A Hybrid SIR-Fuzzy Model for Epidemic Dynamics:A Numerical Study

This study focuses on the urgent requirement for improved accuracy in diseasemodeling by introducing a newcomputational framework called the Hybrid SIR-Fuzzy Model.By integrating the traditional Susceptible-Infectious-Recovered(SIR)modelwith fuzzy logic,ourmethod effectively addresses the complex nature of epidemic dynamics by accurately accounting for uncertainties and imprecisions in both data and model parameters.The main aim of this research is to provide a model for disease transmission using fuzzy theory,which can successfully address uncertainty in mathematical modeling.Our main emphasis is on the imprecise transmission rate parameter,utilizing a three-part description of its membership level.This enhances the representation of disease processes with greater complexity and tackles the difficulties related to quantifying uncertainty in mathematical models.We investigate equilibrium points for three separate scenarios and perform a comprehensive sensitivity analysis,providing insight into the complex correlation betweenmodel parameters and epidemic results.In order to facilitate a quantitative analysis of the fuzzy model,we propose the implementation of a resilient numerical scheme.The convergence study of the scheme demonstrates its trustworthiness,providing a conditionally positive solution,which represents a significant improvement compared to current forward Euler schemes.The numerical findings demonstrate themodel’s effectiveness in accurately representing the dynamics of disease transmission.Significantly,when the mortality coefficient rises,both the susceptible and infected populations decrease,highlighting the model’s sensitivity to important epidemiological factors.Moreover,there is a direct relationship between higher Holling type rate values and a decrease in the number of individuals who are infected,as well as an increase in the number of susceptible individuals.This correlation offers a significant understanding of how many elements affect the consequences of an epidemic.Our objective is to enhance decision-making in public health by providing a thorough quantitative analysis of the Hybrid SIR-Fuzzy Model.Our approach not only tackles the existing constraints in disease modeling,but also paves the way for additional investigation,providing a vital instrument for researchers and policymakers alike.

Numerical Simulation of Bed Load and Suspended Load Sediment Transport Using Well-Balanced Numerical Schemes

Sediment transport can be modelled using hydrodynamic models based on shallow water equations coupled with the sediment concentration conservation equation and the bed con-servation equation.The complete system of equations is made up of the energy balance law and the Exner equations.The numerical solution for this complete system is done in a seg-regated manner.First,the hyperbolic part of the system of balance laws is solved using a finite volume scheme.Three ways to compute the numerical flux have been considered,the Q-scheme of van Leer,the HLLCS approximate Riemann solver,and the last one takes into account the presence of non-conservative products in the model.The discretisation of the source terms is carried out according to the numerical flux chosen.In the second stage,the bed conservation equation is solved by using the approximation computed for the system of balance laws.The numerical schemes have been validated making comparisons between the obtained numerical results and the experimental data for some physical experiments.The numerical results show a good agreement with the experimental data.

Numerical Simulation of Diffusion Type Traffic Flow Model Using Second-Order Lax-Wendroff Scheme Based on Exponential Velocity Density Function

In order to control traffic congestion, many mathematical models have been used for several decades. In this paper, we study diffusion-type traffic flow model based on exponential velocity density relation, which provides a non-linear second-order parabolic partial differential equation. The analytical solution of the diffusion-type traffic flow model is very complicated to approximate the initial density of the Cauchy problem as a function of x from given data and it may cause a huge error. For the complexity of the analytical solution, the numerical solution is performed by implementing an explicit upwind, explicitly centered, and second-order Lax-Wendroff scheme for the numerical solution. From the comparison of relative error among these three schemes, it is observed that Lax-Wendroff scheme gives less error than the explicit upwind and explicit centered difference scheme. The numerical, analytical analysis and comparative result discussion bring out the fact that the Lax-Wendroff scheme with exponential velocity-density relation of diffusion type traffic flow model is suitable for the congested area and shows a better fit in traffic-congested regions.

