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2021, Vol. 41 No. 2   Published date: 30 June 2021
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• Dynamics and Optimal Control of an Antibody Immune HIV Model with a Saturated Proliferation Rate

  2021, 41(2): 1. doi:

  Abstract ( 1166 )   PDF (561KB) ( 790 )   　　

  In this paper, an antibody immune HIV model with a saturated proliferation rate is established, and some conditions of local stability and global stability for the disease-free equilibrium, the no-immune endemic equilibrium and the immune endemic equilibrium are obtained by the linearization method and the Lyapunov function method, respectively. Moreover, motivated by the success of the latest anti-AIDS vaccine in some animal experiments, the dual action of antibody immunity and drug therapy is builded into the above kinetic model, which forms a optimal control problem to minimize the concentration of infected cells and virus and the cost of control. Using the Pontryagin maximum principle, the optimality conditions for the optimal control problem are gotten. After obtaining the parameters of the model , numerical simulation tests are carried out for the double control problem and the single control problem respectively. Experimental results show that the concentration of the infected cells and viruses can quickly reduce under the condition of the effective vaccine, and the immune control effect is almost as well as the effect of the double control by comparing with double control effect, which illustrates the vaccine is fairly effective in controlling AIDS. It can be expected that vaccine will greatly change the current situation of AIDS spread and may even eliminate AIDS eventually, after the vaccine is put into clinical practice.

• Optimal Dividend Problem for Two Collaborating Insurance Companies

  2021, 41(2): 19. doi:

  Abstract ( 1227 )   PDF (238KB) ( 537 )   　　

  In this paper, we consider the two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes. The surplus process is a piecewise deterministic Markov process (PDMP), so the optimal dividend problem can be studied by virtue of use the PDMP theory. After getting the properties of the optimal value function, we give the definition of admissible strategy and Markov strategy, and prove the necessary and sufficient condition for a strategy to be a stationary Markov strategy. Using the theory of measure-valued generators, we derive the associated measure-valued dynamic programming equation (measure-valued DPE). Finally we prove the verification theorem under the existence of the optimal strategy.

• Stability and Hopf Bifurcation for a Delayed Cooperation-diffusion-advection System with Dirichlet Boundary Conditions

  2021, 41(2): 28. doi:

  Abstract ( 1202 )   PDF (318KB) ( 692 )   　　

  This paper is devoted to a delayed cooperation-diffusion-advection system with Dirichlet boundary conditions. Firstly we discuss the existence and stability of the positive steady state. Secondly we show that an increasing delay will destabilize the positive steady state and lead to the occurrence of Hopf bifurcation when the delay crosses through the critical bifurcation points.

• Ground State Solutions for a Fourth Order Quasilinear Elliptic Equation

  2021, 41(2): 39. doi:

  Abstract ( 1083 )   PDF (214KB) ( 431 )   　　

  This paper studies the following fourth order quasilinear elliptic equation

  \begin{equation*}

  \left\{\begin{aligned}

  &\triangle^{2} u-\triangle u+V(x)u-\frac{1}{2}u\triangle (u^{2})=f(u),&x\in \mathbb{R}^{N},\\

  &u\in H^{2}(\mathbb{R}^{N}),

  \end{aligned}

  \right.

  \end{equation*}

  where $\triangle^{2}:=\triangle(\triangle)$ is the biharmonic operator, $2<N\leq 6$. We prove that the equation admits a ground state solution of the Nehari-Poho\u{z}aev type.

• On Twin Domination Number of Cartesian Product of Directed Cycles

  2021, 41(2): 57. doi:

  Abstract ( 1190 )   PDF (896KB) ( 561 )   　　

  Let $\gamma^{*}(D)$ denote the twin domination number of digraph $D$ and let $C_{m}\square C_{n}$ denote the Cartesian product of the directed cycle $C_{m}$ and $C_{n}$, for $m, n\geq 2$. In this paper, we give a lower bound for $\gamma^{*}(C_{m}\square C_{n})$ and we determine the exact values of $\gamma^{*}(C_{m}\square C_{n})$ when $m,~n\equiv 0~ ({\rm mod} ~3)$ and when $m\equiv 2~({\rm mod}~3)$.

