Study of Fractional Order Dynamical System of Viral Infection Disease under Piecewise Derivative

This research aims to understand the fractional order dynamics of the deadly Nipah virus(NiV)disease.We focus on using piecewise derivatives in the context of classical and singular kernels of power operators in the Caputo sense to investigate the crossover behavior of the considered dynamical system.We establish some qualitative results about the existence and uniqueness of the solution to the proposed problem.By utilizing the Newtonian polynomials interpolation technique,we recall a powerful algorithm to interpret the numerical findings for the aforesaid model.Here,we remark that the said viral infection is caused by an RNA type virus which can transmit from animals and also from an infected person to person.Fruits bats which are also known as flying foxes are one of the sources of transmission of NiV disease.Here in this work,we investigate its transmission mechanism through some new concepts of fractional calculus for further analysis and prediction.We present the approximate results for different compartments using different fractional orders.By using the piecewise derivative concept,we detect the crossover ormulti-steps behavior in the transmission dynamics of the mentioned disease.Therefore,the considered form of the derivative is used to deal with problems exhibiting crossover behaviors.

On Nonlinear Conformable Fractional Order Dynamical System via Differential Transform Method

This article studies a nonlinear fractional order Lotka-Volterra prey-predator type dynamical system.For the proposed study,we consider the model under the conformable fractional order derivative(CFOD).We investigate the mentioned dynamical system for the existence and uniqueness of at least one solution.Indeed,Schauder and Banach fixed point theorems are utilized to prove our claim.Further,an algorithm for the approximate analytical solution to the proposed problem has been established.In this regard,the conformable fractional differential transform(CFDT)technique is used to compute the required results in the form of a series.Using Matlab-16,we simulate the series solution to illustrate our results graphically.Finally,a comparison of our solution to that obtained for the Caputo fractional order derivative via the perturbation method is given.

A Theoretical Investigation of the SARS-CoV-2Model via Fractional Order Epidemiological Model

We propose a theoretical study investigating the spread of the novel coronavirus(COVID-19)reported inWuhan City of China in 2019.We develop a mathematical model based on the novel corona virus’s characteristics and then use fractional calculus to fractionalize it.Various fractional order epidemicmodels have been formulated and analyzed using a number of iterative and numerical approacheswhile the complications arise due to singular kernel.We use the well-known Caputo-Fabrizio operator for the purposes of fictionalization because this operator is based on the non-singular kernel.Moreover,to analyze the existence and uniqueness,we will use the well-known fixed point theory.We also prove that the considered model has positive and bounded solutions.We also draw some numerical simulations to verify the theoretical work via graphical representations.We believe that the proposed epidemic model will be helpful for health officials to take some positive steps to control contagious diseases.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Quasi Controlled K-Metric Spaces over C^(*)-Algebras with an Application to Stochastic Integral Equations

Generally,the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models.C^(∗)-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research.The concept of a C^(∗)-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space.In fact,It is a generalization by replacing the set of real numbers with a C^(∗)-algebra.After that,this line of research continued,where several fixed point results have been obtained in the framework of C^(∗)-algebra valued metric,aswell as(more general)C^(∗)-algebra-valued b-metric spaces andC^(∗)-algebra-valued extended b-metric spaces.Very recently,based on the concept and properties of C^(∗)-algebras,we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case.In this paper,we first introduce the concept of C^(∗)-algebra-valued quasi-controlledK-metric spaces and prove some fixed point theorems that remain valid in this setting.To support our main results,we also furnish some exampleswhichdemonstrate theutility of ourmainresult.Finally,as an application,we useour results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.

Fractional Order Modeling of Predicting COVID-19 with Isolation and Vaccination Strategies in Morocco

In this work,we present a model that uses the fractional order Caputo derivative for the novel Coronavirus disease 2019(COVID-19)with different hospitalization strategies for severe and mild cases and incorporate an awareness program.We generalize the SEIR model of the spread of COVID-19 with a private focus on the transmissibility of people who are aware of the disease and follow preventative health measures and people who are ignorant of the disease and do not follow preventive health measures.Moreover,individuals with severe,mild symptoms and asymptomatically infected are also considered.The basic reproduction number(R0)and local stability of the disease-free equilibrium(DFE)in terms of R0 are investigated.Also,the uniqueness and existence of the solution are studied.Numerical simulations are performed by using some real values of parameters.Furthermore,the immunization of a sample of aware susceptible individuals in the proposed model to forecast the effect of the vaccination is also considered.Also,an investigation of the effect of public awareness on transmission dynamics is one of our aim in this work.Finally,a prediction about the evolution of COVID-19 in 1000 days is given.For the qualitative theory of the existence of a solution,we use some tools of nonlinear analysis,including Lipschitz criteria.Also,for the numerical interpretation,we use the Adams-Moulton-Bashforth procedure.All the numerical results are presented graphically.

