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Modelling the spread of Ebola virus with Atangana-Baleanu fractional operators

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Abstract

The model of Ebola spread within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced by Atangana and Baleanu. It is expected that the proposed model will show better approximation than the models established before. The existence and uniqueness of solutions for the spread of Ebola disease model is given via the Picard-Lindelof method. Finally, numerical solutions for the model are given by using different parameter values.

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... Most of the published work describes the mathematical system of predators and prey as a problem of Cauchy type of a system of classical differential equations [21][22][23][24][25]. However, recently, there has been great interest in studying the behavior of the solution for some biological systems using fractional differential equations involving the Atangana-Baleanu operator by several authors for the purpose of investigating several real-world systems and modeling infectious diseases; see [26][27][28][29][30][31][32][33][34][35][36]. Some fractional-order models have been investigated via the new operators recently. ...
... For instance its use has been suggested for the dynamics of smoking in [32]. Along the same line, the transference model for the Ebola virus together with AB operator was studied in [31]. A fractional-order model of leptospirosis infection was considered in [26]. ...
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Abstract In this manuscript, the fractional Atangana–Baleanu–Caputo model of prey and predator is studied theoretically and numerically. The existence and Ulam–Hyers stability results are obtained by applying fixed point theory and nonlinear analysis. The approximation solutions for the considered model are discussed via the fractional Adams Bashforth method. Moreover, the behavior of the solution to the given model is explained by graphical representations through the numerical simulations. The obtained results play an important role in developing the theory of fractional analytical dynamic of many biological systems.
... The memory effect explains that future state of the fractional operator of a given function depends on its historical behavior and current state. By using fractional derivatives, particularly Caputo fractional derivatives, several real-world problems have been investigated successfully in engineering and biomathematics [14][15][16][17][18]. The definition of the fractional derivative has various approaches with different kernels, so the modelers are inter-ested in choosing the best one. ...
... Existence of the considered model is necessary from the analysis point of view. Therefore, by using the Picard-Lindel method [16], we find the existence and uniqueness of the solution. The Hyers-Ulam type stability of the extended model is discussed. ...
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The transmission dynamics of a COVID-19 pandemic model with vertical transmission is extended to nonsingular kernel type of fractional differentiation. To study the model, Atangana-Baleanu fractional operator in Caputo sense with nonsingular and nonlocal kernels is used. By using the Picard-Lindel method, the existence and uniqueness of the solution are investigated. The Hyers-Ulam type stability of the extended model is discussed. Finally, numerical simulations are performed based on real data of COVID-19 in Indonesia to show the plots of the impacts of the fractional order derivative with the expectation that the proposed model approximation will be better than that of the established classical model.
... As a result of problems that arise from the real world on the basis of statistical analysis and biological experiments, mathematical models of these problems are proposed and most of them studied. These proposed models enable scientists and researchers to study and verify the behavior of such models separately and independently in biological laboratory experiments (see [30,[40][41][42][43][44][45]). After modeling the biological phenomenon mathematically, that is, as a function of time and the parameters involved, the numerical solutions can be found and these solutions can then be represented in tables and figures. ...
... The following numerical schemes after approximating the following expressions using the Lagrange polynomial interpolation: (42), are given by ...
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In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter ρ is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of ρ=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of ρ and k. All calculations in this work are accomplished by using the Mathematica package.
... Atangana and Goufo [17] , studied generalised version of Ebola virus model and also discussed the endemic equilibrium points. In [18] , authors discussed this model using Atangana Baleanu fractional derivative and also discussed existence and uniqueness of the solutions. Some other method on Ebola disease can be found in [19][20][21] . ...
... In this paper we will use an iterative scheme to solve fractional Ebola virus model as given in [18][19][20] . Proposed iterative scheme is based on the discretization of the domain. ...
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Ebola virus is very challenging problem of the world. The main purpose of this work is to study fractional Ebola virus model. An efficient computational method based on iterative scheme is proposed to solve fractional Ebola model numerically. Stability of proposed method is also discussed. Efficiency of proposed method is shown by listing CPU time. Proposed computational method will work for long time domain. Numerical results are presented graphically. The main reason for using this technique is low computational cost and high accuracy. It is also shown how the approximate solution varies for fractional and integer order Ebola virus model.
