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On epidemiological transition model of the Ebola virus in fractional sense
... Consider the right-hand side of the Eqs. (31)(32)(33)(34) as functions X, Y, Z and Q. ...
... Generally, NSFD is preferred for long-term integrations of epidemics since it preserves the qualitative dynamics of a system, while Euler and Runge-Kutta's methods are laborious in tuning in the quest to achieve this stability and accuracy. In the future, we shall extend our analysis of Influenza A, and other types of problems epidemic models in the sense of fractional as presented in [32,33,21]. ...
... By incorporating fractional derivatives into mathematical models, researchers can better represent and understand these complex dynamics through the memory effect, leading to more precise predictions and deeper insight into the underlying mechanisms. This increased accuracy often results in better modeling of physical, biological and engineering systems, making fractional calculus a valuable tool in a wide range of scientific disciplines (see the references [4,6,9,10,14,16,18,19,21,22,23,25,27,29]). ...
... Next, we present a sweeping iterative algorithm to solve the control and adjoint systems (28) and (25) forward and backward in time, respectively. The stability and convergence of this method have also been established in [20]. ...
... They demonstrated that this polynomial approach is significantly more efficient compared to traditional numerical methods [35]. Moreover, several analytical solutions based on the Caputo-Fabrizio operator method have been successfully developed in relation to SIR models for for Ebola virus, tumors and zombies [36][37][38][39]. ...
In this paper, we derive an exact analytical solution in parametric form for an infectious disease model (SIR) with migration and vaccination (SIRVN). Using derivatives and substitutions, we transform the SIRVN model into a nonlinear third-order differential equation, yielding a semi-analytical solution in parametric form. The model’s long-term oscillatory behavior approximates a Van der Pol-like equation with nonlinear damping. An analytic solution is derived using multi-scale expansion analysis and Laplace transform methods. Our comparison between the exact solution and data from the Jakarta outbreak shows a correlation of R² = 0.99. We demonstrate that vaccination effectively reduces the peak of the outbreak, and the system’s asymptotic stability suggests a transition from pandemic to endemic in Jakarta. Lastly, the Van der Pol-like equation reveals that the model can explain the existence of multiple outbreak waves.
... Mathematical models of ebola virus, malaria, and coronavirus disease transmission are studied by some researchers. [6][7][8] The Omnivore-predator-prey model was proposed and analyzed by Majeed et al. 9 Its existence, uniqueness, and solution boundedness are examined. The II-Holling functional response to predator-prey interactions and the nonlinear functional response to omnivore-prey interactions are both present in the model. ...
... 18,19 Its applications span various fields, including epidemiology, physics, biology, engineering, finance, and environmental science, highlighting its significance in enhancing our understanding and management of dynamic systems. 20,21 Fractional calculus finds wideranging applicability in diverse fields, including biology, economics, mathematics, physics, and control systems, 22,23 enabling effective modeling of various real-world problems. In the realm of epidemiology, the transmission dynamics of different diseases January 11, 2025 14:43 ...
... Additionally, this variation and malaria can both result in side effects, namely septic shock, ARDS, and many organ failures. The first step in diagnosing a patient with the omicron variant is to look for symptoms in those who are at very high risk, such as those who are in the healthcare industry or have had a long history of contact with someone who has the disease [13,[20][21][22][23][24]. ...
... In recent years, many researchers have focused on modelling and analysing various problems in life sciences and biological phenomena such as viruses, nervous system, etc [12][13][14][15][16]. Wavelets are utilized to focus a specified function in both space and scaling. ...
This paper presents a mathematical model to examine the effects of the coexistence of predators on single prey. Based on Caputo operators, we present a newly developed system of differential equations for the predator–prey system using wavelet method. It is well known that a system of nonlinear singular models cannot operate smoothly since they are singular and nonlinear. Therefore, with the help of this numerical approach, we have converted the system into a nonlinear system of algebraic equations by extending it through operational matrix of Legendre wavelets. Using the wavelet collocation scheme, we have calculated these unknown coefficients. It has been demonstrated in tables and graphs that the developed approach is consistent and proficient. Further bifurcation diagrams, as well as phase portraits, have been used to study the proposed system numerically and to analyse its behaviour. In addition, a nonlinear functional analysis have used to establish uniformly boundedness for the proposed model. Also we have discussed residual error analysis and Lyapunov exponent. The applicability and efficacy of this methodology have been demonstrated through this nonlinear system. Additionally, a comparison with existing results highlights the advantages of our numerical approach. All calculations have been done using MATLAB.
