A New Fractal Fractional Modeling of the Computer Viruses System (original) (raw)
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Alexandria Engineering Journal, 2020
In this manuscript, a qualitative analysis of the mathematical model of novel coronavirus (COVID-19) involving anew devised fractal-fractional operator in the Caputo sense having the fractional-order q and the fractal dimension p is considered. The concerned model is composed of eight compartments: susceptible, exposed, infected, super-spreaders, asymptomatic, hospitalized, recovery and fatality. When, choosing the fractal order one we obtain fractional order, and when choosing the fractional order one a fractal system is obtained. Considering both the operators together we present a model with fractal-fractional. Under the new derivative the existence and uniqueness of the solution for considered model are proved using Schaefer's and Banach type fixed point approaches. Additionally, with the help of nonlinear functional analysis, the condition for Ulam's type of stability of the solution to the considered model is established. For numerical simulation of proposed model, a fractional type of two-step Lagrange polynomial known as fractional Adams-Bashforth AB ð Þ method is applied to simulate the results. At last, the results are tested with real data from COVID-19 outbreak in Wuhan City, Hubei Province of China from 4 January to 9 March 2020, taken from a source (Ndaı¨rou, 2020). The Numerical results are presented in terms of graphs for different fractional-order q and fractal dimensions p to describe the transmission dynamics of disease infection.
Journal of Function Spaces, 2022
SARS-CoV-2 is a strain of the large coronavirus family that has led to COVID-19 disease. The virus has been one of the deadliest known viruses in the world to date. Rapid mutations and the creation of new strains cause researchers to focus on the dynamic behaviors of the virus and to analyze it accurately through clinical research and mathematical models. In this paper, from the point of view of mathematical modeling, we intend to focus on the dynamic behavior of the system and examine its analytical and numerical aspects in two different structures. In other words, by recalling newly formulated hybrid fractional-fractal operators, we present a fractal-fractional probability-based model of SARS-CoV-2 virus for the first time and extract its equivalent compact fractal-fractional IVP to investigate its existence and stability criteria. A type of special admissible contractions will help us in this regard. Moreover, based on the source data, we simulate our system according to algorithms derived by Adams-Bashforth method and explain the effects of variation of the dimension of fractal and fractional order on dynamics of solutions. Finally, we transform our fractal-fractional model into a Caputo probability-based model of SARS-CoV-2 to derive solutions via the operational matrix method under Taylor's basis. The numerical simulations show close behaviors for both of models.
Results in Physics, 2021
addressing the dynamics of fractal-fractional type modified SEIR model under Atangana-Baleanu Caputo (ABC) derivative of fractional order y and fractal dimension p for the available data in Pakistan. The proposed model has been investigated for qualitative analysis by applying the theory of non-linear functional analysis along with fixed point theory. The fractional Adams-bashforth iterative techniques have been applied for the numerical solution of the said model. The Ulam-Hyers (UH) stability techniques have been derived for the stability of the considered model. The simulation of all compartments has been drawn against the available data of covid-19 in Pakistan. The whole study of this manuscript illustrates that control of the effective transmission rate is necessary for stoping the transmission of the outbreak. This means that everyone in the society must change their behavior towards self-protection by keeping most of the precautionary measures sufficient for controlling covid-19.
Scientific reports, 2024
In this paper, we investigate a fractal-fractional-order mathematical model with the influence of hospitalized patients and the impact of vaccination with fractal-fractional operators. The respective derivatives are considered in the Caputo, Caputo Fabrizio, and Atangana-Baleanu senses of fractional order α and fractal dimension τ. For the proposed problem, some results regarding basic reproduction number and stability are given. Using the next-generation matrix approach, we have investigated the global and local stability of several types of equilibrium points. We provide a detailed analysis of the existence and uniqueness of the solution. Moreover, we fit the model with the real data of Pakistan from June 01, 2020, till March 24, 2021. Then, we use the fractal-fractional derivative to find a numerical solution for the model. MATLAB software is used for numerical illustration. Graphical presentations corresponding to different parameteric values are given as well.
