A Pell–Lucas Collocation Approach for an SIR Model on the Spread of the Novel Coronavirus (SARS CoV-2) Pandemic: The Case of Turkey

Abstract

In this article, we present a study about the evolution of the COVID-19 pandemic in Turkey. The modelling of a new virus named SARS-CoV-2 is considered by an SIR model consisting of a nonlinear system of differential equations. A collocation approach based on the Pell–Lucas polynomials is studied to get the approximate solutions of this model. First, the approximate solution in forms of the truncated Pell–Lucas polynomials are written in matrix forms. By utilizing the collocation points and the matrix relations, the considered model is converted to a system of the nonlinear algebraic equations. By solving this system, the unknown coefficients of the assumed Pell–Lucas polynomial solutions are determined, and so the approximate solutions are obtained. Secondly, two theorems about the error analysis are given and proved. The applications of the methods are made by using a code written in MATLAB. The parameters and the initial conditions of the model are determined according to the reported data from the Turkey Ministry of Health. Finally, the approximate solutions and the absolute error functions are visualized. To demonstrate the effectiveness of the method, our approximate solutions are compared with the approximate solutions obtained by the Runge–Kutta method. The reliable results are obtained from numerical results and comparisons. Thanks to this study, the tendencies of the pandemic can be estimated. In addition, the method can be applied to other countries after some necessary arrangements.

Keywords

• collocation method
• COVID-19 modeling
• error analysis
• mathematical modeling
• nonlinear differential equations
• Pell–Lucas polynomials