Fractional investigations of zoonotic visceral leishmaniasis disease with singular and non-singular kernel

Muhammad Altaf Khan; Olusola Kolebaje; Ahmet Yildirim; Saif Ullah; P. Kumam; P. Thounthong

doi:10.1140/epjp/i2019-12861-1

2023 Impact factor 2.8

EPJ PLUS - Condensed Matter and Complex Systems

Eur. Phys. J. Plus (2019) 134: 481
https://doi.org/10.1140/epjp/i2019-12861-1

Regular Article

Fractional investigations of zoonotic visceral leishmaniasis disease with singular and non-singular kernel

Muhammad Altaf Khan¹^,2^*, Olusola Kolebaje³, Ahmet Yildirim⁴, Saif Ullah⁵, P. Kumam⁶^,7 and P. Thounthong⁸

¹ Department of Mathematics, City University of Science and Information Technology, 25000, Peshawar, KP, Pakistan
² Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, P.O. Box 9301, Bloemfontein, South Africa
³ Department of Physics, Adeyemi College of Education, 350106, Ondo, Nigeria
⁴ Department of Mathematics, Science Faculty, Ege University, 35030, Izmir, Turkey
⁵ Department of Mathematics, University of Peshawar, 25000, Peshawar, KP, Pakistan
⁶ Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkuts University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, 10140, Bangkok, Thailand
⁷ Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung, Taiwan
⁸ Renewable Energy Research Centre, Department of Teacher Training in Electrical Engineering, Faculty of Technical Education, King Mongkuts University of Technology North Bangkok, 1518, Wongsawang, Bangsue, 10800, Bangkok, Thailand

^* e-mail: altafdir@gmail.com

Received: 1 June 2019
Accepted: 30 June 2019
Published online: 2 October 2019

Abstract

Memory has a great impact to study the dynamics of any real epidemic process in a better way. An epidemic model including memory effect is governed by fractional differential equations. In the present paper we explore the dynamics of the zoonotic visceral leishmaniasis (ZVL) disease using fractional derivative in both the Caputo and Atangana-Baleanu sense. The proposed model in the Caputo sense is solved by the well-known method known as modified differential transform method (MDTM), which is efficient and reliable. Further, the solution for the model with Atangana-Baleanu derivative is obtained by the modified Adams-Bashforh method. Numerical simulations are presented by using a different value of the fractional order parameter $\alpha$ . The numerical results obtained through the MDTM and the modified Adams-Bashforh method are reasonable and provide useful information in the non-integer case. Numerical results presented for the fractional order parameter $\alpha$ and in the integer case for the Caputo derivative model are compared with the Runge-Kutta method which gives good agreement. The application of Atangana-Baleanu derivative, the Caputo derivative and the use of the numerical approaches, MDTM, Adams Bashforth and the Runge-Kutta method (for the integer case) for an epidemic model is a novel practice and provides more flexible and deeper information about the complexity of the dynamics of ZVL.