Yazar "Rafiq, Muhammad" seçeneğine göre listele

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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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    A Nonlinear Structure of a Chemical Reaction Model and Numerical Modeling with the New Aspect of Existence and Uniqueness
    (Mdpi, 2023) Shaikh, Tahira Sumbal; Akgul, Ali; Rehman, Muhammad Aziz-ur; Ahmed, Nauman; Iqbal, Muhammad Sajid; Shahid, Naveed; Rafiq, Muhammad
    In this article, a nonlinear autocatalytic chemical reaction glycolysis model with the appearance of advection and diffusion is proposed. The occurrence and unicity of the solutions in Banach spaces are investigated. The solutions to these types of models are obtained by the optimization of the closed and convex subsets of the function space. Explicit estimates of the solutions for the admissible auxiliary data are formulated. An elegant numerical scheme is designed for an autocatalytic chemical reaction model, that is, the glycolysis model. The fundamental traits of the prescribed numerical method, for instance, the positivity, consistency, stability, etc., are also verified. The authenticity of the proposed scheme is ensured by comparing it with two extensively used numerical techniques. A numerical example is presented to observe the graphical behavior of the continuous system by constructing the numerical algorithm. The comparison depicts that the projected numerical design is more productive as compared to the other two schemes, as it holds all the important properties of the continuous model.
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    Analysis of a diffusive chemical reaction model in three space dimensions
    (Taylor & Francis Inc, 2024) Ahmed, Nauman; Ali, Javaid; Akguel, Ali; Hamed, Y. S.; Aljohani, A. F.; Rafiq, Muhammad; Khan, Ilyas
    This article proposes an implicit operator splitting nonstandard finite difference (OS-NSFD) scheme for numerical treatment of two species in three space dimensions reaction-diffusion glycolysis model. Since, the unknown state variables exhibiting the concentrations of species in glycolysis models and they cannot be negative and obtaining their positive solutions is a challenging task. The established theoretical result ensures that our proposed OS-NSFD scheme is unconditionally convergent at equilibrium point and fulfills the condition of positivity of solutions on contrary to other methods. Further, we analyze the existence and uniqueness of the solution obtained for the underlying system. To highlight the effectiveness of OS-NSFD scheme we compare the simulation results of OS-NSFD scheme with three well-known existing operators splitting finite difference (FD) schemes, namely, forward Euler explicit, backward Euler implicit and Crank Nicolson splitting schemes. Many existing techniques provide with the restricted positive solutions which do not work always. These techniques are only applicable if certain conditions on the discretized parameters are considered otherwise; they produce negative solutions, which is not the physical feature of the real system. The current work bridges this gap by catering the unconditional positive solutions to the reaction diffusion models.
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    Analysis of a Modified System of Infectious Disease in a Closed and Convex Subset of a Function Space with Numerical Study
    (Mdpi, 2023) Shaikh, Tahira Sumbal; Akgul, Ali; Rehman, Muhammad Aziz ur; Ahmed, Nauman; Iqbal, Muhammad Sajid; Shahid, Naveed; Rafiq, Muhammad
    In this article, the transmission dynamical model of the deadly infectious disease namedEbola is investigated. This disease identified in the Democratic Republic of Congo (DRC) and Sudan(now South Sudan) and was identified in 1976. The novelty of the model under discussion is theinclusion of advection and diffusion in each compartmental equation. The addition of these two termsmakes the model more general. Similar to a simple population dynamic system, the prescribed modelalso has two equilibrium points and an important threshold, known as the basic reproductive number.The current work comprises the existence and uniqueness of the solution, the numerical analysis ofthe model, and finally, the graphical simulations. In the section on the existence and uniqueness ofthe solutions, the optimal existence is assessed in a closed and convex subset of function space. Forthe numerical study, a nonstandard finite difference (NSFD) scheme is adopted to approximate thesolution of the continuous mathematical model. The main reason for the adoption of this technique isdelineated in the form of the positivity of the state variables, which is necessary for any populationmodel. The positivity of the applied scheme is verified by the concept of M-matrices. Since thenumerical method gives a discrete system of difference equations corresponding to a continuoussystem, some other relevant properties are also needed to describe it. In this respect, the consistencyand stability of the designed technique are corroborated by using Taylor's series expansion and Von Neumann's stability criteria, respectively. To authenticate the proposed NSFD method, two other illustrious techniques are applied for the sake of comparison. In the end, numerical simulations are also performed that show the efficiency of the prescribed technique, while the existing techniques fail to do so.