... In [53,54] Atangana and Owolabi proposed a new form of the Adams-Bashforth approach based on the Mittag-Leffler kernel for the ABC fractional derivative approximation. The purpose of this section of the article is to demonstrate how to apply the numerical technique described in [52], which has recently been proven for accuracy and reliability [55,56] to solve any fractional differential equation. The numerical technique for approximating ABC is defined in (2). ...
... Atangana and Koca 2016 [2] explored new behavior of the chaotic system by using the Atangana-Baleanu fractional derivative. In 2018 Koca [12,13] analyzed the effect of the Atangana-Baleanu fractional operators on the Ebola virus and rubella disease. In 2018 Jajarmi and Baleanu [8] investigated the pathological behavior of HIV-infection using a new model in fractional calculus. ...
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Smoking is the most lethal social poisoning event. The World Health Organization defines smoking as the most important preventable cause of disease. Around 4.9 million people worldwide die from smoking every year. In order to analysis this matter, we aim to investigate an e-cigarette smoking model with Atangana-Baleanu fractional derivative. We obtain the existence conditions of the solution for this fractional model utilizing fixed-point theory. After giving existence conditions, the uniqueness of the solution is proved. Finally, to show the effect of the Atangana-Baleanu fractional derivative on the model, we give some numerical results supported by illustrative graphics.
... Since it is an important matter in a differential calculus to demonstrate the nature and the unicity of the elucidation to a problem, there are several works in this field (see [32][33][34][35] and references therein). The approach described in this analysis is nonlocal and non-linear, no such system gives exact solutions to be observed problem. ...
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In this article, the non-integer equations of the coupled mass-spring system with Atangana Baleanu fractional derivatives is offered. The physical entities of the structure are well-preserved by presenting an supplementary stricture χ. A nonlinear model with damping factor is considered. The existence and uniqueness problem to related model are scanned by fixed point principle. Our consequences spectacle that the mechanical components reveal viscoelastic behaviors generating temporal fractality at diverse scales and exhibit the existence of material heterogeneities in the mechanical modules. The comparison Jajarmi predictor corrector and Caputo methods is also given.
... Several numerical techniques have been suggested to solve the arbitrary-order differential equations (DEs) [2,6,8,11,16]. In recent years, modeling and analysis of infectious disease in bio-mathematical sciences with fractional operators found more attention by the researchers and some remarkable studies can be seen in [4,21,24,31,42,44]. ...
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Nowadays, the complete world is suffering from untreated infectious epidemic disease COVID-19 due to coronavirus, which is a very dangerous and deadly viral infection. So, the major desire of this task is to propose some new mathematical models for the coronavirus pandemic (COVID-19) outbreak through fractional derivatives. The adoption of modified mathematical techniques and some basic explanation in this research field will have a strong effect on progressive society fitness by controlling some diseases. The main objective of this work is to investigate the dynamics and numerical approximations for the recommended arbitrary-order coronavirus disease system. This system illustrating the probability of spread within a given general population. In this work, we considered a system of a novel COVID-19 with the three various arbitrary-order derivative operators: Caputo derivative having the power law, Caputo–Fabrizio derivative having exponential decay law and Atangana–Baleanu-derivative with generalized Mittag–Leffler function. The existence and uniqueness of the arbitrary-order system is investigated through fixed-point theory. We investigate the numerical solutions of the non-linear arbitrary-order COVID-19 system with three various numerical techniques. For study, the impact of arbitrary-order on the behavior of dynamics the numerical simulation is presented for distinct values of the arbitrary power β.
... Koca [2] investigated the Ebola virus spreading within a particular place of the population by AB-derivative. Dokuyucu and Dutta [3] discussed the model for Ebola virus with the Caputo derivative without a singular kernel in the fractional order. ...
