Caputo Sense Fractional Order Derivative Model of Cholera

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Sani Fakai Abubakar, Mohammed Olarenwaju Ibrahim
1. Department of Mathematics, Faculty of Physical Science, Kebbi State University of Science and Technology, Aliero, Nigeria; 2. Department of Mathematics, Faculty of Physical Science, University of Ilorin, Ilorin, Nigeria

American Journal of Mathematical and Computer Modelling (Science Publishing Group) 2022
7 : 2
31-36
10.11648/j.ajmcm.20220702.12
English
Sani Fakai Abubakar, Mohammed Olarenwaju Ibrahim. Caputo Sense Fractional Order Derivative Model of Cholera, American Journal of Mathematical and Computer Modelling. Volume 7, Issue 2, June 2022 , pp. 31-36. doi: 10.11648/j.ajmcm.20220702.12. Share Research.
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.
Hygiene, Fractional Order, Numerical, Infection

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.

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