An epidemiological model for computer virus with Atangana-Baleanu fractional derivative

The Era of data is transubstantiating into a Big Data model in this technological world in the early 21st century. In 2005, Roger Mougalas coined a combination of data for this future world of the human race. The information helps to find specific solutions for any physical problem under Catastrophic circumstances in high populations such as Covid-19. To store massive data and historical events in a computer, the possibility of damage occurred to the complete data. Hence, viruses are a crucial threat to such data worth millions and billions. For this purpose, we spend enormous costs and efforts to build defensive strategies to save that information. Analyzing the expansion and extension of viruses helps to protect data and prevent viruses. In this manuscript, we study optimal control analysis for the suggested model in the sense of the Atangana-Baleanu derivative (AB-derivative). We employed a fixed point theorem to analyze the solutions for the fractional order computer virus model. We verified the results numerically and expressed them graphically.

SIRS epidemic modeling using fractional-ordered differential equations: Role of fear effect

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ FROM AUTHOR] Copyright of International Journal of Biomathematics is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)

A review on epidemic models in sight of fractional calculus

Biomathematics has become one of the most significant areas of research as a result of interdisciplinary study. Chronic diseases sometimes referred to as non-communicable and communicable diseases, are conditions that develop over an extended period as a result of different factors like genetics, lifestyle, and environment. The most important common types of disease are cardiovascular, alcohol, cancer, and diabetes. More than three-quarters of the world's (31.4 million) deaths occur in low- and middle-income nations, which are disproportionately affected by different infections. Fractional Calculus is a prominent topic for research within the discipline of Applied Mathematics due to its usefulness in solving problems in many different branches of science, engineering, and medicine. Recent researchers have identified the importance of mathematical tools in various disease models as being very useful to study the dynamics with the help of fractional and integer calculus modeling. Due to the complexity of the underlying connections, both deterministic and stochastic epidemiological models are founded on an inadequate understanding of the infectious network. Over the past several years, the use of different fractional operators to model the problem has grown, and it is now a common way to study how epidemics spread. Recently, researchers have actively considered fractional calculus to study different diseases like COVID-19, cancer, TB, HIV, dengue fever, diabetes, cholera, pine welts, smoking and heart attacks, etc. With the help of fractional operator, we modified a mathematical model for the dynamical transmission, analysis, treatment, vaccination, and precaution leveling necessary to mitigate the negative impact of illness on society in the long run, overcoming the memory effect without defining or considering others parameters. In this review paper, we considered all the recent studies based on the fractional modeling of infectious and non-infectious diseases with different fractional operators such as Caputo, Caputo Fabrizio, ABC, and constant proportional with Caputo, etc. This review paper aims to bring all the information together by considering different fractional operators and their uses in the field of infectious disease modeling. The steps taken to accomplish the goal were developing a mathematical model, identifying the equilibrium point, figuring out the minimal reproductive number, and assessing the stability around the equilibrium point. For future direction, we consider the cancer model to study the growth cells of cancer and the impact of therapy to control infections. An equilibrium solution and an analysis of the behavior dynamics of the cell spread with treatment in the form of chemotherapy were obtained. The simulation shows that the population of cancer cells is influenced by the pace of cancer cell growth with the Caputo fractional derivative. The acquired results show how effective and precise the suggested approach is in helping to better understand how chemotherapy works. Chemotherapy medications have been found to increase immunity against particular cancer by reducing the number of tumor cells. Further, we suggested some future work directions with the help of the new hybrid fractional operator. Our innovative methodology might have significant effects on global stakeholders, policymakers, and national health systems. The current strategies for controlling outbreaks and the vaccination and prevention policies that have been implemented would benefit from a more accurate representation of the dynamics of contagious diseases, which necessitates the development of highly complex mathematical models. Microorganisms, interactions between individuals or groups, and environmental, social, economic, and demographic factors on a broader scale are all examples.