Identification of Forcing Mechanisms of Convective Initiation over Mountains through High-Resolution Numerical Simulations

Convection and its ensuing severe weather, such as heavy rainfall, hail, tornado, and high wind, have significant im- pacts on our society and economy （e.g., Cao et al., 2004; Fritsch and Carbone, 2004; Verbout et al., 2006; Ashley and Black, 2008; Cao, 2008; Cao and Ma, 2009; Zhang et al., 2014）. Due to its localized and transient nature, the initiation of convection or convective initiation remains one of the least understood aspects of convection in the scientific communi- ties, and it is a significant challenge to accurately predict the exact timing and location of convective initiation （e.g., Cai et al., 2006; Wilson and Roberts, 2006; Xue and Martin, 2006; Cao and Zhang, 2016）.

NUMERICAL SIMULATION OF SUPERSONIC AXISYMMETRIC FLOW OVER MISSILE AFTERBODY WITH JET EXHAUST USING POSITIVE SCHEMES

Supersonic axisymmetric jet flow over a missile afterbody containing exhaust jet is simulated using the second order accurate positive schemes method developed for solving the axisymmetric Euler equations based on the 2-D conservation laws.Comparisons between the numerical results and the experimental measurements show excellent agreements.The computed results are in good agreement with the numerical solutions obtained by using third order accurate RKDG finite element method.The results show larger gradient at discontinuous points compared with those obtained by second order accurate TVD schemes.It indicates that the presented method is efficient and reliable for solving the axisymmetric jet with external freestream flows,and shows that the method captures shocks well without numerical noise.

An Application of Finite Volume WENO Schemeto Numerical Modeling of Tidal Current

A depth-averaged 2-D numerical model for unsteady tidal flow in estuaries is established by use of the finite volume WENO scheme which maintains both uniform high order accuracy and an essentially non-oscillatory shock transition on unstructured triangular grid. The third order TVD Range-Kutta method is used for time discretization. The model has been firstly tested against four cases： 1） tidal forcing, 2） seiche oscillation, 3） wind setup in a closed bay, and 4） onedimensional dam-break water flow. The results obtained in the present study compare well with those obtained from the corresponding analytic solutions idealized for the above four cases. The model is then applied to the simulation of tidal circulation in the Yangpu Bay, and detailed model calibration and verification have been conducted with measured tidal current in the spring tide, middle tide, and neap tide. The overall performance of the model is in qualitative agreement with the data observed in 2005, and it can be used to calculate the flow in estuaries and coastal waters.

The Sensitivity of Numerical Simulation of the East Asian Monsoon to Different Cumulus Parameterization Schemes

In this paper, a 5-level spectral AGCM is used to examine the sensitivity of simulated East Asian summer monsoon circulation and rainfall to cumulus parameterization schemes. From the simulated results of East Asian monsoon circulations and rainfalls during the summers of 1987 and 1995, it is shown that the Kuo′s convective parameterization scheme is more suitable for the numerical simulation of East Asian summer monsoon rainfall and circulation. This may be due to that the cumulus in the rainfall system is not strong in the East Asian monsoon region.

Construction of a Computational Scheme for the Fuzzy HIV/AIDS Epidemic Model with a Nonlinear Saturated Incidence Rate

This work aimed to construct an epidemic model with fuzzy parameters.Since the classical epidemic model doesnot elaborate on the successful interaction of susceptible and infective people,the constructed fuzzy epidemicmodel discusses the more detailed versions of the interactions between infective and susceptible people.Thenext-generation matrix approach is employed to find the reproduction number of a deterministic model.Thesensitivity analysis and local stability analysis of the systemare also provided.For solving the fuzzy epidemic model,a numerical scheme is constructed which consists of three time levels.The numerical scheme has an advantage overthe existing forward Euler scheme for determining the conditions of getting the positive solution.The establishedscheme also has an advantage over existing non-standard finite difference methods in terms of order of accuracy.The stability of the scheme for the considered fuzzy model is also provided.From the plotted results,it can beobserved that susceptible people decay by rising interaction parameters.

NUMERICAL SCHEMES WITH HIGH ORDER OF ACCURACY FOR THE COMPUTATION OF SHOCK WAVES

High order accurate scheme is highly desirable for Slow computation with shocks. After analysis has been made for the reason of the generation of non-physical oscillations around the shock in numerical computations, a third-order, upwind biased, shock capturing scheme was proposed. Also, a new shock fitting method, called pseudo shock fitting method, was suggested, which in principle can be with any order of accuracy. Test cases for one dimensional flows show that the new method is very satisfactory.

