应用数学和力学

Optimal Vaccination Strategies for a Time-Varying SEIR Epidemic Model With Latent Delay

WANG Xinwei, PENG Haijun, ZHONG Wanxie

2019, 40(7): 701-712. doi: 10.21656/1000-0887.400048

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Abstract:
On the basis of the classic SEIR compartmental model, a time-delayed term was introduced to characterize the latent delay. Furthermore, a controlled time-varying SEIR model with delay was established in view of the vaccination, the successfully immune rate and the seasonally varying incidence coefficient. Meanwhile, the optimal vaccination strategy was determined under the frame of the optimal control problem with the vaccination rate taken as the control variable. In the formulated optimal control problem, 3 kinds of constraints (i.e., the constraints on control, the upper limit on the susceptible population and the time-varying upper limit on the vaccination supply) were considered. The optimal control problem was numerically solved with a multi-interval symplectic pseudospectral method. Numerical results demonstrate that the obtained vaccination strategy can effectively suppress the spread of the disease, and the comparison between different cases suggests that omitting time-varying factors may result in unreasonable vaccination strategies.

Existence of Critical Traveling Waves for Nonlocal Dispersal SIR Models With Delay and Nonlinear Incidence

ZHANG Qiu, CHEN Guangsheng

2019, 40(7): 713-727. doi: 10.21656/1000-0887.390208

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Abstract:
The existence of traveling wave solutions for nonlocal dispersal SIR epidemic models with delay was studied. Firstly, the boundedness of I was proved by contradiction. Then according to the boundedness of I, the existence of traveling waves with c>c^* was established. Secondly, through further analysis of traveling waves with super-critical speeds, the existence of traveling waves with the critical speed was derived. Finally, the influence of basic reproduction number R⁰ on the existence of c>c^* was discussed.

Dynamic Analysis of a Class of HIV-1 Infection Models With Pulsed Immunotherapy

WANG Xiaoe, LIN Xiaolin, LI Jianquan

2019, 40(7): 728-740. doi: 10.21656/1000-0887.390334

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Abstract:
Based on a class of HIV-1 infection immunotherapy models, an HIV-1 infection model with pulsed immunotherapy was addressed. The non-negativity and uniform boundedness of the solutions to the pulsed immunotherapy model were studied with the pulsed differential equation theory. According to the Floquet multiplier theory and the comparison theorem for differential equations, the threshold conditions for the local and global asymptotic stability of the infection-free periodic solution as well as uniform persistence of HIV-1 were obtained. Through numerical simulation, the therapeutic effects of 3 different treatment regimens were compared, and the effectiveness of the pulsed immunotherapy was verified. The numerical results show that, the virus can be effectively controlled or eliminated theoretically when the drug input is large enough or the dosing interval is short properly.

A Combined Artificial Neural Network Method for Solving Time Fractional Diffusion Equations

WANG Jiang, CHEN Wen

2019, 40(7): 741-750. doi: 10.21656/1000-0887.390288

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Abstract:
A combined artificial neural network method was proposed to solve 1D time fractional diffusion equations. The combined artificial neural network is a new network structure constructed through combination of the radial basis function (RBF) neural network and the power series feed-forward neural network. First, the proposed new model was applied to create a numerical solution conforming to the conditions of the time fractional diffusion equation. Meanwhile, an error function was defined and the original differential equation was transformed into a minimization problem. Afterwards, the gradient descent learning algorithm was used to obtain the optimal weights of the neural network and other optimal parameters. Finally, a numerical example was given to illustrate the validity of the proposed method. The work makes a new way to the solution of 1D time fractional diffusion equations.

A Parallel Algorithm for Super-Quadric Discrete Elements Based on the CUDA-GPU Architecture

WANG Siqiang, JI Shunying

2019, 40(7): 751-767. doi: 10.21656/1000-0887.390267

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Abstract:
The traditional parallel computing of large-scale discrete elements was suitable for spherical particles. However, in natural fields or industrial applications, the granular systems commonly comprise non-spherical particles. Meanwhile, the dynamic behaviors and mechanical properties of non-spherical particles are strongly different from those of spherical particles at different spatial scales. Super-quadric elements based on the continuous function envelope were used to effectively describe the geometric shapes of irregular particles, and accurately calculate the contact forces between elements with the non-linear Newtonian method. In view of the complexity of the contact detection between non-spherical particles and the large-scale computational requirements of the discrete element method, a CUDA-GPU parallel algorithm was developed for super-quadric elements. Based on the parallel calculation of spherical particles, the rough contact list of the element envelope and the accurate contact list of the Newtonian method were established with the kernel function. Meanwhile, the parallel algorithm and the memory access mode were optimized to improve the computation efficiency. To examine the reliability of the parallel algorithm, the flow process of non-spherical particles was simulated with the discrete element method and compared with the experimental results. Furthermore, the influences of the aspect ratio and the sharpness parameter of elements on the flow characteristics of non-spherical particles were analyzed. This study provides an effective numerical method for large-scale simulation of non-spherical granular materials.

