A finite difference scheme to solve a fractional order epidemic model of computer virus

Zafar Iqbal; Muhammad Aziz-ur Rehman; Muhammad Imran; Nauman Ahmed; Umbreen Fatima; Ali Akgül; Muhammad Rafiq; Ali Raza; Ali Asrorovich Djuraev; Fahd Jarad; Zafar Iqbal; Muhammad Aziz-ur Rehman; Muhammad Imran; Nauman Ahmed; Umbreen Fatima; Ali Akgül; Muhammad Rafiq; Ali Raza; Ali Asrorovich Djuraev; Fahd Jarad

doi:10.3934/math.2023121

AIMS Mathematics

2023, Volume 8, Issue 1: 2337-2359. doi: 10.3934/math.2023121

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A finite difference scheme to solve a fractional order epidemic model of computer virus

1.
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
2.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
3.
Department of Computer Science, The University of Lahore, Lahore, Pakistan
4.
Department of Mathematics, Art and Science Faculty, Siirt University, Siirt, Turkey
5.
Near East University, Mathematics Research Center, Department of Mathematics, Near East Boulevard, PC: 99138, Nicosia /Mersin 10 Turkey
6.
Department of Mathematics, Faculty of Science and Technology, University of Central Punjab, Lahore, Pakistan
7.
Department of Mathematics, Govt. Maulana Zafar Ali Khan Graduate College Wazirabad, Punjab Higher Education Department (PHED) Lahore, Govt. of Punjab, Lahore, 54000, Pakistan
8.
Department of Training of Pedagogues of Vocal and Instrumental Performance, Uzbek National Music Art Institute named after Yunus Rajabi, Tashkent, Uzbekistan
9.
Cankaya University, Ankara, Turkey
10.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
11.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 15 May 2022 Revised: 01 October 2022 Accepted: 16 October 2022 Published: 01 November 2022
MSC : 65P99, 65Z05, 37M15

In this article, an analytical and numerical analysis of a computer virus epidemic model is presented. To more thoroughly examine the dynamics of the virus, the classical model is transformed into a fractional order model. The Caputo differential operator is applied to achieve this. The Jacobian approach is employed to investigate the model's stability. To investigate the model's numerical solution, a hybridized numerical scheme called the Grunwald Letnikov nonstandard finite difference (GL-NSFD) scheme is created. Some essential characteristics of the population model are scrutinized, including positivity boundedness and scheme stability. The aforementioned features are validated using test cases and computer simulations. The mathematical graphs are all detailed. It is also investigated how the fundamental reproduction number $ \mathfrak{R}_0 $ functions in stability analysis and illness dynamics.
- computer virus,
- fractional order system,
- Grunwald Letinkov technique,
- nonstandard finite differences,
- simulations
Citation: Zafar Iqbal, Muhammad Aziz-ur Rehman, Muhammad Imran, Nauman Ahmed, Umbreen Fatima, Ali Akgül, Muhammad Rafiq, Ali Raza, Ali Asrorovich Djuraev, Fahd Jarad. A finite difference scheme to solve a fractional order epidemic model of computer virus[J]. AIMS Mathematics, 2023, 8(1): 2337-2359. doi: 10.3934/math.2023121

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Abstract

In this article, an analytical and numerical analysis of a computer virus epidemic model is presented. To more thoroughly examine the dynamics of the virus, the classical model is transformed into a fractional order model. The Caputo differential operator is applied to achieve this. The Jacobian approach is employed to investigate the model's stability. To investigate the model's numerical solution, a hybridized numerical scheme called the Grunwald Letnikov nonstandard finite difference (GL-NSFD) scheme is created. Some essential characteristics of the population model are scrutinized, including positivity boundedness and scheme stability. The aforementioned features are validated using test cases and computer simulations. The mathematical graphs are all detailed. It is also investigated how the fundamental reproduction number $ \mathfrak{R}_0 $ functions in stability analysis and illness dynamics.

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