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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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    Comparative analysis of numerical with optical soliton solutions of stochastic Gross-Pitaevskii equation in dispersive media
    (Elsevier, 2023) Baber, Muhammad Zafarullah; Ahmed, Nauman; Yasin, Muhammad Waqas; Iqbal, Muhammad Sajid; Akgul, Ali; Riaz, Muhammad Bilal; Rafiq, Muhammad
    This article deals with the stochastic Gross-Pitaevskii equation (SGPE) perturbed with multiplicative time noise. The numerical solutions of the governing model are carried out with the proposed stochastic non-standard finite difference (SNSFD) scheme. The stability of the scheme is proved by using the Von-Neumann criteria and the consistency is shown in the mean square sense. To seek exact solutions, we applied the Sardar subequation (SSE) and modified exponential rational functional (MERF) techniques. The exact solutions are constructed in the form of exponential, hyperbolic, and trigonometric forms. Finally, the comparison of the exact solutions with numerical solutions is drawn in the 3D and line plots for the different values of parameters.
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    Computational aspects of an epidemic model involving stochastic partial differential equations
    (World Scientific Publ Co Pte Ltd, 2023) Ahmed, Nauman; Yasin, Muhammad W.; Ali, Syed Mansoor; Akguel, Ali; Raza, Ali; Rafiq, Muhammad; Shar, Muhammad Ali
    This paper deals with the study of the reaction-diffusion epidemic model perturbed with time noise. It has various applications such as disease in population models of humans, wildlife, and many others. The stochastic SIR model is numerically investigated with the proposed stochastic backward Euler scheme and proposed stochastic implicit finite difference (IFD) scheme. The stability of the proposed methods is shown with Von Neumann criteria and both schemes are unconditionally stable. Both schemes are consistent with systems of the equations in the mean square sense. The numerical solution obtained by the proposed stochastic backward Euler scheme and solutions converges towards an equilibrium but it has negative and divergent behavior for some values. The numerical solution gained by the proposed IFD scheme preserves the positivity and also solutions converge towards endemic and disease-free equilibrium. We have used two problems to check our findings. The graphical behavior of the stochastic SIR model is much adjacent to the classical SIR epidemic model when noise strength approaches zero. The three-dimensional plots of the susceptible and infected individuals are drawn for two cases of endemic equilibrium and disease-free equilibriums. The results show the efficacy of the proposed stochastic IFD scheme.
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    Computational study of a co-infection model of HIV/AIDS and hepatitis C virus models
    (Nature Portfolio, 2023) Dayan, Fazal; Ahmed, Nauman; Bariq, Abdul; Akgul, Ali; Jawaz, Muhammad; Rafiq, Muhammad; Raza, Ali
    Hepatitis C infection and HIV/AIDS contaminations are normal in certain areas of the world, and because of their geographic overlap, co-infection can't be precluded as the two illnesses have a similar transmission course. This current work presents a co-infection model of HIV/AIDS and Hepatitis C virus with fuzzy parameters. The application of fuzzy theory aids in tackling the issues associated with measuring uncertainty in the mathematical depiction of diseases. The fuzzy reproduction number and fuzzy equilibrium points have been determined in this context, focusing on a model applicable to a specific group defined by a triangular membership function. Furthermore, for the model, a fuzzy non-standard finite difference (NSFD) technique has been developed, and its convergence is examined within a fuzzy framework. The suggested model is numerically validated, confirming the dependability of the devised NSFD technique, which successfully retains all of the key properties of a continuous dynamical system.
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    Construction and numerical analysis of a fuzzy non-standard computational method for the solution of an SEIQR model of COVID-19 dynamics
    (Amer Inst Mathematical Sciences-Aims, 2022) Dayan, Fazal; Ahmed, Nauman; Rafiq, Muhammad; Akgul, Ali; Raza, Ali; Ahmad, Muhammad Ozair; Jarad, Fahd
    This current work presents an SEIQR model with fuzzy parameters. The use of fuzzy theory helps us to solve the problems of quantifying uncertainty in the mathematical modeling of diseases. The fuzzy reproduction number and fuzzy equilibrium points have been derived focusing on a model in a specific group of people having a triangular membership function. Moreover, a fuzzy non-standard finite difference (FNSFD) method for the model is developed. The stability of the proposed method is discussed in a fuzzy sense. A numerical verification for the proposed model is presented. The developed FNSFD scheme is a reliable method and preserves all the essential features of a continuous dynamical system.