An L-Stable Method for Differential-Algebraic Equations of Multibody System Dynamics

LI Bowen, DING Jieyu, LI Yanan

2019, 40(7): 768-779. doi: 10.21656/1000-0887.400038

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Abstract:
An L-stable method over time intervals for differential-algebraic equations of multibody system dynamics was presented. The solution scheme was established based on equidistant nodes and non-equidistant nodes such as Chebyshev and Legendre nodes. According to Ehle’s theorem and conjecture, the unknown matrix and vector in the L-stable solution formula were obtained through comparison with the Padé approximation. The Newtonian iteration method was used during the solution process. The planar 2-link manipulator system was taken as an example, and the results from the L-stable method were compared for different node numbers in the time interval and different steps in the simulation, with those from the classic Runge-Kutta method. The comparison shows that, the proposed method has the advantages of good stability and high precision, and is suitable for multibody system dynamics simulation under long-term conditions.

Dynamic Analysis and Simulation of the Lower Extremity Exoskeleton Based on Human-Machine Interaction

ZHANG Yan, LI Fanru, LI Wei, LIU Zuojun

2019, 40(7): 780-790. doi: 10.21656/1000-0887.390212

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Abstract:
A human body-exoskeleton model with human-machine interaction was established. The human body and the exoskeleton were modeled respectively with a 7-link rigid body mechanism, and the D-H coordinate system was introduced to obtain the change vectors of the human-machine model during motion. The Newton-Euler equations were used for dynamic analysis, and the body-exoskeleton interaction was simplified as elastic forces. According to the distance changes between the centroids of the body and the exoskeleton in motion, the relative displacements and the interaction forces were determined. Finally, the dynamic model was simulated with the ADAMS (automatic dynamic analysis of mechanical system), and the joint torques obtained from the dynamic equations were substituted into the simulation. The results verify the correctness of the body-exoskeleton model.

Two High-Order Difference Schemes for Solving Time Distributed-Order Diffusion Equations

HU Jiahui, WANG Jungang, NIE Yufeng

2019, 40(7): 791-800. doi: 10.21656/1000-0887.390358

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Abstract:
Based on the composite Simpson’s quadrature rule and the composite 2-point Gauss-Legendre quadrature rule, 2 high-order finite difference schemes were proposed for solving time distributed-order diffusion equations. Other than the existing methods whose convergence rates are only 1st-order or 2nd-order in the temporal domain, the proposed 2 schemes both have 3rd-order convergence rates in the temporal domain, and 4th-order rates in the spatial domain and the distributed order, respectively. Such high-order convergence rates were further verified with numerical examples. The results show that, both of the proposed 2 schemes are stable, and have higher accuracy and efficiency compared with existing algorithms.

Adaptive RBF-Network Dynamic Surface Tracking Control of Sprayer Boom Systems

LU Zeyang, LI Shujiang, WANG Xiangdong

2019, 40(7): 801-809. doi: 10.21656/1000-0887.390270

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Abstract:
To realize fast and accurate servo tracking of the canopy height, an electro-hydraulic servo system was used for the position adjustment device. The electro-hydraulic servo system was modeled with the load of the sprayer boom. First, the strong non-linearity and parameters’ uncertainties were well considered and the complete system model was built, and the controller was designed with the dynamic surface control method. Then the RBF network was used to approximate the non-linearity and uncertainty functions, the control law was applied with damping terms to compensate for the disturbance influences on the system, and all the signals in the closed loop system were proved to be uniformly bounded based on the Lyapunov stability method. Finally, simulation and verification of a sprayer boom with the electro-hydraulic servo system were conducted. The results show that the designed controller has good copy tracking performances.

Lie Symmetry of Constrained Hamiltonian Systems and Its Application in Field Theory

ZHOU Jingrun, FU Jingli

2019, 40(7): 810-822. doi: 10.21656/1000-0887.390218

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Abstract:
The Lie symmetry method was studied for constrained Hamiltonian systems, and the conservation laws of the field theory systems were obtained. Firstly, the generalized canonical equations for constrained Hamiltonian systems were derived. Secondly, the determining equations and structural equations about the Lie symmetry of the constrained Hamiltonian systems were deduced. Thirdly, the Lie theorems and the conserved quantities for constrained Hamiltonian systems were given. Finally, the Lie symmetry for the system of the complex scalar field coupled to the Chern-Simons term was discussed. Two examples in the field theory illustrate the validity of this method.

2019 Vol. 40, No. 7