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In the Nidovirales order of the Coronaviridae family, where the coronavirus (crown‐like spikes on the surface of the virus) causing severe infections like acute lung injury and acute respiratory distress syndrome. The contagion of this virus categorized as severed, which even causes severe damages to human life to harmless such as a common cold. In this manuscript, we discussed the SARS‐CoV‐2 virus into a system of equations to examine the existence and uniqueness results with the Atangana–Baleanu derivative by using a fixed‐point method. Later, we designed a system where we generate numerical results to predict the outcome of virus spreadings all over India.
... Most (if not all) of these mathematical models that arise from many real-life problems are proposed and studied on the basis of biological experiences or statistical analysis. It is through some of these models that the interested scientist can studied and verify the behavior of these models in isolation in a modern laboratory-type biology experiment (see [3,8,20,24] and [31] ). ...
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This article investigates a family of approximate solutions for the fractional model (in the Liouville-Caputo sense) of the Ebola virus via an accurate numerical procedure (Chebyshev spectral collocation method). We reduce the proposed epidemiological model to a system of algebraic equations with the help of the properties of the Chebyshev polynomials of the third kind. Some theorems about the convergence analysis and the existence-uniqueness solution are stated. Finally, some numerical simulations are presented for different values of the fractional-order and the other parameters involved in the coefficients. We also note that we can apply the proposed method to solve other models
... Recently, the A-B derivative has been used in modeling various real world phenomena, for example see [28][29][30][31][32][33][34]. Further, A-B derivative has also used to model various infectious diseases like Ebola virus, dynamics of smoking, Leptospirosis, etc [35][36][37][38] in more comprehensive way. To the best of our knowledge, there is very less literature to consider fractional order evolution equation with A-B fractional derivative for existence theory as well stability analysis. ...
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The current work is devoted to investigate evolution problem of nonlocal Cauchy type under “Mittag–Leffler” type fractional derivative which works as nonsingular and nonlocal kernel. For the considered problem, we establish some appropriate results devoted to the qualitative theory of existence and stability analysis. Upon using Banach and Krasnoselskii’s fixed point theorems, we establish the required analysis. Further stability theory of Ulam’s type is constructed by sing nonlinear analysis. Some interesting examples are provided to support our established results.
... Recently, the AB derivative has been used in modeling various real world phenomena, for example see [42,43]. Further, AB derivative has also used to model various infectious diseases like Ebola virus, dynamics of smoking, Leptospirosis, etc [44][45][46][47][48][49][50][51] in more comprehensive way. ...
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In the current article, we studied the novel corona virus (2019-nCoV or COVID-19) which is a threat to the whole world nowadays. We consider a fractional order epidemic model which describes the dynamics of COVID-19 under nonsingular kernel type of fractional derivative. An attempt is made to discuss the existence of the model using the fixed point theorem of Banach and Krasnoselskii’s type. We will also discuss the Ulam-Hyers type of stability of the mentioned problem. For semi analytical solution of the problem the Laplace Adomian decomposition method (LADM) is suggested to obtain the required solution. The results are simulated via Matlab by graphs. Also we have compare the simulated results with some reported real data for Commutative class at classical order.
... The existence results are investigated by using Schauders and Weissingers fixed point theorems. Many papers have been developed under various types of fractional derivatives, see [32][33][34][35][36][37][38][39][40][41][42][43][44][45] and references cited therein. ...
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In this paper, a mathematical model of generalized fractional-order is constructed to study the problem of human immunodeficiency virus (HIV) infection of CD$4^+$ T-cells with treatment. The model consists of a system of four nonlinear differential equations under the generalized Caputo fractional derivative sense. The existence results for the fractional-order HIV model are investigated via Banach’s and Leray-Schauder nonlinear alternative fixed point theorems. Further, we also established different types of Ulam’s stability results for the proposed model. The effective numerical scheme so-called predictor-corrector algorithm has been employed to analyze the approximated solution and dynamical behavior of the model under consideration. It is worth noting that, unlike many discusses recently conducted, dimensional consistency has been taken into account during the fractionalization process of the classical model.
... This definition has great advantage, especially when using Laplace transforms to solve some initial conditional physical problems. In the last three years, by taking this definition into account, better modeling and investigation of problems in different fields has been achieved successfully [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]. ...