Theory and Semi-Analytical Study of Micropolar Fluid Dynamics through a Porous Channel

In this work,We are looking at the characteristics of micropolar flow in a porous channel that’s being driven by suction or injection.The working of the fluid is described in the flowmodel.We can reduce the governing nonlinear partial differential equations(PDEs)to a model of coupled systems of nonlinear ordinary differential equations using similarity variables(ODEs).In order to obtain the results of a coupled system of nonlinear ODEs,we discuss a method which is known as the differential transform method(DTM).The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs.To observe beast agreement between analytical method and numerical method,we compare our result with the Rung-Kutta method of order four(RK4).We also provide simulation plots to the obtained result by using Mathematica.Onthese plots,we discuss the effect of different parameters which arise during the calculation of the flow model equations.

Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials

In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices.Further,operational matrices are constructed using variable order differentiation and integration.We are finding the operationalmatrices of variable order differentiation and integration by omitting the discretization of data.With the help of aforesaid matrices,considered FDEs are converted to algebraic equations of Sylvester type.Finally,the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions.Some examples are given to check the proposed method’s accuracy and graphical representations.Exact and numerical solutions are also compared in the paper for some examples.The efficiency of the method can be enhanced further by increasing the scale level.

Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

In this manuscript,our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces.We establish some fixed point theorems in this setting.Also,we plot some graphs of an example of obtained result for better understanding.We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space.Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions.Triangular conorms are known as dual operations of triangular norms.The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.

On Riemann-Type Weighted Fractional Operators and Solutions to Cauchy Problems

In this paper,we establish the new forms of Riemann-type fractional integral and derivative operators.The novel fractional integral operator is proved to be bounded in Lebesgue space and some classical fractional integral and differential operators are obtained as special cases.The properties of new operators like semi-group,inverse and certain others are discussed and its weighted Laplace transform is evaluated.Fractional integro-differential freeelectron laser(FEL)and kinetic equations are established.The solutions to these new equations are obtained by using the modified weighted Laplace transform.The Cauchy problem and a growth model are designed as applications along with graphical representation.Finally,the conclusion section indicates future directions to the readers.

On Fuzzy Conformable Double Laplace Transform with Applications to Partial Differential Equations

The Laplace transformation is a very important integral transform,and it is extensively used in solving ordinary differential equations,partial differential equations,and several types of integro-differential equations.Our purpose in this study is to introduce the notion of fuzzy double Laplace transform,fuzzy conformable double Laplace transform(FCDLT).We discuss some basic properties of FCDLT.We obtain the solutions of fuzzy partial differential equations(both one-dimensional and two-dimensional cases)through the double Laplace approach.We demonstrate through numerical examples that our proposed method is very successful and convenient for resolving partial differential equations.

MHD Maxwell Fluid with Heat Transfer Analysis under Ramp Velocity and Ramp Temperature Subject to Non-Integer Differentiable Operators

The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic(MHD)time-dependent Maxwell fluid over an unbounded plate embedded in a permeable medium.Non-dimensional parameters along with Laplace transformation and inversion algorithms are used to find the solution of shear stress,energy,and velocity profile.Recently,new fractional differential operators are used to define ramped temperature and ramped velocity.The obtained analytical solutions are plotted for different values of emerging parameters.Fractional time derivatives are used to analyze the impact of fractional parameters(memory effect)on the dynamics of the fluid.While making a comparison,it is observed that the fractional-order model is best to explain the memory effect as compared to classical models.Our results suggest that the velocity profile decrease by increasing the effective Prandtl number.The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.The incremental value of the M is observed for a decrease in the velocity field,which reflects to control resistive force.Further,it is noted that the Atangana-Baleanu derivative in Caputo sense(ABC)is the best to highlight the dynamics of the fluid.The influence of pertinent parameters is analyzed graphically for velocity and energy profile.Expressions for skin friction and Nusselt number are also derived for fractional differential operators.