... Other interesting works can be found in [1,2,5,22,23], where several approaches have been used to establish stability theorems for various fractional-orders. Many studies on fractional-order systems have been devoted to modelling and analyzing various biological challenges, including viruses [17]. Scientists have also delved into tumor models [16]. ...
The paper investigates the behavior of a four-dimensional fractional-order mathematical model for cancer dynamics, focusing on the interactions between cancer cells, immune cells, host cells, and endothelial cells. This model provides new insights into the complex dynamics of cancer progression. A stability analysis of the system is performed across various fractional-order values using both Caputo and Caputo-Fabrizio fractional derivatives. The obtained results are then compared and validated by solving the governing system using the Adams-Bashforth-Moulton method. This study highlights the impact of the type fractional operator and the variation in fractional order on the system stability, which cannot be obtained from classical integer-order systems.
... Several works were proposed to mathematically analyze infectious diseases, and numerous optimization strategies have been developed to control specific diseases such as the ZIKA virus [2], HIV [3,4], malaria [5,6], tuberculosis [7,8], COVID-19 [9,10], Ebola [11,28], the influenza pandemic [12], and others [13,14,[29][30][31]. These infectious diseases generally spread from place to place, sometimes across continents. ...
In this paper, we propose a multi-cell discrete-time SIRI epidemic model that describes the spatial-temporal spread of an infectious disease in a geographical domain represented as a grid of cells (regions) divided into cells. We assume that people can move between these cells due to their connectivity. To investigate the efficacy of travel restrictions and vaccination in preventing epidemic spread, we include two control variables in our model. We aim to reduce the number of infected individuals in each cell designated as infectious by the health authority, while also minimizing the expenses associated with implementing the travel ban and vaccination efforts. We develop an optimal control approach using the definition of a supplementary function that identifies the automatic activation of travel restrictions and vaccination in every cell according to health authority decisions. We present numerical results testing the spread of epidemics without any intervention, applying travel restrictions only, implementing vaccination only, and combining the two strategies.
... Sene [49] explored the SIR epidemic model with the Mittag-Leffler fractional derivative, showing how such models can better capture the dynamics of disease spread, including stability and equilibrium analysis. Additionally, Masti, Sayevand, and Jafari [38] applied Bernstein's operational matrices to the fractional modeling of Ebola virus dynamics, demonstrating the superior accuracy of fractional derivatives over classical models. ...
... In recent years, the development of fractional models has gathered increasing attention. Mobile-immobile advection-dispersion model [5], brain tumor model [6], chaotic financial model [7], optimal control problems [8], non-linear KDV equation [9], Ebola virus model [10], and Evans model for dynamic economics [11] are some examples of * Corresponding author. ...
... Because fractional derivatives capture nonlocal and memory effects, they offer a more detailed understanding of complex biological systems than integer derivatives. By including fractional calculus into epidemiological models, researchers can get more insights into long-range interactions, anomalous diffusion, sub diffusion, and other phenomena that are commonly observed in biological processes [4][5][6][7][8][9][10][11][12][13][14][15]. Mathematical models have been used to simulate the dynamics of diseases and epidemics, including malaria [16]. ...
Diabetes is a chronic disease and a major public health concern all over the world, even when made feasible to learn about the root cause of the disease by awareness along with the methods of prevention. This study introduces a mathematical deterministic model that describes the progression of type 2 diabetes, integrating a component of awareness and a saturation treatment function named Holling type II. The presented work investigates and numerically analyzes the impact of various treatment strategies, such as consistent physical activity, a healthy diet, access to medical services, and the efficacy of treatments. The research findings indicate that the use of Holling type II treatment functions can prevent minor and major complications related to diabetes.
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.
In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro‐differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique.
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.