Lagrangian model of zooplankton dispersion:numerical schemes comparisons and parameter sensitivity tests

This paper presents two comparisons or tests for a Lagrangian model of zooplankton dispersion:numerical schemes and time steps.Firstly,we compared three numerical schemes using idealized circulations.Results show that the precisions of the advanced Adams-Bashfold-Moulton(ABM) method and the Runge-Kutta(RK) method were in the same order and both were much higher than that of the Euler method.Furthermore,the advanced ABM method is more efficient than the RK method in computational memory requirements and time consumption.We therefore chose the advanced ABM method as the Lagrangian particle-tracking algorithm.Secondly,we performed a sensitivity test for time steps,using outputs of the hydrodynamic model,Symphonie.Results show that the time step choices depend on the fluid response time that is related to the spatial resolution of velocity fields.The method introduced by Oliveira et al.in 2002 is suitable for choosing time steps of Lagrangian particle-tracking models,at least when only considering advection.

Optimal Evacuation Scheme Based on Dam-Break Flood Numerical Simulation

The optimal evacuation scheme is studied based on the dam-break flood numerical simulation. A three- dimensional dam-break mathematical model combined with the volume of fluid （VOF） method is adopted. According to the hydraulic information obtained from numerical simulation and selecting principles of evacuation emergency scheme, evacuation route analysis model is proposed, which consists of the road right model and random degree model. The road right model is used to calculate the consumption time in roads, and the random degree model is used to judge whether the roads are blocked. Then the shortest evacuation route is obtained based on Dijstra algorithm. Gongming Reservoir located in Shenzhen is taken as a case to study. The results show that industrial area I is flooded at 2 500 s, and after 5 500 s, most of industrial area II is submerged. The Hushan, Loucun Forest and Chaishan are not flooded around industrial area I and II. Based on the above analysis, the optimal evacuation scheme is determined.

A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS

In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem.The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order.The concentration equation is solved by the method of mixed finite element,where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method.The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation.In actual computations the characteristics adopts a large time step without any loss of accuracy.The scalar unknowns and its adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux.In order to derive the optimal 3/2-order error estimate in L^(2) norm,a post-processing technique is included in the approximation to the scalar unknowns.Numerical experiments are illustrated finally to validate theoretical analysis and efficiency.This method can be used to solve such an important problem.

A New Second Order Numerical Scheme for Solving Forward Backward Stochastic Differential Equations with Jumps

In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.

Efficient Finite Difference WENO Scheme for Hyperbolic Systems withNon-conservativeProducts

Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Riemann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(i)It can capture jumps in stationary linearly degenerate wave families exactly.(i)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and twolayer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.

A CLASS OF TWO-STEP TVD MACCORMACK TYPE NUMERICAL SCHEME FOR MHD EQUATIONS

In this paper, a new numerical scheme of Total Variation Diminishing (TVD) MacCormack type for MagnetoHydroDynamic (MHD) equations is proposed by taking into account of the characteristics such as convergence, stability, resolution. This new scheme is established by solving the MHD equations with a TVD modified MacCormack scheme for the purpose of developing a scheme of quick convergence as well as of TVD property. To show the validation, simplicity and practicability of the scheme for modelling MHD problems, a self-similar Cauchy problem with the discontinuous initial data consisting of constant states, and the collision of two fast MHD shocks, and two-dimensional Orszag and Tang's MHD vortex problem are discussed with the numerical results conforming to the existing results obtained by the Roe type TVD, the high-order Godunov scheme,and Weighted Essentially Non-Oscillatory (WENO) scheme. The numerical tests show that this two-step TVD MacCormack numerical scheme for MHD system is of robust operation in the presence of very strong waves, thin shock fronts, thin contact and slip surface discontinuities.

