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

Najat Almutairi; Sayed Saber; Hijaz Ahmad; Najat Almutairi; Sayed Saber; Hijaz Ahmad

doi:10.3934/math.20231504

AIMS Mathematics

2023, Volume 8, Issue 12: 29382-29410. doi: 10.3934/math.20231504

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Research article

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

1.
Department of Mathematics, College of Science, Qassim University, Buraidah, Saudi Arabia
2.
Department of Mathematics and Statistics, Faculty of Science, Beni-Suef University, Egypt
3.
Department of Mathematics, Faculty of Science and Arts in Baljurashi, Al-Baha University, Saudi Arabia
4.
Near East University, Operational Research Center in Healthcare, Nicosia, Turkey
5.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
6.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, 39, 00186, Rome, Italy

Received: 03 September 2023 Revised: 02 October 2023 Accepted: 11 October 2023 Published: 30 October 2023
MSC : 34C60, 92B05, 92C42, 92D25, 92D30

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.
- fractional derivatives,
- nonlinear equations,
- simulation,
- numerical results,
- infectious disease,
- time varying control system
Citation: Najat Almutairi, Sayed Saber, Hijaz Ahmad. The fractal-fractional Atangana-Baleanu operator for pneumonia disease: stability, statistical and numerical analyses[J]. AIMS Mathematics, 2023, 8(12): 29382-29410. doi: 10.3934/math.20231504

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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.

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