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The newly constructed optimal perturbation iteration procedure with Laplace transform is applied to obtain the new approximate semi-analytical solutions of the fractional type of damped Burgers’ equation. The classical damped Burgers’ equation is remodeled to fractional differential form via the Atangana–Baleanu fractional derivatives described with the help of the Mittag–Leffler function. To display the efficiency of the proposed optimal perturbation iteration technique, an extended example is deeply analyzed.
... Khan et al. [40] , studied EU of solution and stability for the Lotka-Volterra system involving the ABC fractional derivative. Koca in [41] , investigated ABC fractional spread Ebola virus model for EU of solutions and demonstrated the results numerically. ...
... In this sense, Atangana and Koca [3] analyzed a nonlinear chaotic system and demonstrated new chaotic behavior under nonlocal derivative, Alkahtani et al. [1] redefined N1H1 spread model by replacing the time derivative by nonlocal fractional derivative. Koca [18,19] handled rubella and Ebola disease models under the Mittag-Leffler function as a non-local kernel and investigated system response. Toufik and Atangana [32] observed fractional nonlinear chaotic models with a newly defined numerical approximation method. ...
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In this study, we aim to comprehensively investigate a drinking model connected to immigration in terms of Atangana-Baleanu derivative in Caputo type. To do this, we firstly extend the model describing drinking model by changing the derivative with time fractional derivative having Mittag-Leffler kernel. The existence and uniqueness of the drinking model solutions together with the stability analysis is shown by the help of Banach fixed point theorem. The special solution of the model is investigated using the Sumudu transformation and then, we present some numerical simulations for the different fractional orders to emphasize the effectiveness of the used derivative. © 2021 American Institute of Mathematical Sciences. All rights reserved.
... The studies conducted over the last years according to the this improvement show us that the new fractional derivative with Mittag-Leffler kernel can be used as an effective mathematical tool for modelling the complex real-life problems, [15][16][17][18][19][20] . ...
Article
In this study, we aim to comprehensively investigate a drinking model connected to immigration in terms of Atangana-Baleanu derivative in Caputo type. To do this, we firstly extend the model describing drinking model by changing the derivative with time fractional derivative having Mittag-Leffler kernel. The existence and uniqueness of the drinking model solutions together with the stability analysis is shown by the help of Banach fixed point theorem. The special solution of the model is investigated using the Sumudu transformation and then, we present some numerical simulations for the different fractional orders to emphasize the effectiveness of the used derivative.
... Fractional-order operators have successfully been applied to model a number of mathematical problems arising from the fields like physics, chemistry, biology, ecology, finance, and engineering. Many such mathematical models have been proposed and analyzed by using different fractional-order operators as can be found in the recent studies [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. It is well known that Riemann-Liouville and Caputo-type fractional operators have singular type of kernels in the integrands of their definitions. ...
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In this research work, a mathematical model related to HIV-I cure infection therapy consisting of three populations is investigated from the fractional calculus viewpoint. Fractional version of the model under consideration has been proposed. The proposed model is examined by using the Atangana–Baleanu fractional operator in the Caputo sense (ABC). The theory of Picard–Lindelöf has been employed to prove existence and uniqueness of the special solutions of the proposed fractional-order model. Further, it is also shown that the non-negative hyper-plane R+3 is a positively invariant region for the underlying model. Finally, to analyze the results, some numerical simulations are carried out via a numerical technique recently devised for finding approximate solutions of fractional-order dynamical systems. Upon comparison of the numerical simulations, it has been demonstrated that the proposed fractional-order model is more accurate than its classical version. All the necessary computations have been performed using MATLAB R2018a with double precision arithmetic.
... A new version of the Adams-Bashforth framework based on the Mittag-Leffler Kernel was suggested by Atangana and Owolabi [47,48] to approximate the ABC derivative. In this section, our objective is to use the numerical scheme established in [46], which has been recently tested in terms of accuracy and reliability [49,50] to deal with any given fractional differential equations. For the approximation of ABC, the numerical scheme is given in Definition (3) To view the deep analysis on this numerical technique, our readers are referred to [46]. ...