Analytical study of transmission dynamics of 2019-nCoV pandemic via fractal fractional operator
Results in Physics, 2021
Through this paper, we aim to study the dynamics of 2019-nCoV transmission using fractal-ABC type fractional differential equations by incorporating population self-protection behavior changes. The basic parameters of disease dynamics spread differently from country to country due to the different sensitive parameters. The proposed model in this study links the infection rate, the marginal value of the infection force for the population, the recovery rate, the rate of decomposition of the 2019-nCoV in the environment, and what methods are needed to stop the spread of the virus. We give a detailed analysis of the proposed model in this study by analyzing disease-free equilibrium point, the number of reproduction and the positivity of the model solutions, in addition to verifying the existence, uniqueness and stability of this disease using fixed point theories. Further on exploiting Adam Bash's numerical scheme, we compute some numerical results for the required model. The concerned results have been simulated against some real initial data of three different counties including China, Brazil, and Italy.
Fractals
In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal–fractional derivatives in which fractional derivative is taken in Atangana–Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam–Hyers stable under the n...
Role of fractal-fractional operators in modeling of rubella epidemic with optimized orders
Open Physics
Fractal-fractional (FF) differential and integral operators having the capability to subsume features of retaining memory and self-similarities are used in the present research analysis to design a mathematical model for the rubella epidemic while taking care of dimensional consistency among the model equations. Infectious diseases have history in their transmission dynamics and thus non-local operators such as FF play a vital role in modeling dynamics of such epidemics. Monthly actual rubella incidence cases in Pakistan for the years 2017 and 2018 have been used to validate the FF rubella model and such a data set also helps for parameter estimation. Using nonlinear least-squares estimation with MATLAB function lsqcurvefit, some parameters for the classical and the FF model are obtained. Upon comparison of error norms for both models (classical and FF), it is found that the FF produces the smaller error. Locally asymptotically stable points (rubella-free and rubella-present) of the...
Numerical solution of a fractal-fractional order chaotic circuit system
Revista Mexicana De Fisica, 2021
The dynamical system has an important research area and, due to its wide applications, many researchers and scientists are working to develop new models and techniques for their solution. In this work, we present in this work the dynamics of a chaotic model in the presence of newly introduced fractal-fractional operators. The model is formulated initially in ordinary differential equations, and then we utilize the fractal-fractional (FF) in power law, exponential, and Mittag-Leffler to generalize the model. For each fractal-fractional order model, we briefly study its numerical solution via the numerical algorithm. We present some graphical results with arbitrary order of fractal and fractional orders, and present that these operators provide different chaotic attractors for different fractal and fractional order values. The graphical results demonstrate the effectiveness of the fractal-fractional operators.
Analysis of fractal fractional differential equations
Alexandria Engineering Journal, 2020
Nonlocal differential and integral operators with fractional order and fractal dimension have been recently introduced and appear to be powerful mathematical tools to model complex real world problems that could not be modeled with classical and nonlocal differential and integral operators with single order. To stress further possible application of such operators, we consider in this work an advection-dispersion model, where the velocity is considered to be 1. We consider three cases of the models, when the kernels are power law, exponential decay law and the generalized Mittag-Leffler kernel. For each case, we present a detailed analysis including, numerical solution, stability analysis and error analysis. We present some numerical simulation.
Fractals
Waterborne diseases are illnesses caused by pathogenic bacteria that spread through water and have a negative inuence on human health. Due to the involvement of most countries in this vital issue, accurate analysis of mathematical models of such diseases is one of the rst priorities of researchers. In this regard, in this paper, we turn to a waterborne disease model for solution's existence, HU-stability, and computational analysis. We transform the model to an analogous fractal-fractional integral form and study its qualitative analysis using an iterative convergent sequence and xed point technique to see whether there is a solution. We use Lagrange's interpolation to construct numerical algorithms for the fractal-fractional waterborne disease model in terms of computations. The approach is then put to the test in a case study, yielding some interesting outcomes.