THE DYNAMICS OF HIV/AIDS MODEL WITH FRACTAL-FRACTIONAL CAPUTO DERIVATIVE

Saif Ullah (), Mohamed Altanji (), Muhammad Altaf Khan, Ahmed Alshaheri () and Wojciech Sumelka
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Saif Ullah: Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa 25120, Pakistan
Mohamed Altanji: Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
Muhammad Altaf Khan: Faculty of Natural and Agricultural Sciences, University of the Free State, South Africa4Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya 60115, Indonesia
Ahmed Alshaheri: Department of Mathematics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Wojciech Sumelka: Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5 Street, 60-965 Poznan, Poland

FRACTALS (fractals), 2023, vol. 31, issue 02, 1-20

Abstract: 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 ℛ0 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 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, q1 and q2, 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.

Keywords: HIV/AIDS Epidemic Model; Fractal-Fractional Operator; Power-Law; Stability; Numerical Results (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X23400157

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