In this paper, we define a new type of string stability based on Mittag-Leffler function so-called Mittag-Leffler (p\alpha)-string stability. This kind of stability for a class of singularly perturbed stochastic systems of fractional order will be considered. The fractional derivative in these systems is in the local sense. String stability indicates uniform boundedness of the interconnected system if the initial cases of interconnected system be uniformly bounded. The deduction of the sufficient conditions of stability is based on a mixture of the concept of the Mittag-Leffler stability with the notion of p-mean string stability of singularly perturbed stochastic systems. In the sequel, our purpose is to investigate the full order system in their lower order subsystems, i.e., the reduced order system and the boundary layer correction.
Mathematical Modeling has emerged as a vital tool for understanding the dynamics of the spread of many infectious diseases, one amongst is Ebola virus. The main focus of this paper is to model mathematically the transmission dynamics of Ebola virus. For this purpose we tend to use basic SIR model of Ebola Virus to predict the outbreak of the diseases. As we cannot fully solve the 3 basic equations of SIR model with a certain formula solution, we introduce Euler and fourth-order Runge-Kutta methods (RK4). These two proposed strategies are quite efficient and practically well suited for solving initial value problem (IVP) for ordinary differential equations (ODE).We discuss the numerical comparisons between Euler method and Runge-Kutta methods and also discuss regarding their performances with the actual data. The population that we used for this model had roughly a similar number of individuals as the number was living in Republic of Liberia during 2014.
In the present case, we propose the correct version of the fractional Adams-Bashforth methods which take into account the nonlinearity of the kernels including the power law for the Riemann-Liouville type, the exponential decay law for the Caputo-Fabrizio case and the Mittag-Leffler law for the Atangana-Baleanu scenario. The Adams-Bashforth method for fractional differentiation suggested and are commonly use in the literature nowadays is not mathematically correct and the method was derived without taking into account the nonlinearity of the power law kernel. Unlike the proposed version found in the literature, our approximation, in all the cases, we are able to recover the standard case whenever the fractional power .
In this work, we introduce fractional Mellin transform of order α; 0 < α ≤ 1 on a function
which belongs to the Lizorkin space. Further, some properties and applications of fractional
Mellin transform are given.
Ebola is a world health problem and with a recent outbreak. There exist different models in the literature to predict its behavior, most of them based on data coming from previous outbreaks or using restricted number of persons in the population variable. This paper deals both with classical and fractional order SEIR (susceptible, exposed, infections, removed) Ebola epidemic model and its comparison with real data extracted from the reports periodically published by the World Health Organization (WHO), starting from March 27th, 2014. As it has been shown in the literature, one physical meaning of the fractional order in fractional derivatives is that of index of memory; and therefore, it seems to be useful for epidemic models, as in this paper. The number of confirmed cases by the WHO in its reports is used for our analysis and estimation of the parameters in our classical and fractional SEIR models. Our approach gives a good approximation to real data. Following our results, the current outbreak will continue for approximately two years, assuming that no new outbreak appears at a different community or country. Our estimates give a number of the order nine million confirmed cases.
The Ebola virus is currently one of the most virulent pathogens for humans. The latest major outbreak occurred in Guinea, Sierra Leone, and Liberia in 2014. With the aim of understanding the spread of infection in the affected countries,
it is crucial to modelize the virus and simulate it. In this paper, we begin by studying a simple mathematical model that describes the 2014 Ebola outbreak in Liberia. Then, we use numerical simulations and available data provided by the World Health Organization to validate the obtained mathematical model. Moreover, we develop a new mathematical model including vaccination of individuals.
We discuss different cases of vaccination in order to predict the effect of vaccination on the infected individuals over time. Finally, we apply optimal control to study the impact of vaccination on the spread of the Ebola virus. The optimal control problem is solved numerically by using a direct multiple shooting method.
For a given West African country, we constructed a model describing the spread of the
deathly disease called Ebola hemorrhagic fever. The model was first constructed using the
classical derivative and then converted to the generalized version using the beta- derivative.
We studied in detail the endemic equilibrium points and provided the Eigen values
associated using the Jacobian method. We furthered our investigation by solving the model
numerically using an iteration method. The simulations were done in terms of time and beta.
The study showed that, for small portion of infected individuals, the whole country could die
out in a very short period of time in case there is not good prevention
In Fractional Calculus (FC), the Laplace and the Fourier integral transforms are traditionally employed for solving different problems. In this paper, we demonstrate the role of the Mellin integral transform in FC. We note that the Laplace integral transform, the sin- and cos-Fourier transforms, and the FC operators can all be represented as Mellin convolution type integral transforms. Moreover, the special functions of FC are all particular cases of the Fox H-function that is defined as an inverse Mellin transform of a quotient of some products of the Gamma functions.
