebyshev polynomial solution by MATLAB, and so the approximate solution is obtained. In order to check the robustness of the method, residual error of the solution is reviewed. The results show that the method is efficient and accurate.

Keywords

SIR model, collocation method, COVID-19 modeling, error analysis, mathematical modeling

Ethical Statement

There is no conflict of interest

Supporting Institution

Izmir University of Economics

References

• [1] “The 1918 Flu Pandemic: Why It Matters 100 Years Later,” Website of Centers for Disease Control and Prevention, Public Health Matters Blog, 14-05-2018. [Online]. Available: https://blogs.cdc.gov/publichealthmatters/2018/05/1918-flu [Accessed: 14-08-2023]
• [2] C.A. Taylor, C. Boulos and M.J. Memoli. “The 1968 Influenza Pandemic and COVID-19 Outcomes,” MedRxiv. The Preprint Server for Health Sciences, October 25, 2021. [Online]. Available: https://www.medrxiv.org/content/10.1101/2021.10.23.21265403v1.full-text. [Accessed: 25-12-2023].
• [3] “Coronavirus (COVID-19) outbreak,” WHO Homepage, World Health Organization. [Online]. Available: https://www.who.int/westernpacific/emergencies/covid-19. [Accessed: 14-08-2023].
• [4] L. Akin and M.G. Gözel, “Understanding dynamics of pandemics,” Turkish Journal of Medical Sciences, vol. 50, pp. 515-519, April 2020. Doi: 10.3906/sag2004-133
• [5] W.O. Kermack and A.G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society of London A, vol. 115, 1927, pp. 700–721, August 1927. Doi:10.1098/rspa.1927.0118
• [6] F.S. Akinboro, S. Alao and F.O. Akinpelu, “Numerical solution of SIR model using differential transformation method and variational iteration method,” General Mathematics Notes, vol. 22, no. 2, pp. 82–92, June 2014.
• [7] T. Harko, F.S. Lobo and M. Mak, “Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates,” Applied Mathematics and Computation, vol. 236, pp. 184–194, June 2014. doi:10.1016/j.amc.2014.03.030
• [8] S. Hasan, A. Al-Zoubi, A. Freihet, M. Al-Smadi and S. Momani. “Solution of fractional SIR epidemic model using residual power series method,” Applied Mathematics and Information Sciences, vol. 13, pp. 153–161, March 2019. Doi: 10.18576/amis/130202
• [9] S.U. Khan and I. Ali. “Numerical analysis of stochastic SIR model by Legendre spectral collocation method,” Advances in Mechanical Engineering, vol. 11, pp. 1-10, July 2019. Doi: 10.1177/168781401986
• [10] A. Secer, N. Ozdemir and M.A. Bayram, “Hermite polynomial approach for solving the SIR model of epidemics,” Mathematics, vol. 6, no. 12, pp. 305, December 2018 . doi:10.3390/math6120305
• [11] S. Side, A.M. Utami and M.I. Pratama, “Numerical solution of SIR model for transmission of tuberculosis by Runge-Kutta method,” Journal of Physics: Conference Series, vol. 1040, pp. 012021, May 2018. doi:10.1088/1742-6596/1040/1/012021
• [12] D. Uçar and E. Çelik, “Analysis of Covid 19 disease with SIR model and Taylor matrix method,” AIMS Mathematics, vol. 7, pp. 11188-11200, April 2022. doi:10.3934/math.2022626
• [13] Ş. Yüzbaşı and G. Yıldırım, “A Pell-Lucas Collocation Approach for an SIR Model on the Spread of the Novel Coronavirus (SARS CoV-2) Pandemic: The Case of Turkey,” Mathematics, vol. 11, pp. 697, January 2023 . doi: 10.3390/math11030697
• [14] B.M. Ndiaye, L. Tendeng and D. Seck, “Comparative prediction of confirmed cases with COVID-19 pandemic by machine learning, deterministic and stochastic SIR models,” ArXiv:2004, Quantitative Biology: Populations ans Evolution, vol. 2004, pp.13489, April 24, 2020. [Online]. Available: https://arxiv.org/abs/2004.13489. [Accessed: 25-12-2023].
• [15] P.L. Chebyshev, Mémoires présentés à l'Académie impériale des Sciences de St. Petersbourg par divers Savans et dans ses assemblées (in French). 9th Edition. Saint- Pétersbourg: Imperatorskaja akademija nauk, 1854.
• [16] J.C. Mason and D.C. Handscomb, Chebyshev Polynomials. New York: Chapman & Hall/CRC, 2003.
• [17] T.J. Rivlin, Pure and Applied Mathematics: Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory, 2nd Edition. New York: Wiley, 1990, pp. 249.
• [18] M.R. Ahmadi and H. Adibi. “The Chebyshev Tau Technique for the Solution of Laplace’s Equation,” Applied Mathematics and Computation, vol. 184, pp. 895-900, January 2007. doi:10.1016/j.amc.2006.05.212
• [19] “Genel Koronavirus Tablosu”, The Turkey Ministry of Health COVID 19 Information Platform, [Online]. Available: https://covid19.saglik.gov.tr/TR-66935/genel-koronavirus-tablosu.html. [Accessed: 27-05-2023].
• [20] “Coronavirus disease 2019 (COVID-19): Situation report 73,” World Health Organization Homepage, WHO Coronavirus disease (COVID-19) Situation reports, 2 April 2020. [Online]. Available: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports. [Accessed: 27-05-2023].
• [21] C.N. Ngonghala, E. Iboi, S. Eikenberry, M. Scotch, C.R. MacIntyre, H.M. Bonds and A.B. Gumel. “Mathematical Assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus,” Mathematical Biosciences, vol. 325, pp. 108364, May 2020. Doi:10.1016/j.mbs.2020.108364.