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A mosquito born viral disease (Dengue) becoming endemic around the globe which including cause of severe illness and death in various Asian and Latin American countries. It needs proper management by researchers and medicine professionals. The current research work is a step towards the prevention/reduction of such deadly disease in the society. More precisely, this work addressing various mathematical proofs interconnected to the existence and stability along with numerical findings by using mathematical modeling techniques. Further, the existence results have been established for the proposed model under the Atangana-Baleanu derivative in Caputo sense (ABC) with fractional order. In continuation, we find the deterministic stability for the proposed model. Lastly, the new version of numerical approximation’s framework for the approximation of ABC fractional derivative is used to carried out the numerical simulation for the obtained results.
... More importantly, due to its nonlocal and nonsingular kernel, the accuracy superseded the other kinds of fractional operators like Caputo, Riemann-Liouville, Caputo-Fabrizio, and many others [37]. The aforementioned operator has been successfully applied to several complicated models in the field of science and engineering, we refer to [7,19,21,22,25,27,[30][31][32] and the references therein. ...
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The whole world is still shaken by the new corona virus and many countries are starting opting for the lockdown again after the first wave that already killed thousands of people. New observations also show that the virus spreads quickly during the cold period closer to winter season. On the other side, the number of new infections decreases considerably during hot period closer to summer time. The geographic structure of our planet is such that when some countries (in a hemisphere) are in their winter season, others in the other hemisphere are in their summer season. However, we have observed in the world some countries undertaking national lockdown during their summer time, which result in their economy to be hugely hit. Other factors, beside the lockdown, have also impacted negatively the socio-economic situation in affected countries. These include, among others, the human immunodeficiency virus (HIV) susceptible to combine to the new corona virus. The new corona virus is indeed recent and many of its effect and impact on the society are still unknown and are still to be uncovered. Hence we use here the of Atangana-Baleanu fractional derivative to mathematically express and analyses a model of HIV disease combined with COVID-19 to assess the pandemic situation in many countries affected, such as South Africa, United Kingdom (UK), China, Spain, United States of America (USA), and Italy. A way to achieve that is to perform stability and bifurcation analysis. It is also possible to investigate in which conditions the combined model contains a forward and a backward bifurcation. Moreover, utilizing the techniques of Schaefer and Banach contraction principle, existence and uniqueness of solutions of the generalized fractional model were presented. Also, the Atangana-Baleanu fractional (generalized) HIV-COVID-19 con-infection model is solved numerically via well-known and effective numerical scheme and a predicted prevalence for the COVID-19 is provided. The global trend shown by the numerical simulation proves that the disease will stabilize at a later stage when adequate measures are taken.
... Some disease models which are an important area in mathematical modelling are discussed [1,9,10,13]. In our paper, we have investigated the system of equations involving fractional derivatives. ...
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In our paper, the spread of SIQR model with fractional order differential equation is considered. We have evaluated the system with fractional way and investigated stability of the non-virus equilibrium point and virus equilibrium points. Also, the existence of the solutions are proved. Finally, the efficient numerical method for finding solutions of system is given.
... Khan et al. [17] provided an HIV-TB model including ABfractional derivative and analyzed the model for well-posedness, stability analysis, and numerical solutions. Koca in [18] studied the AB-fractional spread Ebola virus model for the existence of solutions and illustrated the results numerically. Khan et al. [19] considered the AB-fractional-order HIV/AIDS model and applied the fixed point theorem for the existence results and studied the stability analysis. ...
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In the present paper, we study a mathematical model of an imperfect testing infectious disease model in the sense of the Mittage-Leffler kernel. The Banach contraction principle has been used for the existence and uniqueness of solutions of the suggested model. Furthermore, a numerical method equipped with Lagrangian polynomial interpolation has been utilized for the numerical outcomes. Diagramming and discussion are used to clarify the effects of related parameters in the fractional-order imperfect testing infectious disease model.
... This is because of the powerful tools (see, e.g., [5]) that are not available in the classical calculus. In particular, FC enable researches to model in an efficient way many complicated real-world problems like COVID-19 (see [9]), HIV (see [10]), Rubella disease (see [11]), Ebola virus (see [12]), and HBV infection (see [13]). ...