In this paper, several known and some new applications of the Mellin integral transform to different problems in FC are exemplarily presented. The Mellin integral transform is employed to derive the inversion formulas for the FC operators and to evaluate some FC related integrals and in particular, the Laplace transforms and the sin- and cos-Fourier transforms of some special functions of FC. We show how to use the Mellin integral transform to prove the Post-Widder formula (and to obtain its new modi-fication), to derive some new Leibniz type rules for the FC operators, and to get new completely monotone functions from the known ones.
Fractional (non-integer order) calculus can provide a concise model for the description of the dynamic events that occur in biological tissues. Such a description is important for gaining an understanding of the underlying multiscale processes that occur when, for example, tissues are electrically stimulated or mechanically stressed. The mathematics of fractional calculus has been applied successfully in physics, chemistry, and materials science to describe dielectrics, electrodes and viscoelastic materials over extended ranges of time and frequency. In heat and mass transfer, for example, the half-order fractional integral is the natural mathematical connection between thermal or material gradients and the diffusion of heat or ions. Since the material properties of tissue arise from the nanoscale and microscale architecture of subcellular, cellular, and extracellular networks, the challenge for the bioengineer is to develop new dynamic models that predict macroscale behavior from microscale observations and measurements. In this paper we describe three areas of bioengineering research (bioelectrodes, biomechanics, bioimaging) where fractional calculus is being applied to build these new mathematical models.
The taxonomy of the family Filoviridae (marburgviruses and ebolaviruses) has changed several times since the discovery of its members, resulting in a plethora of species and virus names and abbreviations. The current taxonomy has only been partially accepted by most laboratory virologists. Confusion likely arose for several reasons: species names that consist of several words or which (should) contain diacritical marks, the current orthographic identity of species and virus names, and the similar pronunciation of several virus abbreviations in the absence of guidance for the correct use of vernacular names. To rectify this problem, we suggest (1) to retain the current species names Reston ebolavirus, Sudan ebolavirus, and Zaire ebolavirus, but to replace the name Cote d'Ivoire ebolavirus [sic] with Taï Forest ebolavirus and Lake Victoria marburgvirus with Marburg marburgvirus; (2) to revert the virus names of the type marburgviruses and ebolaviruses to those used for decades in the field (Marburg virus instead of Lake Victoria marburgvirus and Ebola virus instead of Zaire ebolavirus); (3) to introduce names for the remaining viruses reminiscent of jargon used by laboratory virologists but nevertheless different from species names (Reston virus, Sudan virus, Taï Forest virus), and (4) to introduce distinct abbreviations for the individual viruses (RESTV for Reston virus, SUDV for Sudan virus, and TAFV for Taï Forest virus), while retaining that for Marburg virus (MARV) and reintroducing that used over decades for Ebola virus (EBOV). Paying tribute to developments in the field, we propose (a) to create a new ebolavirus species (Bundibugyo ebolavirus) for one member virus (Bundibugyo virus, BDBV); (b) to assign a second virus to the species Marburg marburgvirus (Ravn virus, RAVV) for better reflection of now available high-resolution phylogeny; and (c) to create a new tentative genus (Cuevavirus) with one tentative species (Lloviu cuevavirus) for the recently discovered Lloviu virus (LLOV). Furthermore, we explain the etymological derivation of individual names, their pronunciation, and their correct use, and we elaborate on demarcation criteria for each taxon and virus.
The aim of this article is to investigate an efficient computational method for solving distributed‐order fractional optimal control problems. In the proposed method, a new Riemann‐Liouville fractional integral operator for the Bernstein wavelet is given. This approach is based on a combination of the Bernstein wavelets basis, fractional integral operator, Gauss‐Legendre numerical integration, and Newton's method for solving obtained system. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed method. The error analysis of the proposed method is carried out. Examples reveal the applicability of the proposed technique.
