Caputo Sense Fractional Order Derivative Model of Cholera

Sani Fakai Abubakar; Mohammed Olarenwaju Ibrahim

doi:doi:10.11648/j.ajmcm.20220702.12

| Peer-Reviewed

Caputo Sense Fractional Order Derivative Model of Cholera

Sani Fakai Abubakar, Mohammed Olarenwaju Ibrahim

Published in American Journal of Mathematical and Computer Modelling (Volume 7, Issue 2)

Received: 30 June 2022 Accepted: 26 July 2022 Published: 27 September 2022

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

A deterministic mathematical cholera model is formulated using ordinary differential equations. The formulated system of equations was then transformed into fractional derivative of Caputo sense, with order λ that ranges between 0 and 1. The transformed equations were displayed in Caputo sense fractional order derivative using the fractional derivative operator. These equations were then interpreted and the numerical Adams-Bashforth-Moulton kind of predictor-corrector method was used on maple 18 software to obtain the model’s outcome. Dynamics of cholera disease controls, comprising treatment, hygiene consciousness and vaccine were analyzed and the results were produced in graphs. The graphs show the dynamics of the susceptible, effects of vaccine on the susceptible and the rate of cholera infection. After studying and interpretation of the graphs, the result show that lower fractional order values in the range 0.25 to 0.5 gives lower values of susceptible and vaccinated individuals but gives higher number of infected individuals. To test efficiency of the obtained result, we compared it with the integer order derivative result, and found that the fractional order results gave a better and efficient, portray of the successful useable controls. Caputo sense fractional order derivative using Adams-Bashforth-Moulton kind of predictor-corrector numerical method, guaranteed getting result similar to Runge-Kutta fourth-order numerical method.

Published in	American Journal of Mathematical and Computer Modelling (Volume 7, Issue 2)
DOI	10.11648/j.ajmcm.20220702.12
Page(s)	31-36
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Hygiene, Fractional Order, Numerical, Infection

References

[1]	M. Awais, F. S. Alshammar, S. Ullah, M. A. Khan and S. Islam. Modelling and Simulation of the Novel Corona Virus in Caputo Derivative. Result Phys. 2020 Dec. 19: 103588, doi: 10.1016/j.rinp.2020. 103588. Epub 2020 Nov. 17, PMD: 33224721; PMCID: PMC7671651.
[2]	N. R. O. Bastos, Calculus of Variations Involving Caputo-Fabrizio Fractional Differentiation. Statistics, Optimization and Information Computing, 6 (2018) 12-21.
[3]	I. Bazhlekov and E. Bazhlekova, Fractional Derivative Modelling of Bioreaction Diffusion Processes. AIP Conference Proceedings, 2333, 060006(2021), https://doi.org/10.1063/5.0041611.
[4]	E. Bonyah, A. Atangana and M. A. Khan. Modelling the Spread of Computer Virus via Caputo Fractional Derivative and the Beta-Derivative. Asia Pacific Journal on Computational Engineering, 4:1 (2017). DOI 10.1186/s40540.016.0019-1.
[5]	S. Bushnaq, S. A. Khan, K. Shah, and G. Zaman, Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative. Cogent, Mathematics and Statistics, 5 no. 1, (2018) 1-12.
[6]	K. Diethelm, N. J. Ford, and A. D. Freed, A Predictor-Corrector Approach for the Numerical Solutions of Fractional Differential Equations. Nonlinear Dynamics, 29 no. 1, (2002) 3-22.
[7]	E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional Derivative without Singular Kernel to Kortewweg-de Vries-Burges Equation. Mathematical Modelling and Analysis, 21 no. 2, (2016) 188-198.
[8]	F. Evirgen and M. Yavuz, An Alternative Approach for Nonlinear Optimization Problem with Caputo-Fabrizio Derivative. ITM web of Conferences, 22 (01009) (2018).
[9]	F. S. Khan, M. Khalid, O. Bazighifan and A. El-Mesady. Euler’s Numerical Method on Fractional DSEK Model Under ABC Derivative. Hindawi, Complexity, vol. 2020, 447591, (2020). https://doi.org/10.1155/2020/4475491.
[10]	NCDC (Nigeria Centre for Disease Control), Cholera Situation Report No. 38. Nigeria: Cholera Outbreak Emergency Plan of Action (EPoA) DREF Operation n MDRNG033, IFRC, (2021).
[11]	E. Okyere, F. T. Oduro, S. K. Amponash, I. K. Dontwi, and N. K. Frempong, Fractional Order SIR Model with Constant Population. British Journal of Mathematics and Computer Science, 14 no. 2, (2016) 1-12.
[12]	K. M. Owolabi, J. F. Gomez-Aguilar, G. Fernandez-Anaya, J. E. Lavin-Delgado and E. Hernandez-Castillo. Modelling of Chaotic Processes with Caputo Fractional Order Derivative. MDPI Entropy 22, 1027, (2020).
[13]	M. Ozair, U. Saeed, and T. Hussain, Fractional Order SEIRS Model. Advanced Studies in Biology, 6 no. 2, (2014) 47-56.
[14]	E. C. Oliveira and J. A. T. Machado, A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering, Hindawi Publishing Corporation, 2014 (238459), (2014).
[15]	I. Podlubny, Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Mathematics in Science and engineering. Academic Press, San Diago, (1999) 198.
[16]	R. K. Shukla and P. Sapra, Fractional calculus and its applications for scientific professionals: A literature review. International Journal of Modern Sciences, 17 no. 2, (2019) 111-137.

