Browsing by Author "Iqbal, Zafar"

Now showing 1 - 8 of 8

Citation - WoS: 20
Citation - Scopus: 26
A finite difference scheme to solve a fractional order epidemic model of computer virus
(Amer inst Mathematical Sciences-aims, 2023) Jarad, Fahd; Rehman, Muhammad Aziz-ur; Imran, Muhammad; Ahmed, Nauman; Fatima, Umbreen; Akgul, Ali; Jarad, Fahd; Matematik
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.
Citation - WoS: 12
Citation - Scopus: 13
Analysis of the Fractional Diarrhea Model With Mittag-Leffler Kernel
(Amer inst Mathematical Sciences-aims, 2022) Jarad, Fahd; Ahmed, Nauman; Akgul, Ali; Raza, Ali; Shahzad, Muhammad; Iqbal, Zafar; Jarad, Fahd; Matematik
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.
Citation - WoS: 12
Citation - Scopus: 11
Design, analysis and comparison of a nonstandard computational method for the solution of a general stochastic fractional epidemic model
(Mdpi, 2022) Ahmed, Nauman; Baleanu, Dumitru; Macias-Diaz, Jorge E.; Raza, Ali; Baleanu, Dumitru; Rafiq, Muhammad; Iqbal, Zafar; Ahmad, Muhammad Ozair; 56389; Matematik
Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R-0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R-0 < 1. A similar result is obtained for the endemic equilibrium when R-0 > 1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge-Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.
Citation - WoS: 3
Citation - Scopus: 6
Mathematical and numerical investigations of the fractional-order epidemic model with constant vaccination strategy
(Editura Acad Romane, 2021) Iqbal, Zafar; Baleanu, Dumitru; Rehman, Muhammad Aziz Ur; Baleanu, Dumitru; Ahmed, Nauman; Raza, Ali; Rafiq, Muhammad; 56389; Matematik
This work is devoted to find the reliable numerical solution of an epidemic model with constant vaccination strategy. For this purpose, a structure preserving numerical scheme called the Grunwald-Letnikov nonstandard finite difference scheme is designed. The proposed technique retains all the important properties of the continuous epidemic model like boundedness, positivity, and stability. This behavior of the proposed numerical scheme is validated mathematically and graphically. The role of the vaccination in controlling the disease dynamics in the population is verified through numerical simulations. The stability of the system under discussion is also examined at the disease free equilibrium point and the endemic equilibrium point. Finally, the outcome of this study is furnished with concluding remarks and future directions of research.
Citation - WoS: 28
Citation - Scopus: 32
New applications related to Covid-19
(Elsevier, 2021) Akgul, Ali; Baleanu, Dumitru; Ahmed, Nauman; Raza, Ali; Iqbal, Zafar; Rafiq, Muhammad; Baleanu, Dumitru; Rehman, Muhammad Aziz-ur; 56389; Matematik
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.
Citation - WoS: 5
Citation - Scopus: 7
New applications related to hepatitis C model
(Amer inst Mathematical Sciences-aims, 2022) Ahmed, Nauman; Jarad, Fahd; Raza, Ali; Akgul, Ali; Iqbal, Zafar; Rafiq, Muhammad; Ahmad, Muhammad Ozair; Jarad, Fahd; 234808; Matematik
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.
Citation - WoS: 51
Citation - Scopus: 62
Positivity and boundedness preserving numerical algorithm for the solution of fractional nonlinear epidemic model of HIV/AIDS transmission
(Pergamon-elsevier Science Ltd, 2020) Iqbal, Zafar; Baleanu, Dumitru; Ahmed, Nauman; Baleanu, Dumitru; Adel, Waleed; Rafiq, Muhammad; Rehman, Muhammad Aziz-ur; Alshomrani, Ali Saleh; 56389; Matematik
In this article, an integer order nonlinear HIV/AIDS infection model is extended to the non-integer nonlinear model. The Grunwald Letnikov nonstandard finite difference scheme is designed to obtain the numerical solutions. Structure preservence is one of the main advantages of this scheme. Reproductive number R-0 is worked out and its key role in disease dynamics and stability of the system is investigated with the following facts, if R-0 < 1 the disease will be diminished and it will persist in the community for R-0 > 1. On the other hand, it is sought out that system is stable when R-0 < 1 and R-0 > 1 implicates that system is locally asymptotically stable. Positivity and boundedness of the scheme is also proved for the generalized system. Two steady states of the system are computed and verified by computer simulations with the help of some suitable test problem. (C) 2020 Elsevier Ltd. All rights reserved.
Citation - WoS: 14
Citation - Scopus: 14
Structure preserving computational technique for fractional order Schnakenberg model
(Springer Heidelberg, 2020) Iqbal, Zafar; Baleanu, Dumitru; Ahmed, Nauman; Baleanu, Dumitru; Rafiq, Muhammad; Iqbal, Muhammad Sajid; Rehman, Muhammad Aziz-ur; 56389; Matematik
The current article deals with the analysis and numerical solution of fractional order Schnakenberg (S-B) model. This model is a system of autocatalytic reactions by nature, which arises in many biological systems. This study is aiming at investigating the behavior of natural phenomena with a more realistic and practical approach. The solutions are obtained by applying the Grunwald-Letnikov (G-L) finite difference (FD) and the proposed G-L nonstandard finite difference (NSFD) computational schemes. The proposed formulation is explicit in nature, strongly structure preserving as well as it is independent of the time step size. One very important feature of our proposed scheme is that it preserves the positivity of the solution of continuous fractional order S-B model because the unknown variables involved in this system describe the chemical concentrations of different substances. The comparison of the proposed scheme with G-L FD method reflects the significance of the said method.