SIMULATIONS AND ANALYSIS OF COVID-19 AS A FRACTIONAL MODEL WITH DIFFERENT KERNELS

Recently, Atangana proposed new operators by combining fractional and fractal calculus. These recently proposed operators, referred to as fractal–fractional operators, have been widely used to study complex dynamics. In this paper, the COVID-19 model is considered via Atangana–Baleanu fractal-fractional operator. The Lyapunov stability for the model is derived for first and second derivative. Numerical results have developed through Lagrangian-piecewise interpolation for the different fractal–fractional operators. We develop numerical outcomes through different differential and integral fractional operators like power-law, exponential law, and Mittag-Leffler kernel. To get a better outcome of the proposed scheme, numerical simulation is made with different kernels having the memory effects with fractional parameters. [ FROM AUTHOR] Copyright of Fractals is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)

On the Modeling of COVID-19 Spread via Fractional Derivative: A Stochastic Approach

: The coronavirus disease (COVID-19) pandemic has caused more harm than expected in developed and developing countries. In this work, a fractional stochastic model of COVID-19 which takes into account the random nature of the spread of disease, is formulated and analyzed. The existence and uniqueness of solutions were established using the fixed-point theory. Two different fractional operators', namely, power-law and Mittag–Leffler function, numerical schemes in the stochastic form, are utilized to obtain numerical simulations to support the theoretical results. It is observed that the fractional order derivative has effect on the dynamics of the spread of the disease. © 2023, Pleiades Publishing, Ltd.

Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate

In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties of the SIR model are discussed in detail. The local and global stability of the model are analyzed. Moreover, some conditions are developed to guarantee local and global asymptotic stability. Finally, numerical simulations are provided to support the theoretical results and used to analyze the impact of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of the disease, and limitation in treatment resources on COVID-19. The graphical results show that face masks, social distancing, quarantine, lockdown, immigration, and effective treatment rates significantly reduce the infected population over time. In contrast, limitation in the availability of treatment raises the infected population. © 2023 The Author(s). Published by IOP Publishing Ltd.

A fractional PI observer for incommensurate fractional order systems under parametric uncertainties.

The problem of state observation in incommensurate fractional order systems has been poorly studied. Currently some observers that have been proposed are based on a copy of the system, which causes them to be highly dependent on the system parameters, additionally they are redundant (estimate variables that are available). So this paper proposes a novel fractional observer against parametric uncertainties for a certain type of incommensurate fractional order systems. The fractional observer design is based on a property concerning observability in incommensurate fractional order systems which allows us to construct the observer only considering the available output and its fractional derivatives. On the other hand, the convergence analysis of the observation error is carried out using a particular approach of fractional order systems related to the Global Mittag-Leffler boundedness. We prove that there is a compact set GMLA (Globally Mittag-Leffler Attractive, according to Definition 4) where the system that represents the observation error dynamics is attractive and we also prove that the observation error is uniformly bounded. Additionally, the fractional observer is model-free i.e., a system copy is not required, this gives robustness in spite of parametric uncertainties and it is also reduced order therefore one observer must be designed for each variable that we want to estimate consequently the observer is non-redundant (no estimation of variables that are already available). Moreover, our proposed fractional observer can be designed for commensurate fractional order systems and we also show that if we consider integer derivative order, the proposed fractional observer presents certain properties. Finally, in order to show the effectiveness of the proposed fractional observer, an incommensurate fractional order Rössler hyperchaotic system is considered as a numerical example and an incommensurate fractional model of the COVID-19 pandemic as a real-world application.

A Mathematical Model for COVID-19 Image Enhancement based on Mittag-Leffler-Chebyshev Shift

The lungs CT scan is used to visualize the spread of the disease across the lungs to obtain better knowledge of the state of the COVID-19 infection. Accurately diagnosing of COVID-19 disease is a complex challenge that medical system face during the pandemic time. To address this problem, this paper proposes a COVID-19 image enhancement based on Mittag-Leffler-Chebyshev polynomial as pre-processing step for COVID-19 detection and segmentation. The proposed approach comprises the MittagLeffler sum convoluted with Chebyshev polynomial. The idea for using the proposed image enhancement model is that it improves images with low gray level changes by estimating the probability of each pixel. The proposed image enhancement technique is tested on a variety of lungs computed tomography (CT) scan dataset of varying quality to demonstrate that it is robust and can resist significant quality fluctuations. The blind/referenceless image spatial quality evaluator (BRISQUE), and the natural image quality evaluator (NIQE) measures for CT scans were 38.78, and 7.43 respectively. According to the findings, the proposed image enhancement model produces the best image quality ratings. Overall, this model considerably enhances the details of the given datasets, and it may be able to assist medical professionals in the diagnosing process.

