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    A finite difference scheme to solve a fractional order epidemic model of computer virus
    (Amer Inst Mathematical Sciences-Aims, 2023) Iqbal, Zafar; Rehman, Muhammad Aziz-ur; Imran, Muhammad; Ahmed, Nauman; Fatima, Umbreen; Akgul, Ali; Rafiq, Muhammad
    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 R0 functions in stability analysis and illness dynamics.
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    A fractal fractional model for computer virus dynamics
    (Pergamon-Elsevier Science Ltd, 2021) Akgul, Ali; Fatima, Umbreen; Iqbal, Muhammad Sajid; Ahmed, Nauman; Raza, Ali; Iqbal, Zafar; Rafiq, Muhammad
    The gist behind this study is to extend the classical computer virus model into fractal fractional model and subsequently to solve the model by Atangana-Toufik method. This method solve nonlinear model under consideration very efficiently. We use the Mittag-Leffler kernels on the proposed model. Atangana-Baleanu integral operator is used to solve the set of fractal-fractional expressions. In this model, three types of equilibrium points are described i.e trivial, virus free and virus existing points. These fixed points are used to establish some standard results to discuss the stability of the system by calculating the Jacobian matrices at these points. Routh-Hurwitz criteria is used to verify that the system is locally asymptotically stable at all the steady states. The emphatic role of the basic reproduction number R-0 is also brought into lime light for stability analysis. Sensitivity analysis of R-0 is also discussed. Optimal existence and uniqueness of the solution is the nucleus of this study. Computer simulations and patterns and graphical patterns illustrate reliability and productiveness of the proposed method. (C) 2021 Elsevier Ltd. All rights reserved.
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    Analysis of the fractional diarrhea model with Mittag-Leffler kernel
    (Amer Inst Mathematical Sciences-Aims, 2022) Iqbal, Muhammad Sajid; Ahmed, Nauman; Akgul, Ali; Raza, Ali; Shahzad, Muhammad; Iqbal, Zafar; Rafiq, Muhammad
    In this article, we have introduced the diarrhea disease dynamics in a varying population. For this purpose, a classical model of the viral disease is converted into the fractional-order model by using Atangana-Baleanu fractional-order derivatives in the Caputo sense. The existence and uniqueness of the solutions are investigated by using the contraction mapping principle. Two types of equilibrium points i.e., disease-free and endemic equilibrium are also worked out. The important parameters and the basic reproduction number are also described. Some standard results are established to prove that the disease-free equilibrium state is locally and globally asymptotically stable for the underlying continuous system. It is also shown that the system is locally asymptotically stable at the endemic equilibrium point. The current model is solved by the Mittag-Leffler kernel. The study is closed with constraints on the basic reproduction number R-0 and some concluding remarks.
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    Analysis of the fractional polio model with the Mittag-Leffler kernels
    (Elsevier, 2023) Iqbal, Muhammad Sajid; Ahmed, Nauman; Akgul, Ali; Satti, Ammad Mehmood; Iqbal, Zafar; Raza, Ali; Rafiq, Muhammad
    This article investigates the transmission of polio-virus disease in the human population. The classical model is considered for studying fatal disease. First of all, the model is converted into the fractal fractional epidemic model. Then, the existence of the solution for the said model is ensured with the help of the fixed point theory. Points of equilibria for the model are worked out. The basic reproduction number is described and its role in the disease communication and sta-bility of the model is examined by some standard results. Simulated graphs are also plotted to sup-port the pre-results and claims. Lastly, the findings of the study are presented.(c) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
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    New applications related to Covid-19
    (Elsevier, 2021) Akgul, Ali; Ahmed, Nauman; Raza, Ali; Iqbal, Zafar; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-ur
    Analysis of mathematical models projected for COVID-19 presents in many valuable outputs. We analyze a model of differential equation related to Covid-19 in this paper. We use fractal-fractional derivatives in the proposed model. We analyze the equilibria of the model. We discuss the stability analysis in details. We apply very effective method to obtain the numerical results. We demonstrate our results by the numerical simulations.
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    New applications related to hepatitis C model
    (Amer Inst Mathematical Sciences-Aims, 2022) Ahmed, Nauman; Raza, Ali; Akgul, Ali; Iqbal, Zafar; Rafiq, Muhammad; Ahmad, Muhammad Ozair; Jarad, Fahd
    The main idea of this study is to examine the dynamics of the viral disease, hepatitis C. To this end, the steady states of the hepatitis C virus model are described to investigate the local as well as global stability. It is proved by the standard results that the virus-free equilibrium state is locally asymptotically stable if the value of R-0 is taken less than unity. Similarly, the virus existing state is locally asymptotically stable if R-0 is chosen greater than unity. The Routh-Hurwitz criterion is applied to prove the local stability of the system. Further, the disease-free equilibrium state is globally asymptotically stable if R-0 < 1. The viral disease model is studied after reshaping the integer-order hepatitis C model into the fractal-fractional epidemic illustration. The proposed numerical method attains the fixed points of the model. This fact is described by the simulated graphs. In the end, the conclusion of the manuscript is furnished.
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    Optimal existence of fractional order computer virus epidemic model and numerical simulations
    (Wiley, 2021) Akgul, Ali; Iqbal, Muhammad Sajid; Fatima, Umbreen; Ahmed, Nauman; Iqbal, Zafar; Raza, Ali; Rafiq, Muhammad
    Aim of this article is to analyze the fractional order computer epidemic model. To this end, a classical computer epidemic model is extended to the fractional order model by using the Atangana-Baleanu fractional differential operator in Caputo sense. The regularity condition for the solution to the considered system is described. Existence of the solution in the Banach space is investigated and some benchmark results are presented. Steady states of the system is described and stability of the model at these states is also studied, with the help of Jacobian matrix method. Some results for the local stability at disease free equilibrium point and endemic equilibrium point are presented. The basic reproduction number is mentioned and its role on stability analysis is also highlighted. The numerical design is formulated by applying the Atangana-Baleanu integral operator. The graphical solutions are also presented by computer simulations at both the equilibrium points.