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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 and Modeling of Fractional Order Model for Hepatitis B at Different Stages
    (Natural Sciences Publishing, 2023) Raza, Ali; Farman, Muhammad; Ahmad, Aqeel; Akgul, Ali; Sultan, Muhammad; Al Bayatti, Hilal
    Fractional operator is used to construct the framework of complex hepatitis B by using Caputo and Caputo Fabrizio fractional order derivative. Examination the uniqueness and stability to test the viability of the fractional order model with the proposed numerical plan as well as analyzes qualitatively. Union of different parts behind iterative approach on account of Fabrizio offers a bounded solution that accomplished required outcomes. The fractional system of differential equations which has four parts, susceptible individuals A(t), acute infected B(t),C(t) is chronic hepatitis and I(t) represents individuals who have retrieve after the infection with a life time freedom. At the end, the impact of the framework parameter on the spread of the ailment are begun to analyze using the numerical simulations. © 2023 NSP Natural Sciences Publishing Cor.
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    Analysis and Modelling of HIV/AIDS Model with Fractional Order Parameter Estimation
    (Natural Sciences Publishing, 2022) Farman, Muhammad; Raza, Ali; Akgul, Ali; Saleem, Muhammad Umer; Ahmad, Aqeel; Iqbal, Muhammad Sajid
    In this paper, nonlinear fractional order HIV/AIDS mathematical model is discussed epidemic problems for the complex transmission of the disease. It is accepted that susceptible wind up contaminated by means of sexual contacts with infective eventually create AIDS. The point of this task was to amend transmission models recently created to represent HIV transmission and AIDS related mortality. The Caputo-Fabrizio fractional derivative operator of order ? ? (0,1) is used to obtain fractional differential equations structure. The stability fractional order model was developed and the unique non-negative solution was tested. The numerical simulations are performed using an iterative technique. Some new results are being viewed with the help of Sumudo transform. Nonetheless, according to Banach, the related findings are given nonlinear functional analysis and fixed point theory. However, mathematical simulations are also acknowledged to evaluate the impact of the model’s parameter by decreasing the fractional values and showing the effect of the b fractional parameter on our obtained solutions. The impact of various parameters is represented graphically. © 2022. NSP Natural Sciences Publishing Cor.
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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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    Biological Nitrogen Fixation: An Analysis of Intoxicating Tribulations from Pesticides for Sustainable Legume Production
    (Springer Nature, 2022) Ahmad, Zahoor; Tariq, Rana Muhammad Sabir; Ramzan, Musarrat; Bukhari, Muhammad Adnan; Raza, Ali; Iqbal, Muhammad Aamir; Meena, Ram Swaroop
    A score of pedo-environmental factors serves as limiting elements for the biological nitrogen fixation (BNF) process in root nodules of leguminous plants. Since the advent of the green revolution, pesticides have been considered indispensable for keeping crop pests below the economic threshold level to ensure sustainable production of field crops for the rapidly increasing world population. However, pesticide application has also been associated with adverse effects on plant growth and development besides causing a detrimental reduction in microbial community dynamics. Rhizobium strains that are host-specific are no exception to this threat and are negatively influenced by different pesticides, especially fungicides, which seriously affect the functioning of the nitrogen (N) fixation process. Pesticides containing different synthetic chemicals affect symbiotic nitrogen fixation (SNF), and consequently, the amount of N fixed. This leads to reliance on crop plants primarily on N available in soil solution. Ultimately, reduced soil fertility leads to deteriorate crop productivity and quality of the produce. The objective of this review has been to synthesize, explore and critically analyze the effects of pesticide applications and their physiological impacts on BNF in legumes for sustainable crop production to strengthen food security for the increasing world population. Our review elucidates that indiscriminate use of agrochemicals could result in an undesirable environment for the healthy survival of symbiotic and symbiotic organisms leading to a corresponding reduction in exopolysaccharide (EPS) synthesis leading to poor atmospheric N fixation and thus affecting the whole agroecosystems. Therefore, by giving due consideration to the harmful effects of pesticides, farmers’ awareness about the safe usage of agrochemicals might be among the top priorities to conserve the environment besides harmonically preserving living organisms. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022.
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    Changing Climate Scenario: Perspectives of Camelina sativa as Low-Input Biofuel and Oilseed Crop
    (Springer International Publishing, 2023) Ahmad, Muhammad; Waraich, Ejaz Ahmad; Hafeez, Muhammad Bilal; Zulfiqar, Usman; Ahmad, Zahoor; Iqbal, Muhammad Aamir; Raza, Ali
    High population shifts and climate change are putting thrust on the food industry, especially edible oil production. Monoculture of high-input crops certainly affects the crop yield and soil health. The import of edible oil is increasing in the major part of the world, putting some burden on the national exchequer of the countries. The current oil crops are unable to meet the deficit to address the problems; a crop with distinct features must be incorporated in the cropping system. [Camelina sativa (L.) Crantz], a unique profiled biodiesel crop, is famous as gold of pleasure, and its oil is famous as a golden liquid. Camelina oil is an outstanding feedstock for the bio-based industry since its unique composition allows multiple applications. It is a rich source of oil >43%, which comprises a huge amount of unsaturated fatty acids, which accounts for 90%, containing 30-40% of alpha-linolenic acid and 15-25% of linoleic acid. The revival of this unique oilseed crop was based on (a) numerous inherent promising physiognomies, vigorous agronomic characteristics, eye-catching oil profile, genetic continuity with Arabidopsis, and the comfort of genetic remodeling by floral dip; (b) the investment in camelina which is understood as it merits serious considerations as potential biodiesel and oilseed and which shares a big role toward the sustainability along with increasing the diversity and production of plant oils; and (c) a univocal and descriptive portrayal of the different growth stages of camelina which will be used as an important apparatus for agronomy and research. In this review, the extended BBCH (Biologische Bundesanstalt, Bundessortenamt, and Chemische Industrie) scale was used to describe the phenological stages. The best use of camelina in the industrial sector as a drop-in product of packing materials, coatings, and adhesions can be achieved by further research to enlarge the camelina market. © The Editor(s) (ifapplicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022.
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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.
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    Numerical simulations of nonlinear stochastic Newell-Whitehead-Segel equation and its measurable properties
    (Elsevier, 2023) Iqbal, Muhammad Sajid; Yasin, Muhammad Waqas; Ahmed, Nauman; Akgul, Ali; Rafiq, Muhammad; Raza, Ali
    In this article, the stochastic form of the Newell-Whitehead-Segel equation has been investigated. This is a fully nonlinear partial differential equation and has huge applications. The nonlinearity of the underlying problem leads to the fact that one has to do the nonlinear analysis of the problem. So, firstly this article describes the regularity of the solution in the context of existing theory and a new approach has been applied to show the existence of the solution and corresponding explicit a-priori estimates of the Schauder type have been proposed. Secondly, in the next part, we have proposed two numerical schemes for the solution of the underlying problem and both schemes are very fighting for consistency and stability. The obtained numerical results are reliable, time-efficient, and very much adjacent to the exact state of the unknown function. (c) 2022 Elsevier B.V. All rights reserved.