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
Ebola virus disease (EVD) is a severe and frequently lethal disease caused by Ebola virus (EBOV). EVD outbreaks typically start from a single case of probable zoonotic transmission, followed by human-to-human transmission via direct contact or contact with infected bodily fluids or contaminated fomites. EVD has a high case–fatality rate; it is characterized by fever, gastrointestinal signs and multiple organ dysfunction syndrome. Diagnosis requires a combination of case definition and laboratory tests, typically real-time reverse transcription PCR to detect viral RNA or rapid diagnostic tests based on immunoassays to detect EBOV antigens. Recent advances in medical countermeasure research resulted in the recent approval of an EBOV-targeted vaccine by European and US regulatory agencies. The results of a randomized clinical trial of investigational therapeutics for EVD demonstrated survival benefits from two monoclonal antibody products targeting the EBOV membrane glycoprotein. New observations emerging from the unprecedented 2013–2016 Western African EVD outbreak (the largest in history) and the ongoing EVD outbreak in the Democratic Republic of the Congo have substantially improved the understanding of EVD and viral persistence in survivors of EVD, resulting in new strategies toward prevention of infection and optimization of clinical management, acute illness outcomes and attendance to the clinical care needs of patients. Ebola virus disease (EVD) is caused by the filovirus Ebola virus (EBOV). Although the natural host of EBOV is undefined, a single zoonotic transmission is the probable start of most EVD outbreaks. EVD is characterized by gastrointestinal manifestations and multiple organ dysfunction syndrome and has a high case–fatality rate.
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.
The narrative of Ebola's arrival in the United States has been overwhelmed by our fear of a West African-style epidemic. The real story of Ebola's arrival is about our healthcare system's failure to identify, treat, and contain healthcare associated infections. Having long been willfully ignorant of the path of fatal infectious diseases through our healthcare facilities, this paper considers why our reimbursement and quality reporting systems made it easy for this to be so. West Africa's challenges in controlling Ebola resonate with our own struggles to standardize, centralize, and enforce infection control procedures in American healthcare facilities.
Let E1, E2 be two Banach spaces, and let f: E1 → E2 be a mapping, that is “approximately linear”. S. M. Ulam posed the problem: ”Give conditions in order for a linear mapping near an approximately linear mapping to exist”. The purpose of this paper is to give an answer to Ulam’s problem.
We review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics. The problems in continuum mechanics concern mathematical modelling of viscoelastic bodies (§1), and unsteady motion of a particle in a viscous fluid, i.e. the Basset problem (§2). In the former analysis fractional calculus leads us to introduce intermediate models of viscoelasticity which generalize the classical spring-dashpot models. The latter analysis leads us to introduce a hydrodynamic model suitable to revisit in §3 the classical theory of the Brownian motion, which is a relevant topic in statistical mechanics. By the tools of fractional calculus we explain the long tails in the velocity correlation and in the displacement variance. In §4 we consider the fractional diffusion-wave equation, which is obtained from the classical diffusion equation by replacing the first-order time derivative by a fractional derivative of order α with 0 α α α
An algorithm for approximating solutions to differential equations in a modified new Bernstein polynomial basis is introduced. The algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the Galerkin method to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of Bernstein polynomials and the procedure is much simpler compared to the piecewise B spline method for solving differential equations. A recursive definition of the Bernstein polynomials and their derivatives are also presented. The current procedure is implemented to solve three linear equations and one nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations.
In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.
Ebola continues to attract worldwide attention as a highly lethal virus of unknown origin that leaves victims bleeding to death and has no known vaccine or cure. The purpose of this historical research was to review and analyze the primary and secondary sources available on Ebola for use by primary care nurses in the event of future outbreaks. A rich resource of history has been well documented by some of the original physicians, virologists, and members of international teams, but nothing was found to be documented by nurses during these outbreaks. Multiple themes emerged including the origins of the viral strains of Ebola, transmission factors, epidemiology, virology, nonhuman and genetic research, treatment, and clinical implications. This research will provide primary care nurses with historical information about Ebola to help in future treatment options and algorithm development.
solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.
Mathematical model of the spread and control of Ebola virus disease
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Analysis of a ten compartmental mathematical model of malaria transmission
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Fractional Calculus in Bbioengineering
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Insect-eating bat may be origin of Ebola outbreak, new study suggests
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Fractional treatment: An accelerated mass-spring system
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Nonlinear oscillation with fractional derivative and its applications
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