Dynamics of a Fractional-Order Hepatitis B Epidemic Model and Its Solutions by Nonstandard Numerical Schemes

Abstract

In this chapter, we propose and analyze a fractional-order hepatitis B epidemic model, which is an extension of a recognized ordinary differential model. Our main objective is to study dynamical properties of the proposed fractional-order model and its numerical solutions. Firstly, we establish positivity and boundedness of the proposed model. Secondly, the asymptotic stability of the model is investigated rigorously by the Lyapunov stability theorem for fractional dynamical systems and numerical simulations. Finally, we construct positivity-preserving nonstandard finite difference (NSFD) schemes for the fractional-order model. It should be emphasized that the constructed NSFD schemes preserve the positivity of solutions and the stability of the continuous model for all finite step sizes and therefore, they reflect exactly dynamics of the continuous model. Meanwhile, the standard Grunwald-Letnikov scheme fails to preserve the correct behavior of the continuous model for a given step size. Actually, the standard scheme provides numerical approximations that are completely different from the exact solution of the fractional-order model. Some numerical simulations are performed to confirm the validity of the theoretical results and to show advantages and superiority of NSFD schemes over the standard ones. The numerical experiments indicate that there is a good agreement between numerical simulations and the theoretical results.