The dynamics of HIV/AIDS model with fractal-fractional caputo derivative

The human immunodeficiency virus (HIV) is a major global public health issue and causes millions of deaths around the globe. The most severe phase of HIV infection is known as AIDS. In recent years, a number of mathematical models based on classical integer-order derivative have been developed to analyze the insight dynamics of HIV/AIDS. This paper presents the transmission dynamics of HIV/AIDS using fractional order (FO) and a fractal-fractional order compartmental model with the power-law kernel. In the first phase, the proposed model is formulated using the Caputo-type fractional derivative. The basic properties such as the solution positivity and existence as well as uniqueness of the fractional model are presented. The equilibria and the basic reproductive number Script capital R-0 are evaluated. Further, using fractional stability concepts the stability of the model (both local and global) around the equilibrium is presented in the disease-free case. In addition, the fractional model is solved numerically, and the graphical results with many values of q1 are shown. In the second phase, the concept of a fractal-fractional (FF) operator is applied to obtain a more generalized model that addresses the dynamics of HIV/AIDS. The uniqueness and existence of the solutions of the FF-based model are shown via a the Picard-Lindelof approach while the modified Adams-Bashforth method is utilized to present the numerical solution. Detailed numerical simulations are presented for various values fractional as well as the fractal orders, q(1) and q(2), respectively. The graphical results reveal that the FF-based model provides biologically more feasible results than the models in fractional and classical integer-order cases.