A fractal fractional order vaccination model of COVID-19 pandemic using Adam’s moulton analysis

The pandemic caused by coronaviruses (SARS-COV-2) is a zoonotic disease targeting the respiratory tract of active humans. Few mild symptoms of fever and tiredness get cured without any medicinal aid , whereas some severe symptoms of dry cough with breathing illness led to perceived risk of secondary transmission. This paper studies the effectiveness of vaccination in Covid -19 pandemic disease by modelling three compartments susceptible, vaccinated and infected (SVI) of Atangana Baleanu of Caputo (ABC) type derivatives in non-integer order. The disease dynamics is analysed and its stability is performed. Numerical approximation is derived using Adam’s Moulton method and simulated to forecast the results for controllability of pandemic spread.

Generalized Mittag-Leffler Kernel Form Solutions of Free Convection Heat and Mass Transfer Flow of Maxwell Fluid with Newtonian Heating: Prabhakar Fractional Derivative Approach

In this article, the effects of Newtonian heating along with wall slip condition on temperature is critically examined on unsteady magnetohydrodynamic (MHD) flows of Prabhakar-like non integer Maxwell fluid near an infinitely vertical plate under constant concentration. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on a newly introduced Prabhakar fractional operator with generalized Fourier’s law and Fick’s law. This fractional model has been solved analytically and exact solutions for dimensionless velocity, concentration, and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. Physical impacts of different parameters such as α, Pr, β, Sc, Gr, γ, and Gm are studied and demonstrated graphically by Mathcad software. Furthermore, to validate our current results, some limiting models such as classical Maxwell model, classical Newtonian model, and fractional Newtonian model are recovered from Prabhakar fractional Maxwell fluid. Moreover, we compare the results between Maxwell and Newtonian fluids for both fractional and classical cases with and without slip conditions, showing that the movement of the Maxwell fluid is faster than viscous fluid. Additionally, it is visualized that both classical Maxwell and viscous fluid have relatively higher velocity as compared to fractional Maxwell and viscous fluid.

Sensitivity Analysis of COVID-19 with Quarantine and Vaccination: A Fractal-Fractional Model

To eradicate most infectious diseases, mathematical modelling of contagious diseases has revealed that a combination of quarantine, vaccination, and cure is frequently required. However, eradicating the disease will remain a difficult task if they aren't provided at the appropriate time and in the right quantity. Control analysis has been shown to be an effective way for discovering the best approaches to preventing the spread of contagious diseases through the development of disease preventive interventions. The method comprises reducing the cost of infection, implementing control measures, or both. In order to gain a better understanding of COVID-19's future dynamics, this study presents a compartmental mathematical model. The problem is modelled as a highly nonlinear coupled system of classical order ODEs, which is then generalised using the Mittag-Leffler kernel's fractal-fractional derivative. The uniqueness of the fractional model under discussion has also been demonstrated. The boundedness and non-negativity of the considered model are also established. The next generation technique is used to examine basic reproduction, anddisease free and endemic equilibrium. We used reported cases from Australia in this investigation due to the high risk of infection. The reported cases are considered between 1st July 2021 and 20th August 2021. On the basis of previous data, the spread of infection is predicted for the next 600 days which is shown through different graphs. The graphical solution of the considered nonlinear model is obtained via numerical scheme by implementing the MATLAB software. Based on the fitted values of parameters, the basic reproduction number R0 is calculated as R0≈1.58276. Furthermore, the impact of fractional and fractal parameter on the disease spread among different classes is demonstrated. In addition, the impact of quarantine and vaccination on infected people is dramatically depicted. It's been argued that public awareness of the quarantine and effective vaccination can drastically reduce infection rates in the population.

Fractional optimal control of COVID-19 pandemic model with generalized Mittag-Leffler function.

In this paper, we consider a fractional COVID-19 epidemic model with a convex incidence rate. The Atangana-Baleanu fractional operator in the Caputo sense is taken into account. We establish the equilibrium points, basic reproduction number, and local stability at both the equilibrium points. The existence and uniqueness of the solution are proved by using Banach and Leray-Schauder alternative type theorems. For the fractional numerical simulations, we use the Toufik-Atangana scheme. Optimal control analysis is carried out to minimize the infection and maximize the susceptible people.

Generalized form of fractional order COVID-19 model with Mittag-Leffler kernel.

An important advantage of fractional derivatives is that we can formulate models describing much better systems with memory effects. Fractional operators with different memory are related to the different type of relaxation process of the nonlocal dynamical systems. Therefore, we investigate the COVID-19 model with the fractional derivatives in this paper. We apply very effective numerical methods to obtain the numerical results. We also use the Sumudu transform to get the solutions of the models. The Sumudu transform is able to keep the unit of the function, the parity of the function, and has many other properties that are more valuable. We present scientific results in the paper and also prove these results by effective numerical techniques which will be helpful to understand the outbreak of COVID-19.