Analysis of Twitter Data Using Evolutionary Clustering during the COVID-19 Pandemic

People started posting textual tweets on Twitter as soon as the novel coronavirus(COVID-19)emerged.Analyzing these tweets can assist institutions in better decision-making and prioritizing their tasks.Therefore,this study aimed to analyze 43 million tweets collected between March 22 and March 30,2020 and describe the trend of public attention given to the topics related to the COVID-19 epidemic using evolutionary clustering analysis.The results indicated that unigram terms were trended more frequently than bigram and trigram terms.A large number of tweets about the COVID-19 were disseminated and received widespread public attention during the epidemic.The high-frequency words such as“death”,“test”,“spread”,and“lockdown”suggest that people fear of being infected,and those who got infection are afraid of death.The results also showed that people agreed to stay at home due to the fear of the spread,and they were calling for social distancing since they become aware of the COVID-19.It can be suggested that social media posts may affect human psychology and behavior.These results may help governments and health organizations to better understand the psychology of the public,and thereby,better communicate with them to prevent and manage the panic.

Utilization of Machine Learning Methods in Modeling Specific Heat Capacity of Nanofluids

Nanofluids are extensively applied in various heat transfer mediums for improving their heat transfer characteristics and hence their performance.Specific heat capacity of nanofluids,as one of the thermophysical properties,performs principal role in heat transfer of thermal mediums utilizing nanofluids.In this regard,different studies have been carried out to investigate the influential factors on nanofluids specific heat.Moreover,several regression models based on correlations or artificial intelligence have been developed for forecasting this property of nanofluids.In the current review paper,influential parameters on the specific heat capacity of nanofluids are introduced.Afterwards,the proposed models for their forecasting and modeling are proposed.According to the reviewed works,concentration and properties of solid structures in addition to temperature affect specific heat capacity to large extent and must be considered as inputs for the models.Moreover,by using other effective factors,the accuracy and comprehensive of the models can be modified.Finally,some suggestions are offered for the upcoming works in the relevant topics.

Qualitative Analysis of a Fractional Pandemic Spread Model of the Novel Coronavirus (COVID-19)

In this study,we classify the genera of COVID-19 and provide brief information about the root of the spread and the transmission from animal(natural host)to humans.We establish a model of fractional-order differential equations to discuss the spread of the infection from the natural host to the intermediate one,and from the intermediate one to the human host.At the same time,we focus on the potential spillover of bat-borne coronaviruses.We consider the local stability of the co-existing critical point of the model by using the Routh–Hurwitz Criteria.Moreover,we analyze the existence and uniqueness of the constructed initial value problem.We focus on the control parameters to decrease the outbreak from pandemic form to the epidemic by using both strong and weak Allee Effect at time t.Furthermore,the discretization process shows that the system undergoes Neimark–Sacker Bifurcation under specific conditions.Finally,we conduct a series of numerical simulations to enhance the theoretical findings.

Modeling of Heart Rate Variability Using Time-Frequency Representations

The heart rate variability signal is highly correlated with the respiration even at high workload exercise.It is also known that this phenomenon still exists during increasing exercise.In the current study,we managed to model this correlation during increasing exercise using the time varying integral pulse frequency modulation(TVIPFM)model that relates the mechanical modulation(MM)to the respiration and the cardiac rhythm.This modulation of the autonomic nervous system(ANS)is able to simultaneously decrease sympathetic and increase parasympathetic activity.The TVIPFM model takes into consideration the effect of the increasing exercise test,where the effect of a time-varying threshold on the heart period is studied.Our motivation is to analyze the heart rate variability(HRV)acquired by time varying integral pulse frequency modulation using time frequency representations.The estimated autonomic nervous system(ANS)modulating signal is filtered throughout the respiration using a time varying filtering,during exercise stress testing.And after summing power of the filtered signal,we compare the power of the filtered modulation of the ANS obtained with different time frequency representations:smoothed pseudo Wigner–Ville representation,spectrogram and their reassignments.After that,we used a student t-test p<0.01 to compare the power of heart rate variability in the frequency band of respiration and elsewhere.

