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

Supporting Institution

Izmir University of Economics

Ethical Statement

There is no conflict of interest

Covid 19 için SIR modelinin Chebyshev Polinom Çözümleri

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

Supporting Institution

İzmir Ekonomi Üniversitesi

Ethical Statement

Makalede etik açıdan herhangi bir sorun bulunmamaktadır.

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