Cite This Article

Plain Text BibTeX RIS

APA Style

Sani Fakai Abubakar, Mohammed Olarenwaju Ibrahim. (2022). Caputo Sense Fractional Order Derivative Model of Cholera. American Journal of Mathematical and Computer Modelling, 7(2), 31-36. https://doi.org/10.11648/j.ajmcm.20220702.12

Copy | Download

ACS Style

Sani Fakai Abubakar; Mohammed Olarenwaju Ibrahim. Caputo Sense Fractional Order Derivative Model of Cholera. Am. J. Math. Comput. Model. 2022, 7(2), 31-36. doi: 10.11648/j.ajmcm.20220702.12

Copy | Download

AMA Style

Sani Fakai Abubakar, Mohammed Olarenwaju Ibrahim. Caputo Sense Fractional Order Derivative Model of Cholera. Am J Math Comput Model. 2022;7(2):31-36. doi: 10.11648/j.ajmcm.20220702.12

Copy | Download

@article{10.11648/j.ajmcm.20220702.12,
  author = {Sani Fakai Abubakar and Mohammed Olarenwaju Ibrahim},
  title = {Caputo Sense Fractional Order Derivative Model of Cholera},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {7},
  number = {2},
  pages = {31-36},
  doi = {10.11648/j.ajmcm.20220702.12},
  url = {https://doi.org/10.11648/j.ajmcm.20220702.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20220702.12},
  abstract = {A deterministic mathematical cholera model is formulated using ordinary differential equations. The formulated system of equations was then transformed into fractional derivative of Caputo sense, with order λ that ranges between 0 and 1. The transformed equations were displayed in Caputo sense fractional order derivative using the fractional derivative operator. These equations were then interpreted and the numerical Adams-Bashforth-Moulton kind of predictor-corrector method was used on maple 18 software to obtain the model’s outcome. Dynamics of cholera disease controls, comprising treatment, hygiene consciousness and vaccine were analyzed and the results were produced in graphs. The graphs show the dynamics of the susceptible, effects of vaccine on the susceptible and the rate of cholera infection. After studying and interpretation of the graphs, the result show that lower fractional order values in the range 0.25 to 0.5 gives lower values of susceptible and vaccinated individuals but gives higher number of infected individuals. To test efficiency of the obtained result, we compared it with the integer order derivative result, and found that the fractional order results gave a better and efficient, portray of the successful useable controls. Caputo sense fractional order derivative using Adams-Bashforth-Moulton kind of predictor-corrector numerical method, guaranteed getting result similar to Runge-Kutta fourth-order numerical method.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Caputo Sense Fractional Order Derivative Model of Cholera
AU  - Sani Fakai Abubakar
AU  - Mohammed Olarenwaju Ibrahim
Y1  - 2022/09/27
PY  - 2022
N1  - https://doi.org/10.11648/j.ajmcm.20220702.12
DO  - 10.11648/j.ajmcm.20220702.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 31
EP  - 36
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20220702.12
AB  - A deterministic mathematical cholera model is formulated using ordinary differential equations. The formulated system of equations was then transformed into fractional derivative of Caputo sense, with order λ that ranges between 0 and 1. The transformed equations were displayed in Caputo sense fractional order derivative using the fractional derivative operator. These equations were then interpreted and the numerical Adams-Bashforth-Moulton kind of predictor-corrector method was used on maple 18 software to obtain the model’s outcome. Dynamics of cholera disease controls, comprising treatment, hygiene consciousness and vaccine were analyzed and the results were produced in graphs. The graphs show the dynamics of the susceptible, effects of vaccine on the susceptible and the rate of cholera infection. After studying and interpretation of the graphs, the result show that lower fractional order values in the range 0.25 to 0.5 gives lower values of susceptible and vaccinated individuals but gives higher number of infected individuals. To test efficiency of the obtained result, we compared it with the integer order derivative result, and found that the fractional order results gave a better and efficient, portray of the successful useable controls. Caputo sense fractional order derivative using Adams-Bashforth-Moulton kind of predictor-corrector numerical method, guaranteed getting result similar to Runge-Kutta fourth-order numerical method.
VL  - 7
IS  - 2
ER  -

