A mathematical analysis of a system of Caputo–Fabrizio fractional differential equations for the anthrax disease model in animals
Shahram Rezapour, Sina Etemad, Hakimeh Mohammadi
Abstract
Abstract We study a fractional-order model for the anthrax disease between animals based on the Caputo–Fabrizio derivative. First, we derive an existence criterion of solutions for the proposed fractional $\mathcal {CF}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>CF</mml:mi></mml:math> -system of the anthrax disease model by utilizing the Picard–Lindelof technique. By obtaining the basic reproduction number $\mathcal{R}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> of the fractional $\mathcal{CF}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>CF</mml:mi></mml:math> -system we compute two disease-free and endemic equilibrium points and check the asymptotic stability property. Moreover, by applying an iterative approach based on the Sumudu transform we investigate the stability of the fractional $\mathcal{CF}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>CF</mml:mi></mml:math> -system. We obtain approximate series solutions of this system by means of the homotopy analysis transform method, in which we invoke the linear Laplace transform. Finally, after the convergence analysis of the numerical method HATM, we present a numerical simulation of the $\mathcal{CF}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>CF</mml:mi></mml:math> -fractional anthrax disease model and review the dynamical behavior of the solutions of this $\mathcal {CF}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>CF</mml:mi></mml:math> -system during a time interval.