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Differential equations with fractional derivative are being extensively used in the modelling of the transmission of many infective diseases like HIV, Ebola, and COVID-19. Analytical solutions are unreachable for a wide range of such kind of equations. Stability theory in the sense of Ulam is essential as it provides approximate analytical solutions. In this article, we utilize some fixed point theorem (FPT) to investigate the stability of fractional neutral integrodifferential equations with delay in the sense of Ulam-Hyers-Rassias (UHR). This work is a generalized version of recent interesting works. Finally, two examples are given to prove the applicability of our results.
... In 2016 Atangana-Baleanu with Caputo generalize the non-singular kernel to Mittag-Leffler kernel in the definition of ABC derivative . Infectious problems and many biological problems have been studied by this operator (Baleanu and Fernandez 2018;Koca 2018;Kumar et al. 2019;2020;Rahman et al. 2020;Danane et al. 2021;Rahman et al., 2021). Qualitative analysis of solution for blood alcohol and HBV arbitrary order model has been investigated in (Salman and Yousef 2017;Singh 2020). ...
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2022): Modeling and analysis of a fractional anthroponotic cutaneous leishmania model with Atangana-Baleanu derivative, ABSTRACT Very recently, Atangana and Baleanu defined a novel arbitrary order derivative having a kernel of non-locality and non-singularity, known as AB derivative. We analyze a non-integer order Anthroponotic Leshmania Cutaneous (ACL) problem exploiting this novel AB derivative. We derive equilibria of the model and compute its threshold quantity, i.e. the so-called reproduction number. Conditions for the local stability of the no-disease as well as the disease endemic states are derived in terms of the threshold quantity. The qualitative analysis for solution of the proposed problem have derived with the aid of the theory of fixed point. We use the predictor cor-rector numerical approach to solve the proposed fractional order model for approximate solution. We also provide, the numerical simulations for each of the compartment of considered model at different fractional orders along with comparison with integer order to elaborate the importance of modern derivative. The fractional investigation shows that the non-integer order derivative is more realistic about the inner dynamics of the Leishmania model lying between integer order. ARTICLE HISTORY
... This is due to the existence of many nice tools (see e.g., [1,2]) that are not available in the classical calculus. In particular, FC enables researches to model in an efficient way many complicated real world problems, e.g., COVID-19 (see [3]), Ebola virus (see [4]), and HIV (see [5]). Moreover, it has recent interesting applications in image processing (see [6]) and in diabetes (see [7]). ...
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Fractional derivatives are used to model the transmission of many real world problems like COVID-19. It is always hard to find analytical solutions for such models. Thus, approximate solutions are of interest in many interesting applications. Stability theory introduces such approximate solutions using some conditions. This article is devoted to the investigation of the stability of nonlinear differential equations with Riemann-Liouville fractional derivative. We employed a version of Banach fixed point theory to study the stability in the sense of Ulam-Hyers-Rassias (UHR). In the end, we provide a couple of examples to illustrate our results. In this way, we extend several earlier outcomes.
... The Atangana-Baleanu derivative [36] is a fractional derivative having a nonsingular and nonlocal kernel that is used to simulate physical and biological phenomena and became the pioneer to employ a fractional-order derivative in the component of a non-singular having the Mittag-Leffler function in the kernel. In several real-world situations, the ABC-fractional derivative yields more accurate results [37]. Additionally, employing the Atangana-Baleanu derivative to describe the transmission dynamics involving delay is a novelty in the research. ...
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A susceptible diabetes comorbidity model was used in the mathematical treatment to explain the predominance of mellitus. In the susceptible diabetes comorbidity model, diabetic patients were divided into three groups: susceptible diabetes, uncomplicated diabetics, and complicated diabetics. In this research, we investigate the susceptible diabetes comorbidity model and its intricacy via the Atangana-Baleanu fractional derivative operator in the Caputo sense (ABC). The analysis backs up the idea that the aforesaid fractional order technique plays an important role in predicting whether or not a person will develop diabetes after a substantial immunological assault. Using the fixed point postulates, several theoretic outcomes of existence and Ulam's stability are proposed for the susceptible diabetes comorbidity model. Meanwhile, a mathematical approach is provided for determining the numerical solution of the developed framework employing the Adams type predictor-corrector algorithm for the ABC-fractional integral operator. Numerous mathematical representations correlating to multiple fractional orders are shown. It brings up the prospect of employing this structure to generate framework regulators for glucose metabolism in type 2 diabetes mellitus patients.
