Different Epidemic Models on Complex Networks

Models for diseases spreading are not just limited to SIS or SIR. For instance, for the spreading of AIDS/HIV, the susceptible individuals can be classified into different cases according to their immunity, and similarly, the infected individuals can be sorted into different classes according to their infectivity. Moreover, some diseases may develop through several stages. Many authors have shown that the individuals＇ relation can be viewed as a complex network. So in this paper, in order to better explain the dynamical behavior of epidemics, we consider different epidemic models on complex networks, and obtain the epidemic threshold for each ease. Finally, we present numerical simulations for each case to verify our results.     

Influence of Selective Edge Removal and Refractory Period in a Self-Organized Critical Neuron Model

A simple model for a set of integrate-and-fire neurons based on the weighted network is introduced. By considering the neurobiological phenomenon in brain development and the difference of the synaptic strength, we construct weighted networks develop with link additions and followed by selective edge removal. The network exhibits the small-world and scale-free properties with high network efficiency. The model displays an avalanche activity on a power-law distribution. We investigate the effect of selective edge removal and the neuron refractory period on the self-organized criticality of the system.     

Pure Multiplicative Noises Induced Population Extinction in an Anti-tumor Model under Immune Surveillance

The dynamical characters of a theoretical anti-tumor model under immune surveillance subjected to a pure multiplicative noise are investigated. The effects of pure multiplicative noise on the stationary probability distribution （SPD） and the mean first passage time （MFPT） are analysed based on the approximate Fokker-Planck equation of the system in detail. For the anti-tumor model, with the multiplieative noise intensity D increasing, the tumor population move towards to extinction and the extinction rate can be enhanced. Numerical simulations are carried out to check the approximate theoretical results. Reasonably good agreement is obtained.     

Charge and Energy Dependences of Ionization and Transfer for Helium in Collisions with Fast Charged Projectiles

The classical method within the independent electron model is employed to investigate （i） charge dependences of single and double ionization for helium by various charged ions A^q＋ （q = 1 - 8） at impact energies of 0.64 and 1.44 MeV/u, respectively, （ii） energy dependences of transfer ionization for helium by 0.5-3 MeV/u A^8.9＋ ions impact. The Lenz-Jensen model of the atom is applied instead of the Bohr model of the atom, and the impact-parameter dependences are also introduced into the calculations. Satisfactory agreement is found between theoretical and experimental data.     

Asymmetric Heat Conduction in One-Dimensional Hard-Point Model with Mass Gradient

The heat conduction in a one-dimensional （1D） hard-point model with mass gradient is studied. Using numerical simulation, we find an asymmetric heat conduction in this model with greater heat current in the direction of mass increase. The increase of temperature gradient, mass gradient and system size axe found to enhance the asymmetric heat conduction. Based on the collision dynamic of a haxd-point particle, we give a qualitative explanation for the underlying mechanism of asymmetric effect.     

A Weakly Nonlinear Model for Kelvin-Helmholtz Instability in Incompressible Fluids

A weakly nonlinear model is proposed for the Kelvin-Helmholtz instability in two-dimensional incompressible fluids by expanding the perturbation velocity potential to third order. The third-order harmonic generation effects of single-mode perturbation are analyzed, as well as the nonlinear correction to the exponential growth of the fundamental modulation. The weakly nonlinear results are supported by numerical simulations. Density and resonance effects exist in the development of mode coupling.     

注重函数模型构建,探究实际应用问题

随着社会科技的发展,数学与其他学科的联系日趋密切,数学的应用越来越广泛,因此构建数学模型、解决实际问题已成为近几年的考试热点之一.这类问题贴近生活,贴近社会,有利于体现数学的人文价值和社会价值,有利于考查学生分析、猜想、建模和综合应用等各方面的能力.     

ON THE EXPECTED DISCOUNTED PENALTY FUNCTION IN A MARKOV-DEPENDENT RISK MODEL WITH CONSTANT DIVIDEND BARRIER

This article considers a Markov-dependent risk model with a constant dividend barrier. A system of integro-differential equations with boundary conditions satisfied by the expected discounted penalty function, with given initial environment state, is derived and solved. Explicit formulas for the discounted penalty function are obtained when the initial surplus is zero or when all the claim amount distributions are from rational family. In two state model, numerical illustrations with exponential claim amounts are given.