Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter: Beta-Derivative

Monday, 25th of July 2016

Modeling the Spread of Rubella Disease

Using the Concept of with Local

Derivative with Fractional Parameter:

Beta-Derivative

ABDON ATANGANA

1

AND BADR SAAD T. ALKAHTANI

2

1

Faculty of Natural and Agricultural Sciences Institute for Groundwater Studies University of the Free

State 9300 Bloemfontein South Africa; and

2

Department of Mathematics College of Science King

Saud University P. O. Box 1142 Riyadh 11989 Saudi Arabia

Received 17 March 2015; accepted 26 May 2015

Our aim in this work was to examine the model underpinning the spread of the Rubella virus using the novel deriv-

ative called beta-derivative. The study of the equilibrium points together with the analysis of the disease free equi-

librium points was presented. Due to the complexity of the modified equation we introduced a new operator based

on the S umudu tr ansform. The prop erties of this oper ator wer e proposed and proved in detail. We made used of

this operator together with the idea of perturbation method to derive a special solution of the extended model. The

st ability of the method for solving this model was presented. The uniqueness of the special solution was pr esen ted

and numeri cal simu lations wer e done. Th e graphic al r epr esentation s show that the model depends on both param-

eters and the fr actional order.

V

C

2015 Wiley Periodicals Inc. Complexity 21: 442–451 2016

Key Words: model of Rubella disease; beta-derivative; novel operator; stability analysi

Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter: Beta-Derivative

Abstract

Our aim in this work was to examine the model underpinning the spread of the Rubella virus using the novel derivative called beta-derivative. The study of the equilibrium points together with the analysis of the disease free equilibrium points was presented. Due to the complexity of the modified equation we introduced a new operator based on the Sumudu transform. The properties of this operator were proposed and proved in detail. We made used of this operator together with the idea of perturbation method to derive a special solution of the extended model. The stability of the method for solving this model was presented. The uniqueness of the special solution was presented and numerical simulations were done. The graphical representations show that the model depends on both parameters and the fractional order

http://onlinelibrary.wiley.com/doi/10.1002/cplx.21704/abstract