Numerical solution of a system of Volterra integral equations in application to the avian human influenza epidemic model

Document Type : Research Article

Author

• R. Katani

Faculty of Sciences, Yasouj University, Yasouj-Iran.

10.22067/ijnao.2021.69416.1021

Abstract

We propose an effcient multistage method for solving a system of linear and nonlinear Volterra integral equations of the second kind. This numerical method is based on the Gauss–Legendre quadrature rule that obtains several values of unknown function at each step, and it will be shown that the order of the convergence is O(M-4), where M is the number of the nodes in the time discretization. The method has also the advantages of simplicity of application, less computational time, and useful performance for large intervals. In order to show the effciency of the method, numerical results for the avian human influenza epidemic model is obtained that is comparable with the fourth-order Runge–Kutta method.
 

Keywords

• The avian human influenza epidemic model
• system of Volterra integral equations
• Gauss–Legendre quadrature rule
• convergence order

20.1001.1.24236977.2022.12.1.2.6

Main Subjects

• Numerical methods for integral equations and its applications

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References

1. Agarwal, R.P., O’Regan, D., Tisdell, C. and Wong, P.J.Y. Constant-sign solutions of a system of Volterra integral equations, Comput. Math. Appl. 54 (1) (2007) 58–75.
2. Brunner, H. and van der Houwen, P.J. The numerical solution of Volterra equations, Amsterdam etc., North-Holland 1986.
3. Caliò, F., Garralda-Guillem, A.I., Marchetti, E. and Ruiz Galán, M. Numerical approaches for systems of Volterra-Fredholm integral equations, Appl. Math. Comput. 225 (2013) 811–821.
4. Gautschi, W. Orthogonal polynomial, computation and approximation, Oxford science publications 2004.
5. Iwami, S., Takeuchi, Y., Korobeinikov, A. and Liu, X. Prevention of the avian influenza epidemic: What policy should we choose?, J. Theor. Biol. 252 (4) (2008) 732–741.
6. Iwami, S., Takeuchi, Y. and Liu, X. The avian-human influenza epidemic model, Math. Biosci. 207 (1) (2007) 1–25.
7. Katani, R. and Shahmorad, S. A block by block method with Romberg quadrature for the system of Urysohn type Volterra integral equations, Comput. Appl. Math. 31 (1) (2012) 1–13.

8. Koekoek, R., Lesky, P. and Swarttouw, R. Hypergeometric orthogonal polynomials and their q-analogues, Springer 2010.
9. Linz, P. Analytical and numerical methods for Volterra equations, SIAM Philadelphia 1985.
10. Maleknejad, K. and Ostadi, A. Numerical solution of system of Volterra integral equations with weakly singular kernels and its convergence analysis, Appl. Numer. Math. 115 (2017) 82–98.
11. Poland, G.A., Jacobson, R.M. and Targonski, P.V. The avian and pandemic influenza: an overview, Vaccine 25 (2007) 3057–3061.
12. Shinya, K., Ebina, M., Yamada, S., Ono, M., Kasai, N. and Kawaoka, Y., The avian flu: Influenza virus receptors in the human airway, Nature 440 (2006) 435–436.
13. Trefethen, L.N. Approximation theory and approximation practice (Applied Mathematics), society for industrial and applied mathematics 2013.