The fractal-fractional Atangana-Baleanu operator for pneumonia disease: stability, statistical and numerical analyses

Abstract

The present paper studies pneumonia transmission dynamics by using fractal-fractional operators in the Atangana-Baleanu sense. Our model predicts pneumonia transmission dynamically. Our goal is to generalize five ODEs of the first order under the assumption of five unknowns (susceptible, vaccinated, carriers, infected, and recovered). The Atangana-Baleanu operator is used in addition to analysing existence, uniqueness, and non-negativity of solutions, local and global stability, Hyers-Ulam stability, and sensitivity analysis. As long as the basic reproduction number $ \mathscr{R}_{0} $ is less than one, the free equilibrium point is local, asymptotic, or otherwise global. Our sensitivity statistical analysis shows that $ \mathscr{R}_{0} $ is most sensitive to pneumonia disease density. Further, we compute a numerical solution for the model by using fractal-fractional. Graphs of the results are presented for demonstration of our proposed method. The results of the Atangana-Baleanu fractal-fractional scheme is in excellent agreement with the actual data.

Keywords

• fractional derivatives
• nonlinear equations
• simulation
• numerical results
• infectious disease
• time varying control system