Copy | Download

Author Information

Sani Fakai Abubakar

Department of Mathematics, Faculty of Physical Science, Kebbi State University of Science and Technology, Aliero, Nigeria
Mohammed Olarenwaju Ibrahim

Department of Mathematics, Faculty of Physical Science, University of Ilorin, Ilorin, Nigeria

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Sani Fakai Abubakar, Mohammed Olarenwaju Ibrahim. (2022). Caputo Sense Fractional Order Derivative Model of Cholera. American Journal of Mathematical and Computer Modelling, 7(2), 31-36. https://doi.org/10.11648/j.ajmcm.20220702.12

Copy | Download

ACS Style

Sani Fakai Abubakar; Mohammed Olarenwaju Ibrahim. Caputo Sense Fractional Order Derivative Model of Cholera. Am. J. Math. Comput. Model. 2022, 7(2), 31-36. doi: 10.11648/j.ajmcm.20220702.12

Copy | Download

AMA Style

Sani Fakai Abubakar, Mohammed Olarenwaju Ibrahim. Caputo Sense Fractional Order Derivative Model of Cholera. Am J Math Comput Model. 2022;7(2):31-36. doi: 10.11648/j.ajmcm.20220702.12

Copy | Download

@article{10.11648/j.ajmcm.20220702.12,
  author = {Sani Fakai Abubakar and Mohammed Olarenwaju Ibrahim},
  title = {Caputo Sense Fractional Order Derivative Model of Cholera},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {7},
  number = {2},
  pages = {31-36},
  doi = {10.11648/j.ajmcm.20220702.12},
  url = {https://doi.org/10.11648/j.ajmcm.20220702.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20220702.12},
  abstract = {A deterministic mathematical cholera model is formulated using ordinary differential equations. The formulated system of equations was then transformed into fractional derivative of Caputo sense, with order λ that ranges between 0 and 1. The transformed equations were displayed in Caputo sense fractional order derivative using the fractional derivative operator. These equations were then interpreted and the numerical Adams-Bashforth-Moulton kind of predictor-corrector method was used on maple 18 software to obtain the model’s outcome. Dynamics of cholera disease controls, comprising treatment, hygiene consciousness and vaccine were analyzed and the results were produced in graphs. The graphs show the dynamics of the susceptible, effects of vaccine on the susceptible and the rate of cholera infection. After studying and interpretation of the graphs, the result show that lower fractional order values in the range 0.25 to 0.5 gives lower values of susceptible and vaccinated individuals but gives higher number of infected individuals. To test efficiency of the obtained result, we compared it with the integer order derivative result, and found that the fractional order results gave a better and efficient, portray of the successful useable controls. Caputo sense fractional order derivative using Adams-Bashforth-Moulton kind of predictor-corrector numerical method, guaranteed getting result similar to Runge-Kutta fourth-order numerical method.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Caputo Sense Fractional Order Derivative Model of Cholera
AU  - Sani Fakai Abubakar
AU  - Mohammed Olarenwaju Ibrahim
Y1  - 2022/09/27
PY  - 2022
N1  - https://doi.org/10.11648/j.ajmcm.20220702.12
DO  - 10.11648/j.ajmcm.20220702.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 31
EP  - 36
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20220702.12
AB  - A deterministic mathematical cholera model is formulated using ordinary differential equations. The formulated system of equations was then transformed into fractional derivative of Caputo sense, with order λ that ranges between 0 and 1. The transformed equations were displayed in Caputo sense fractional order derivative using the fractional derivative operator. These equations were then interpreted and the numerical Adams-Bashforth-Moulton kind of predictor-corrector method was used on maple 18 software to obtain the model’s outcome. Dynamics of cholera disease controls, comprising treatment, hygiene consciousness and vaccine were analyzed and the results were produced in graphs. The graphs show the dynamics of the susceptible, effects of vaccine on the susceptible and the rate of cholera infection. After studying and interpretation of the graphs, the result show that lower fractional order values in the range 0.25 to 0.5 gives lower values of susceptible and vaccinated individuals but gives higher number of infected individuals. To test efficiency of the obtained result, we compared it with the integer order derivative result, and found that the fractional order results gave a better and efficient, portray of the successful useable controls. Caputo sense fractional order derivative using Adams-Bashforth-Moulton kind of predictor-corrector numerical method, guaranteed getting result similar to Runge-Kutta fourth-order numerical method.
VL  - 7
IS  - 2
ER  -

Copy | Download