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    Investigating the impact of stochasticity on HIV infection dynamics in CD4+T cells using a reaction-diffusion model
    (Nature Portfolio, 2024) Ahmed, Nauman; Yasin, Muhammad W.; Ali, Syed Mansoor; Akgul, Ali; Raza, Ali; Rafiq, Muhammad; Muhammad, Shah
    The disease dynamics affect the human life. When one person is affected with a disease and if it is not treated well, it can weaken the immune system of the body. Human Immunodeficiency Virus (HIV) is a virus that attacks the immune system, of the body which is the defense line against diseases. If it is not treated well then HIV progresses to its advanced stages and it is known as Acquired Immunodeficiency Syndrome (AIDS). HIV is typically a disease that can transferred from one person to another in several ways such as through blood, breastfeeding, sharing needles or syringes, and many others. So, the need of the hour is to consider such important disease dynamics and that will help mankind to save them from such severe disease. For the said purpose the reaction-diffusion HIV CD4+ T cell model with drug therapy under the stochastic environment is considered. The underlying model is numerically investigated with two time-efficient schemes and the effects of various parameters used in the model are analyzed and explained in a real-life scenario. Additionally, the obtained results will help the decision-makers to avoid such diseases. The random version of the HIV model is numerically investigated under the influence of time noise in Ito<^> sense. The proposed stochastic backward Euler (SBE) scheme and proposed stochastic Implicit finite difference (SIFD) scheme are developed for the computational study of the underlying model. The consistency of the schemes is proven in the mean square sense and the given system of equations is compatible with both schemes. The stability analysis proves that both schemes and schemes are unconditionally stable. The given system of equations has two equilibria, one is disease-free equilibrium (DFE) and the other is endemic equilibrium. The simulations are drawn for the different values of the parameters. The proposed SBE scheme showed the convergent behavior towards the equilibria for the given values of the parameters but also showed negative behavior that is not biological. The proposed SIFD scheme showed better results as compared with the stochastic SBE scheme. This scheme has convergent and positive behavior towards the equilibria points for the given values of the parameters. The effect of various parameters is also analyzed. Simulations are drawn to evaluate the efficacy of the schemes.
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    Investigation of soliton structures for dispersion, dissipation, and reaction time-fractional KdV-burgers-Fisher equation with the noise effect
    (Taylor & Francis Inc, 2024) Ahmed, Nauman; Baber, Muhammad Z.; Iqbal, Muhammad Sajid; Akguel, Ali; Rafiq, Muhammad; Raza, Ali; Chowdhury, Mohammad Showkat Rahim
    In this manuscript, the soliton structures for the time-fractional KdV-Burgers-Fisher equation with the effect of noise are investigated analytically. This is the dispersion-dissipation-reaction model. The third- and fifth-order time-fractional stochastic KdV-Burgers-Fisher equations are under consideration. These wave structures are constructed with the help of an extended generalized Riccati equation mapping method (EGREM). This method is a combined form of the $G'/G$G '/G expansion method with the generalized Riccati equation mapping method, and it will give the different forms of wave structures like shock, singular, combo, hyperbolic, trigonometric, mixed trigonometric, and rational solutions. These techniques are used symbolically with computational tools like Mathematica to demonstrate the efficiency and simplicity of the proposed strategy. Additionally, with the various relevant parameter values, the sketches of some solutions in the form of 3D and contour representations for the purpose of comprehending physical processes are drawn. These sketches clearly show the random behavior of these wave structures that are appearing in dispersion, dissipation, and reaction concentrations of these mathematical models.
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    Investigation the soliton solutions of mussel and algae model leading to concentration
    (Elsevier, 2023) Islam, Warda; Baber, Muhammad Z.; Ahmed, Nauman; Akgul, Ali; Rafiq, Muhammad; Raza, Ali; Yahia, I. S.
    In this study, the abundant families of soliton solutions are constructed for the mussel -algae model which provides information about represents the concentration of algae in the water layer, and mussel biomass of sediment surface in the per square meter. These solutions are extracted in the form of shock singular, complex solitary-shock, shock-singular and periodic-singular. Also, the rational solutions are also emerged during the derivation. For these solutions we are used the new modified extended direct algebraic (NMEDA) technique. These solutions are gained by the help of Mathematica. Additionally, the physical behaviors of some solutions are shown for different values of parameters, the graphs in 3D-dimensions along with their contour diagrams are also drawn.(c) 2023 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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    Mathematical modeling of Ebola using delay differential equations
    (Springer Heidelberg, 2024) Raza, Ali; Ahmed, Nauman; Rafiq, Muhammad; Akguel, Ali; Cordero, Alicia; Torregrosa, Juan R.
