Existence and Uniqueness Solution for a Mathematical Model with Mittag-Leffler Kernel

Öz

In this work, we analyse the fractional order West Nile Virus model involving the Atangana-Baleanu derivatives. Existence and uniqueness solutions were obtained by the fixed-point theorem. Another impressive aspect of the work is illustrated by simulations of different fractional orders by calculating the numerical solutions of the mathematical model.

Anahtar Kelimeler

• Virus
• Mathematical Modeling
• Fractional derivatives and integrals
• Numerical solution

Kaynakça

1. Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science. 2016; , 20(2), 763-769.
2. Atangana A, Owolabi KM. New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena. 2018;13(1):3.
3. Bagley RL, Torvik PJ. A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology. 1983 Jun 1;27(3):201-10.
4. Bagley RL, Torvik PJ. Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA journal. 1985 Jun;23(6):918-25.
5. Bowman C, Gumel AB, Van den Driessche P, Wu J, Zhu H. A mathematical model for assessing control strategies against West Nile virus. Bulletin of mathematical biology. 2005 Sep 1;67(5):1107-33.
6. Campbell GL, Marfin AA, Lanciotti RS, Gubler DJ. West nile virus. The Lancet infectious diseases. 2002 Sep 1;2(9):519-29.
7. Caputo M. Linear models of dissipation whose Q is almost frequency independent—II. Geophysical Journal International. 1967 Nov 1;13(5):529-39.
8. Dokuyucu MA. Caputo and atangana-baleanu-caputo fractional derivative applied to garden equation. Turkish Journal of Science. 2020 Mar 3;5(1):1-7.

9. Dokuyucu M, Celik E. Analyzing a novel coronavirus model (COVID-19) in the sense of Caputo-Fabrizio fractional operator. Applied and Computational Mathematics. 2021;20(1).
10. Hayes EB, Komar N, Nasci RS, Montgomery SP, O'Leary DR, Campbell GL. Epidemiology and transmission dynamics of West Nile virus disease. Emerging infectious diseases. 2005 Aug;11(8):1167.
11. Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. elsevier; 2006 Feb 16.
12. Koca İ, Akçetin E, Yaprakdal P. Numerical approximation for the spread of SIQR model with Caputo fractional order derivative. Turkish Journal of Science. 2020;5(2):124-39.
13. Koeller R. Applications of fractional calculus to the theory of viscoelasticity. (1984): 299-307.
14. Koksal ME. Stability analysis of fractional differential equations with unknown parameters. Nonlinear Analysis: Modelling and Control. 2019 Feb 1;24(2):224-40.
15. Koksal ME. Time and frequency responses of non-integer order RLC circuits. AIMS Mathematics. 2019 Jan 1;4(1):64-78.
16. Lewis M, Rencławowicz J, den Driessche PV. Traveling waves and spread rates for a West Nile virus model. Bulletin of mathematical biology. 2006 Jan;68:3-23.
17. Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier; 1998 Oct 27.
18. Tarboush AK, Lin Z, Zhang M. Spreading and vanishing in a West Nile virus model with expanding fronts. Science China Mathematics. 2017 May;60:841-60.
19. Wonham MJ, de-Camino-Beck T, Lewis MA. An epidemiological model for West Nile virus: invasion analysis and control applications. Proceedings of the royal society of London. Series B: Biological Sciences. 2004 Mar 7;271(1538):501-7.