• The Number of Connected Cayley Graphs over Dicyclic Group

  2021, 41(2): 64. doi:

  Abstract ( 1245 )   PDF (184KB) ( 740 )   　　

  Let $p$ be an odd prime. In this paper, we obtain the number of (connected)

  Cayley graphs on the dicyclic groups $T_{4p}=\langle a,b~|~a^{p}=b^4=1,b^{-1}ab=a^{-1}\rangle$ up to isomorphism by using the P\'{o}lya enumeration theorem.

• 11-valent Arc-transitive Graphs of Order 4p

  2021, 41(2): 76. doi:

  Abstract ( 1243 )   PDF (186KB) ( 386 )   　　

  In this paper, a complete classification of 11-valent symmetric graphs of $4p$ order is given, where $p$ is a prime.

  According to the results of the classification, there is only one the 11-valent graph of order $4p$, which is a complete graph $K_{12}$.

• Based on the Robust ARMA Residual Control Chart of IGGⅢ Weight Function

  2021, 41(2): 83. doi:

  Abstract ( 1356 )   PDF (4816KB) ( 669 )   　　

  
  

  Since the classical ARMA residual control chart is susceptible to outlier values, this paper first establishes a robust ARMA model based on the Institute of Geodesy & Geophysics III  IGGIII weight function to obtains an independent residual distribution sequence. Then a robust residual control chart is constructed based on the estimations of the mean and standard deviation by the weighted three mean and average absolute deviation, respectively. Simulations and empirical tests show that the robust control chart can resist the outlier interferences better, and the monitoring effect is better than that of the traditional control chart. 

• High Performance Numerical Simulation of Dust Removal Fan Based on OpenFOAM

  2021, 41(2): 96. doi:

  Abstract ( 1243 )   PDF (25462KB) ( 277 )   　　

  In this paper, the computational fluid dynamics numerical simulation software OpenFOAM is used to study the numerical simulation of a fan's dust removal process. A real dust removal fan equipment involves the flow of three phases: the gas phase, the dust particle state and the liquid spray. This paper abstracts a two-phase flow model of the gas phase and dust particles, and focuses on the transient simulation of the flow field changes in the dust removal fan. To this end, a mathematical model of the internal airflow of the wind turbine, which includes a Navier-Stokes partial differential equation that considers the centrifugal force and the coercive force caused by rotation, and a ${k}-\omega-\mathrm{SST}$ turbulence model, is established. The simulation results show that the velocity at the fan's tip is higher than that of the surrounding fluid, the tip moves the fluid to the direction of the tip movement, and the velocity closer to the tip is larger. In addition, under the given boundary conditions and initial conditions, the particle pressure field, velocity vector diagram, etc. are obtained, respectively, to visually and intuitively simulate the fan's dust removal process.

• Statistical Analysis of Inbound Tourism in Shanghai Based on Intervention Model

  2021, 41(2): 109. doi:

  Abstract ( 1076 )   PDF (363KB) ( 397 )   　　

  This paper selects the monthly data of the number of visitors to Shanghai from January 2004 to August 2012. Based on the general time series prediction method, intervention analysis is introduced, and the R software is used to predict the time series. First of all, through analysis, the trend effect and seasonal effect of the series are found, so a product season model is used to fit the time series and predict the the number of visitors to Shanghai for the next 8 months. Secondly, after pre-processing the original data and determining the time point of the Expo impact, the time series is divided into two parts, by the time point then the method of intervention analysis is used to establish an intervention combination model to predict the number of visitors to Shanghai for the next 8 months. Finally, the prediction results of the two models are compared by calculating their relative errors. By comparison, it is found that the prediction effect of the intervention model is better, indicating that in the presence of emergencies or major policies, it is better to use an intervention model to analyze and predict the time series.