... Research shows the widespread use of these types of fractional derivatives and integrals. Various investigations about different disease virus have been done by using fractional Atangana-Baleanu derivatives; For instance, different studies have been reported in the rate of spreading within the population, transmission dynamics, and immune responding to antibodies of virus of some disease, such as Rubella [20] , Ebola [21,22] , Hepatitis B [23] , HIV/AIDS [24] , and the novel Coronavirus (2019-nCov) [25][26][27] . Some researchers used the Atangana-Baleanu derivatives to solve the different applicable equations in physical phenomena, such as the elastic heat conduction equation [28] and oxygen diffusion equation [29] . ...
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Our purpose in this paper is to present some common fixed point results for αC-admissible multi-valued mappings which have greatest lower bound property. By utilizing an extended contraction condition that is more generally of than other contractions that be used in this literature, we deduce some convenient results in complex-valued double controlled metric spaces. Also, we give some significant applications of these results to the existence of the solution of Riemann–Liouville and Atangana–Baleanufractional integral inclusion systems.
... The Atangana-Baleanu successfully described the phenomena of real-world problems particularly in the field of epidemiology. Several epidemic models using the Atangana-Baleanu operator have been analyzed by several researchers recently, for instance, in [16] the author used the concept of fractional and studied a mathematical model of Ebola spread within the population, showed its existence and uniqueness of solution by using the method of Picard-Lindelof. In [17] the authors presented a new formula for the fractional differential equation with Mittag-Leffler kernel with Riemann-Liouville fractional integrals, which is accessible to handle for certain computational purposes. ...
Article
The current paper is the analysis of an Anthroponotic Cutaneous Leishmania infection caused by a parasite known as Leishmania Tropica under the Caputo fractal-fractional operator. The problem has seven compartmental agents having two categories of human and vector population. By using the fixed point theory we show the existence and uniqueness of solutions of the system. Applying the methods of basic theorems of fractal-fractional calculus and the iterative numerical techniques of fractional Adams–Bashforth method for approximate solution. For the simulation of the model, we have considered different values for fractional-order ϱ and fractal dimension χ and compare the results with integer order for real data. The fractal-fractional calculus technique is applied as a beneficial technique to know about the real-world problem and also to controls the whole world situation of the aforesaid pandemic in the different continents and territories of the world. Finally, numerical simulations are performed to elaborate on the significance of the arbitrary order derivative. Our analysis indicates that the fractal-fractional order derivatives are more informative about the complexity of the dynamics of the proposed Leishmania model.
... In [39] , the author studied and analyzed an EVD compartmental model using Atangana-Baleanu fractional derivative operators. The author carried out the existence and uniqueness of solutions for the formulated EVD model and performed numerical experiments via the newly constructed and efficient iterative scheme developed by Toufik and Atangana [40] . ...
Article
The purpose of analysing the transmission dynamism of Q fever (Coxiellosis) in livestock and incorporating ticks is to outline some management practices to minimise the spread of the disease in livestock. Ticks pass coxiellosis from an infected to a susceptible animal through a bite. The faecal matter can also contain coxiellosis, thus contaminating the environment and spreading the disease. First, a nonlinear integer order mathematical model is developed to represent the spread of this infectious disease in livestock. The proposed integer model investigates the positivity and boundedness, disease equilibria, basic reproduction number, bifurcation, and sensitivity analysis. Through mathematical analysis and numerical simulations, it shows that if the environmental transmission and the effective shedding rate of coxiella burnetii into the environment by both asymptomatic and symptomatic livestock are zero, then the usual threshold hold and it produces forward bifurcation. It is noticed that an increase in the recruitment rate of ticks produces backward bifurcation. And also, it is seen that an increase in the natural decay rate of the bacterial in the environment reduces the backward bifurcation point. Furthermore, to take care of the memory aspect of ticks on their host, we modified the initially proposed integer order model by introducing Caputo, Caputo-Fabrizio, Atangana-Baleanu fractional differential operators. The existence and uniqueness of these three newly developed fractional-order differential models are shown using the Banach fixed point theorem. Numerical trajectories are obtained for each of the fractional-order mathematical models. The trajectory of some fractional orders converges to the same endemic equilibrium point as the integer order. Finally, it is seen that the Atangana-Baleanu fractional differential operator captures more susceptibilities and fewer infections than the other operators.