A fuzzy fractional model of coronavirus (COVID-19) and its study with Legendre spectral method.

The virus which belongs to the family of the coronavirus was seen first in Wuhan city of China. As it spreads so quickly and fastly, now all over countries in the world are suffering from this. The world health organization has considered and declared it a pandemic. In this presented research, we have picked up the existing mathematical model of corona virus which has six ordinary differential equations involving fractional derivative with non-singular kernel and Mittag-Leffler law. Another new thing is that we study this model in a fuzzy environment. We will discuss why we need a fuzzy environment for this model. First of all, we find out the approximate value of ABC fractional derivative of simple polynomial function ( t - a ) n . By using this approximation we will derive and developed the Legendre operational matrix of fractional differentiation for the Mittag-Leffler kernel fractional derivative on a larger interval [ 0 , b ] , b ⩾ 1 , b ∈ N . For the numerical investigation of the fuzzy mathematical model, we use the collocation method with the addition of this newly developed operational matrix. For the feasibility and validity of our method we pick up a particular case of our model and plot the graph between the exact and numerical solutions. We see that our results have good accuracy and our method is valid for the fuzzy system of fractional ODEs. We depict the dynamics of infected, susceptible, exposed, and asymptotically infected people for the different integer and fractional orders in a fuzzy environment. We show the effect of fractional order on the suspected, exposed, infected, and asymptotic carrier by plotting graphs.

Fractional Order Model for the Role of Mild Cases in the Transmission of COVID-19.

Most of the nations with deplorable health conditions lack rapid COVID-19diagnostic test due to limited testing kits and laboratories. The un-diagnosticmild cases (who show no critical sign and symptoms) play the role as a route that spread the infection unknowingly to healthy individuals. In this paper, we present a fractional order SIR model incorporating individual with mild cases as a compartment to become SMIR model. The existence of the solutions of the model is investigated by solving the fractional Gronwall's inequality using the Laplace transform approach. The equilibrium solutions (DFE & Endemic) are found to be locally asymptotically stable, and subsequently the basic reproduction number is obtained. Also the global stability analysis is carried out by constructing Lyapunov function. Lastly, numerical simulations that support analytic solution follow. It was also shown that when the rate of infection of the mild cases increases, there is equivalent increase in the overall population of infected individuals. Hence to curtail the spread of the disease there is need to take care of the Mild cases as well.

Applying fixed point methods and fractional operators in the modelling of novel coronavirus 2019-nCoV/SARS-CoV-2.

This study aims to discuss the prevalence of COVID-19 in U.S, Italy, Spain, France and China, where the virus spreads most rapidly and causes tragic outcomes. Thereafter, we present new insights of existence and uniqueness solutions of the 2019-nCoV models via fractional and fractal-fractional operators by using fixed point methods.

A novel mathematical approach of COVID-19 with non-singular fractional derivative.

We analyze a proposition which considers new mathematical model of COVID-19 based on fractional ordinary differential equation. A non-singular fractional derivative with Mittag-Leffler kernel has been used and the numerical approximation formula of fractional derivative of function ( t - a ) n is obtained. A new operational matrix of fractional differentiation on domain [0, a], a ≥ 1, a ∈ N by using the extended Legendre polynomial on larger domain has been developed. It is shown that the new mathematical model of COVID-19 can be solved using Legendre collocation method. Also, the accuracy and validity of our developed operational matrix have been tested. Finally, we provide numerical evidence and theoretical arguments that our new model can estimate the output of the exposed, infected and asymptotic carrier with higher fidelity than the previous models, thereby motivating the use of the presented model as a standard tool for examining the effect of contact rate and transmissibility multiple on number of infected cases are depicted with graphs.

Optimal control of a fractional order model for granular SEIR epidemic with uncertainty.

In this study, we present a general formulation for the optimal control problem to a class of fuzzy fractional differential systems relating to SIR and SEIR epidemic models. In particular, we investigate these epidemic models in the uncertain environment of fuzzy numbers with the rate of change expressed by granular Caputo fuzzy fractional derivatives of order ß ∈ (0, 1]. Firstly, the existence and uniqueness of solution to the abstract fractional differential systems with fuzzy parameters and initial data are proved. Next, the optimal control problem for this fractional system is proposed and a necessary condition for the optimality is obtained. Finally, some examples of the fractional SIR and SEIR models are presented and tested with real data extracted from COVID-19 pandemic in Italy and South Korea.