A Fractal-Fractional Model for the MHD Flow of Casson Fluid in a Channel

An emerging definition of the fractal-fractional operator has been used in this study for the modeling of Casson fluid flow.The magnetohydrodynamics flow of Casson fluid has cogent in a channel where the motion of the upper plate generates the flow while the lower plate is at a static position.The proposed model is non-dimensionalized using the Pi-Buckingham theorem to reduce the complexity in solving the model and computation time.The non-dimensional fractal-fractional model with the power-law kernel has been solved through the Laplace transform technique.The Mathcad software has been used for illustration of the influence of various parameters,i.e.,Hartman number,fractal,fractional,and Casson fluid parameters on the velocity of fluid flow.Through graphs and tables,the results have been implemented and it is shown that the boundary conditions are fully satisfied.The results reveal that the flow velocity is decreasing with the increasing values of the Hartman number and is increasing with the increasing values of the Casson fluid parameter.The findings of the fractal-fractional model have elucidated that the memory effect of the flow model has higher quality than the simple fractional and classical models.Furthermore,to show the validity of the obtained closed-form solutions,special cases have been obtained which are in agreement with the already published solutions.

Effect of Weather on the Spread of COVID-19 Using Eigenspace Decomposition

Since the end of 2019,the world has suffered from a pandemic of the disease called COVID-19.WHO reports show approximately 113M confirmed cases of infection and 2.5 M deaths.All nations are affected by this nightmare that continues to spread.Widespread fear of this pandemic arose not only from the speed of its transmission:a rapidly changing“normal life”became a fear for everyone.Studies have mainly focused on the spread of the virus,which showed a relative decrease in high temperature,low humidity,and other environmental conditions.Therefore,this study targets the effect of weather in considering the spread of the novel coronavirus SARS-CoV-2 for some confirmed cases in Iraq.The eigenspace decomposition technique was used to analyze the effect of weather conditions on the spread of the disease.Our theoretical findings showed that the average number of confirmed COVID-19 cases has cyclic trends related to temperature,humidity,wind speed,and pressure.We supposed that the dynamic spread of COVID-19 exists at a temperature of 130 F.The minimum transmission is at 120 F,while steady behavior occurs at 160 F.On the other hand,during the spread of COVID-19,an increase in the rate of infection was seen at 125%humidity,where the minimum spread was achieved at 200%.Furthermore,wind speed showed the most significant effect on the spread of the virus.The spread decreases with a wind speed of 45 KPH,while an increase in the infectious spread appears at 50 KPH.

Dynamical study of varicella-zoster virus model in sense of Mittag-Leffler kernel

The primary varicella-zoster virus(VzV)infection that causes chickenpox(also known as varicella),spreads quickly among people and,in severe circumstances,can cause to fever and encephalitis.In this paper,the Mittag-Leffler fractional operator is used to examine the mathematical representation of the vzV.Five fractional-order differential equations are created in terms of the disease's dynamical analysis such as S:Susceptible,V:Vaccinated,E:Exposed,I:Infectious and R:Recovered.We derive the existence criterion,positive solution,Hyers-Ulam stability,and boundedness of results in order to examine the suggested fractional-order model's wellposedness.Finally,some numerical examples for the VzV model of various fractional orders are shown with the aid of the generalized Adams-Bashforth-Moulton approach to show the viability of the obtained results.

A Dynamical Study of Modeling the Transmission of Typhoid Fever throughDelayed Strategies

This study analyzes the transmission of typhoid fever caused by Salmonella typhi using a mathematical model thathighlights the significance of delay in its effectiveness.Time delays can affect the nature of patterns and slow downthe emergence of patterns in infected population density.The analyzed model is expanded with the equilibriumanalysis,reproduction number,and stability analysis.This study aims to establish and explore the non-standardfinite difference(NSFD)scheme for the typhoid fever virus transmission model with a time delay.In addition,the forward Euler method and Runge-Kutta method of order 4(RK-4)are also applied in the present research.Some significant properties,such as convergence,positivity,boundedness,and consistency,are explored,and theproposed scheme preserves all the mentioned properties.The theoretical validation is conducted on how NSFDoutperforms other methods in emulating key aspects of the continuous model,such as positive solution,stability,and equilibrium about delay.Hence,the above analysis also shows some of the limitations of the conventional finitedifference methods,such as forward Euler and RK-4 in simulating such critical behaviors.This becomes moreapparent when using larger steps.This indicated that NSFD is beneficial in identifying the essential characteristicsof the continuous model with higher accuracy than the traditional approaches.