... In particular, fractional calculus has been used extensively in the modelling stages in the fields of economics, chemistry, aerodynamics, physics, and polymer rheology. It should be remarked also that a certain kind of fractional derivative has been used recently to model Ebola virus (see [3]) and HIV (see [4]). Fractional differential equations with Caputo and Caputo-Fabrizio derivatives are used recently by the authors in [5] for the model of cancer-immune system. ...
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Modelling some diseases with large mortality rates worldwide, such as COVID-19 and cancer is crucial. Fractional differential equations are being extensively used in such modelling stages. However, exact analytical solutions for the solutions of such kind of equations are not reachable. Therefore, close exact solutions are of interests in many scientific investigations. The theory of stability in the sense of Ulam and Ulam–Hyers–Rassias provides such close exact solutions. So, this study presents stability results of some Caputo fractional differential equations in the sense of Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias. Two examples are introduced at the end to show the validity of our results. In this way, we generalize several recent interesting results.
... As is known, mathematical modelling frequently used to describe many nonlinear problems arising from the most areas of natural sciences such as physics, engineering, economics, biology, epidemiology and so on [2,9,13,[16][17][18]. In particular, nonlinear integral and differential equations are often used in characterization of some problems of real world such as theory of radioactive transfer [16], the modelling of the behaviour of viscoelastic materials in mechanics [29], the modelling of soft tissues like mitral valves or the aorta in the human heart [12], the modelling of tumor growth [8] and so on. ...
... A fractional-order derivative was first roused into operation by Atangana and Baleanu [30] under the rule of a generalized Mittag-Leffler function in the part of a non-singular and non-local kernel. In many real-world problems, the ABC-fractional derivative produces better results [31][32][33][34][35][36][37][38][39][40][41]. ...
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Fundamental Journal of Mathematics and Applications (FUJMA) is an international and peer-reviewed journal which publishes high quality papers on pure and applied mathematics. To be published in this journal, a paper must contain new ideas and be of interest to a wide range of readers. Similarity percentage must be less than 39%. No submission or processing fees are required. The journal appears in 2 numbers per year and has been published since 2018. Survey papers are also welcome.
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Ebola virus (EBOV) causes severe haemorrhagic fever leading to up to 90% lethality. Increasingly frequent outbreaks and the placement of EBOV in the category A list of potential biothreat agents have boosted interest in this virus. Furthermore, development of new technologies (e.g. reverse genetics systems) and extensive studies on Ebola haemorrhagic fever (EHF) in animal models have substantially expanded the knowledge on the pathogenic mechanisms that underlie this disease. Two major factors in EBOV pathogenesis are the impairment of the immune response and vascular dysfunction. Here, we attempt to summarize the current knowledge on EBOV pathogenesis focusing on these two factors and on recent progress in the development of vaccines and potential therapeutics.
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A stochastic discrete-time susceptible-exposed-infectious-recovered (SEIR) model for infectious diseases is developed with the aim of estimating parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. The incidence time series exhibit many low integers as well as zero counts requiring an intrinsically stochastic modeling approach. In order to capture the stochastic nature of the transitions between the compartmental populations in such a model we specify appropriate conditional binomial distributions. In addition, a relatively simple temporally varying transmission rate function is introduced that allows for the effect of control interventions. We develop Markov chain Monte Carlo methods for inference that are used to explore the posterior distribution of the parameters. The algorithm is further extended to integrate numerically over state variables of the model, which are unobserved. This provides a realistic stochastic model that can be used by epidemiologists to study the dynamics of the disease and the effect of control interventions.
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