Numerical Simulations and Analyzing the Dynamics of Hepatitis B Virus With Immune Delay Under Novel Operators

Abstract

Human beings are indeed subject to the hazard of infectious diseases. From the literature, one can see that mathematical modelling is a powerful method for explaining complex biological phenomena accurately. Different operators were used to study the dynamics of infectious diseases. In 2017, more generalized operators are introduced by Atangana through merging fractional and fractal calculus. These newly proposed operators, called fractal-fractional operators, have been commonly utilized to model several physical problems to examine a problem’s complex behaviour and complex structure. In this manuscript, the fractal-fractional mathematical model of HBV with an immune delay has investigated qualitatively and quantitatively. The basic reproduction number and equilibria of the proposed model are given. The Routh Hurtwez stability criterion is used to derive the stability of the equilibria points of the model. Fixed point theorems have been used to verify the model’s existence and uniqueness when using the Atangana-Baleanu fractal-fractional operator. Non-linear analysis was used to calculate the Ulam-Hyres stability of the model. For the different fractal-fractional operators, numerical results are developed using Lagrangian piecewise interpolation. We visualise the achieved numerical solution for several sets of fractional and fractal orders to examine the dynamic behaviour of the proposed model under novel operators in various ways. This research supports the United Nation SDG 3 (Good Health and Well-Being) by developing a fractal-fractional mathematical model of HBV.

Keywords Fractal-fractional operators, Lagrangian piecewise interpolation, Ulam-Hyres stability, UN SDG 3