    Nonlinear delay differential equations (NDDEs) are essential in mathematical epidemiology, computational mathematics, sciences, etc. In this research paper, we have presented a delayed mathematical model of the Ebola virus to analyze its transmission dynamics in the human population. The delayed Ebola model is based on the four human compartments susceptible, exposed, infected, and recovered (SEIR). A time-delayed technique is used to slow down the dynamics of the host population. Two significant stages are analyzed in the said model: Ebola-free equilibrium (EFE) and Ebola-existing equilibrium (EEE). Also, the reproduction number of a model with the sensitivity of parameters is studied. Furthermore, the local asymptotical stability (LAS) and global asymptotical stability (GAS) around the two stages are studied rigorously using the Jacobian matrix Routh-Hurwitz criterion strategies for stability and Lyapunov function stability. The delay effect has been observed in the model in inverse relation of susceptible and infected humans (it means the increase of delay tactics that the susceptibility of humans increases and the infectivity of humans decreases eventually approaches zero which means that Ebola has been controlled into the population). For the numerical results, the Euler method is designed for the system of delay differential equations (DDEs) to verify the results with an analytical model analysis.
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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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    Numerical investigations of stochastic Newell-Whitehead-Segel equation in (2+1) dimensions
    (World Scientific Publ Co Pte Ltd, 2023) Ahmed, Nauman; Yasin, Muhammad Waqas; Iqbal, Muhammad Sajid; Akgul, Ali; Rafiq, Muhammad; Raza, Ali; Baber, Muhammad Zafarullah
    The stochastic Newell-Whitehead-Segel in (2+1) dimensions is under consideration. It represents the population density or dimensionless temperature and it discusses how stripes appear in temporal and spatial dimensional systems. The Newell-Whitehead-Segel equation (NWSE) has applications in different areas such as ecology, chemical, mechanical, biology and bio-engineering. The important thing is if we see the problem in the two-dimensional (2D) manifold, then the whole 3D picture can be included in the model. The 3D space is embedded compactly in the 2D manifolds. So, 2D problems for the Newell-White-Segel equation are very important because they consider the one, two and three dimensions in it. The numerical solutions of the underlying model have been extracted successfully by two schemes, namely stochastic forward Euler (SFE) and the proposed stochastic nonstandard finite difference (SNSFD) schemes. The existence of the solution is guaranteed by using the contraction mapping principle and Schauder's fixed-point theorem. The consistency of each scheme is proved in the mean square sense. The stability of the schemes is shown by using von Neumann criteria. The SFE scheme is conditionally stable and the SNSFD scheme is unconditionally stable. The efficacy of the proposed methods is depicted through the simulations. The 2D and 3D graphs are plotted for various values of the parameters.
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    Numerical modeling of reaction-diffusion e-epidemic dynamics
    (Springer Heidelberg, 2024) Yasin, Muhammad Waqas; Ashfaq, Syed Muhammad Hamza; Ahmed, Nauman; Raza, Ali; Rafiq, Muhammad; Akguel, Ali
    The objective of this paper is to understand the dynamics of virus spread in a computer network by e-epidemic reaction-diffusion model and applying an implicit finite difference (FD) scheme for a numerical solution. The SIR models are used in studies of epidemiology to predict the behavior of the propagation of biological viruses within the population. We divide the population of computer nodes into three parts i.e. susceptible (may catch the virus), infected 1 (infected but not completely), and infected 2 (completely infected). By using Taylor's series expansion the consistency of the implicit scheme is proved. The unconditional stability of the implicit FD model is proved by using the Von Numan stability analysis. The qualitative analysis of the underlying model is also analyzed such as positivity and boundedness of the model. The numerical stability and bifurcation of is also analyzed. Likewise, identical modeling techniques are adopted to analyze the spread of the virus in digital networks. Because the computer virus behaves in the identical way as the biological virus behaves, this paper emphasizes the significance of diffusion in decreasing the gap between reality and theory, offering a more precise depiction of the spread of the virus within the digital network.
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    Numerical scheme and stability analysis of stochastic Fitzhugh-Nagumo model
    (Elsevier, 2022) Yasin, Muhammad W.; Iqbal, Muhammad S.; Ahmed, Nauman; Akgul, Ali; Raza, Ali; Rafiq, Muhammad; Riaz, Muhammad Bilal
    This article deals with the Fitzhugh-Nagumo equation in the presence of stochastic function. A numerical scheme has been developed for the solution of such equations which preserves the certain structure of the unknown functions, also we have given the stability analysis, consistency of the problem, and explicitly optimal a priori estimates for the existence of solutions of such equations. A unique solution has been guaranteed. The corresponding explicit estimates in the function spaces are formulated in the form of theorems. Lastly, one important feature of the article is the simulation of the proposed numerical scheme in the form of the 2D and 3D plots which shows the efficacy of the stochastic analysis of such nonlinear partial differential equations.