Numerical solution of the spread of infectious diseases mathematical model based on shifted Bernstein polynomials

Subject Areas : Statistics

Farshid Mirzaee ¹ , Seyede Fatemeh Hoseini ² , Sahar Alipour ³

1 - Malayer University
2 - Malayer University
3 - Malayer University

Received: 2017-06-28 Accepted : 2020-02-27 Published : 2020-07-01

Keywords: مدل ریاضی انتشار بیماری, چندجملهایهای برنشتاین, روش تقریب کمترین مربعات, معادله انتگرال ولترا- همرشتاین تاخیری,

Abstract :

The Volterra delay integral equations have numerous applications in various branches of science, including biology, ecology, physics and modeling of engineering and natural sciences. In many cases, it is difficult to obtain analytical solutions of these equations. So, numerical methods as an efficient approximation method for solving Volterra delay integral equations are of interest to many researchers. In this paper, a numerical method is developed for solving the Hammerstein–Volterra delay integral equation by least squares (LS) approximation method, which is based on Shifted Bernstein polynomials (BPs). This equation is a mathematical model for the spread of certain infectious diseases with a constant rate that varies seasonally. Least squares method is a mathematical model for data fiting which minimizes the sum of squared the difference between an observed value and the value provided by a model. In this paper, the shifted Bernstein polynomials are introduced and then approximation of an arbitrary function by using these polynomials is presented . Also, the Hammerstein–Volterra delay integral equation is introduced and the details of least squares method for solving a mathematical model is presented. Finally, we show the efficiency of the proposed method by solving two numerical examples and comparing the results with other methods.

References:

 

[1] M. I. Berenguer, D. Gámez, A. l. Guillem, M. R. Galán, M. S. Pérez, Biorthogonal systems for solving Volterra integral equation systems of the second kind, Journal of Computational and Applied Mathematics 235:1875-1883 (2011).

 

[2] Y. Chen, T. Tang, Spectral methods for weakly singular Volterra integral equations with smooth solutions, Journal of Computational and Applied Mathematics, 233:938-950 (2009).

 

[3] H. H. Sorkun, S. Yalçinbaş, Approximate solutions of linear Volterra integral equation systems with variable coefficients, Applied Mathematical. Modelling, 34:3451-3464 (2010).

 

[4] J. P. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numerische Mathematik, 56:409-424 (1989).

 

[5] A. H. Borzabadi, O. S. Fard, A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind, Journal of Computational and Applied Mathematics, 232:449-454 (2009).

 

[6] K. M. Liew, Y. Cheng, S. Kitipornchai, Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems, International Journal for Numerical Methods in Engineering, 65:1310-1332 (2006).

 

[7] Q. Wang, K. Wang, S. Chen, Least squares approximation method for the solution of Volterra-Fredholm integral equations, Journal of Computational and Applied Mathematics, 272:141-147 (2014).

 

[8] Y. Ordokhani, S. Davaei Far, Application of the Bernstein polynomials for solving the nonlinear Fredholm integro-differential equations, Journal of Applied Mathematics and Bioinformatics, 1:13-31 (2011).

 

[9] K. L. Cooke, An epidemic equation with immigration, Mathematical Biosciences, 29:135-158 (1976).

 

[10] M. Mosleh, M. Otadi, Least squares approximation method for the solution of Hammerstein–Volterra delay integral equations, Applied Mathematics and Computation, 258:105-110 (2015).

 

[11] M. Avaji, J. S. Hafshejani, S. S. Dehcheshmeh, D. F. Ghahfarokhi, Solution of delay Volterra integral equations using the Variational iteration method. Journal of Applied Sciences, 12:196-200 (2012).

 

[12] A. Bica, C. IANCU, A numerical method in terms of the third derivative for a delay integral equation from biomathematics, Journal of Inequalities in Pure and Applied Mathematics, 6:1-18 (2005).

 

[13] S. Yalçinbaş, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Aplied Mathematics and Computation, 127:195-206 (2002).