PROGRESS IN THE STUDY OF RETROSPECTIVE NUMERICAL SCHEME AND THE CLIMATE PREDICTION

The retrospective numerical scheme(RNS)is a numerical computation scheme de- signed for multiple past value problems of the initial value in mathematics and considering the self- memory property of the system in physics.This paper briefly presents the historical background of RNS,elaborates the relation of the scheme with other difference schemes and other meteorological prediction methods,and introduces the application of RNS to the regional climatic self-memory model, simplified climate model,barotropic model,spectral model,and mesoscale model.At last,the paper sums up and points out the application perspective of the scheme and the direction for the future study.

Gas-kinetic numerical schemes for one- and two-dimensional inner flows

Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated aad analyzed based on the computed results. The ways of improving the computational efficiency of the gaskinetic numerical method and the computing principles of difference discretization are discussed.

Sensitivity of Numerical Simulations of the East Asian Summer Monsoon Rainfall and Circulation to Different Cumulus Parameterization Schemes

A 5-level spectral AGCM (ImPKU-SLAGCM) is used to examine the sensitivity of the simulated results of the summer monsoon rainfall and circulation in East Asia to different cumulus parameterization schemes in the climatological-mean case and in the cases of weak and strong Asian summer monsoons, respectively. The results simulated with the Arakawa-Schubert's(hereafter A-S's), Kuo's and Manabe's cumulus parameterization schemes show that these simulated distributions of the summer monsoon rainfall and circulation in East Asia depend strongly on the cumulus parameterization schemes either in the climatological-mean case or in the cases of weak and strong Asian summer monsoons. From the simulated results, it might be shown that the Kuo scheme appears to be more suitable for the simulation of the summer monsoon rainfall and circulation in East Asia than the A-S scheme or the Manabe scheme, although the A-S scheme is somewhat better in the simulations of the tropical rainfall. This might be due to that the Kuo's cumulus parameterization scheme is able to reflect well the characteristics of rainfall cloud system in the East Asian summer monsoon region, where the rainfall system used to be a mixing of cumulus and stratus.

High-resolution central difference scheme for the shallow water equations

A two-dimensional nonoscillatory central difference scheme was extended to the shallow water equations. A high-resolution numerical method for solving the shallow water equations was presented. In order to prevent oscillation, the nonlinear limiter is employed to approximate the discrete slopes. The main advantage of the presented method is simplicity comparable with the upwind schemes. This method does not require Riemann solvers or some form of flux difference splitting methods. Furthermore, the discrete derivatives of flux can be approximated by the component-wise approach and thus the computation of Jacobian can be avoided. The method retains high resolution and high accuracy similar to the upwind results. It is applied to simulating several tests, including circular dam-break problem, shock focusing problem and partial dam-break problem. The results are in good agreement with the numerical results obtained by other methods. The simulated results also demonstrate that the presented method is stable and efficient.

One-Dimensional Explicit Tolesa Numerical Scheme for Solving First Order Hyperbolic Equations and Its Application to Macroscopic Traffic Flow Model

In this paper, a new numerical scheme for solving first-order hyperbolic partial differential equations is proposed and is implemented in the simulation study of macroscopic traffic flow model with constant velocity and linear velocity-density relationship. Macroscopic traffic flow model is first developed by Lighthill Whitham and Richards (LWR) and used to study traffic flow by collective variables such as flow rate, velocity and density. The LWR model is treated as an initial value problem and its numerical simulations are presented using numerical schemes. A variety of numerical schemes are available in literature to solve first order hyperbolic equations. Of these the well-known ones include one-dimensional explicit: Upwind, Downwind, FTCS, and Lax-Friedrichs schemes. Having been studied carefully the space and time mesh sizes, and the patterns of all these schemes, a new scheme has been developed and named as one-dimensional explicit Tolesa numerical scheme. Tolesa numerical scheme is one of the conditionally stable and highest rates of convergence schemes. All the said numerical schemes are applied to solve advection equation pertaining traffic flows. Also the one-dimensional explicit Tolesa numerical scheme is another alternative numerical scheme to solve advection equation and apply to traffic flows model like other well-known one-dimensional explicit schemes. The effect of density of cars on the overall interactions of the vehicles along a given length of the highway and time are investigated. Graphical representations of density profile, velocity profile, flux profile, and in general the fundamental diagrams of vehicles on the highway with different time levels are illustrated. These concepts and results have been arranged systematically in this paper.