基于网络连边的谣言传播模型研究

基于Miller对随机网络中SIR传染病模型所做的研究,在谣言传播的过程中引入概率母函数,利用网络连边等图论的相关理论引进θ边和φ边,并且考虑人们的自身认知水平和对于谣言的遗忘因素,建立了一个新的谣言传播模型.借助经典的下一代矩阵方法计算出其基本再生数R_0,对该模型平衡点的性质及动力学特点进行分析,证明了当R_01时,系统有且仅有唯一的边界平衡点;当R_01时,系统存在两个边界平衡点,分别为E0和E*.进一步得到当R_01时,唯一的平衡点是全局渐近稳定的;当R_01时,平衡点E0不稳定,而E*是局部渐近稳定的.最后,结合概率母函数的性质分析了谣言传播最终规模,得到当R_01时,谣言最终不会盛行;当R_01时,谣言将会一直盛行下去.     

一类具有非线性出生率和饱和恢复率的SEIRS传染病模型的后向分支

主要讨论一类具有非线性出生率和饱和恢复率的SEIRS传染病模型的后向分支.当R_01时,存在无病平衡点,且局部渐近稳定;考虑R_0及R_0~c的关系,得到地方病平衡点存在的条件.当R_1~*1,R_0=1时,系统出现后向分支,若R_1~*1,R_0=1,系统出现前向分支.     

一类具有时滞的随机SIS传染病模型

本文研究一类具有时滞的随机SIS传染病模型,并定性分析种群灭绝和持久的充分条件.获得了阈值R_0,当R_01时,种群灭绝.当R_01时,种群持久.并通过了数值模拟验证了上述理论结果.     

一类具有logistic增长的SIR时滞传染病模型的稳定性分析

研究了一类具有logistic增长的时滞SIR传染病模型,得到了决定疾病爆发和消亡的阈值R_0,证明了当R_01时,对于任意的时滞τ,无病平衡点都是全局渐近稳定的,此时疾病消亡;当R_01时,系统会出现一个临界值τ_0,当ττ_0时,地方病平衡点不稳定;当ττ_0,且满足给定的条件时,地方病平衡点局部渐近稳定;当τ=τ_0时,系统发生Hopf分支.通过数值模拟,验证了上述结论的正确性,且做了参数的敏感度分析.     

具有饱和发生率和免疫响应的病毒感染模型的稳定性分析

提出了具有饱和发生率和免疫响应的病毒感染数学模型,得到了基本再生数R_0的表达式.当R_01时,证明了无病平衡点是全局渐近稳定的;当R_01时,得到了免疫耗竭平衡点和持续带毒平衡点局部渐近稳定的条件.     

基于随机与异质网络共存的SEIRS传染病模型

讨论了随机与异质网络共存的SEIRS传染病模型,通过正平衡点的存在性给出基本再生数R_0=((1-η)Aλ+ηβ)/μ.结果表明,当R_01时,无病平衡点(1,0,0,0)局部稳定;当R_01时,无病平衡点(1,0,0,0)不稳定,此时系统存在唯一的地方病平衡点,并且一致持续存在.最后通过数值仿真,验证了理论结果的正确性.     

年龄结构CD4~+T-细胞模型复杂动力学分析

研究了一类年龄结构的CD4~+T-细胞模型.得到了控制HIV病毒扩散的阈值R_0.当R_01时,无病平衡点全局渐近稳定,病毒在人体内消除;当R_01,且-r+(2αrT~*)/(T_(max))0,地方病平衡点局部渐近稳定,病毒在人体内繁殖;当R_01,且-r+(2αrT~*)/(T_(max))0,系统由感染年龄而产生的复杂动力学行为,如Hopf分支,四周期解及混沌等.最后对模型的复杂动力学行为进行了数值模拟.     

一类具有非线性传染率的SEIS传染病模型的稳定性分析

研究了一类具有非线性传染率的SEIS模型,模型中包含常数输入率、自然死亡率、因病死亡率等.定义了模型的基本再生数R_0,并证明了当R_01时,无病平衡点是全局渐近稳定的.当R_01时,得到了唯一的地方平衡点是全局渐近稳定的条件.     

具有一般复发现象的疾病模型的全局稳定性

本文研究了具有一般复发现象和非线性发生率的疾病模型的动力学性质,其中模型是具有无穷分布时滞的微积分方程.该模型描述了包含疱疹等传染病的—般复发现象.利用一致持久性理论和李雅普诺夫函数,我们证明了基本再生数R_0决定的系统的全局动力学性质:当R_0≤1时,疾病灭绝;当R_01时,疾病持久生存,并且正平衡点是全局吸引的.     

具有一般发生率的疟疾传播模型的稳定性研究

研究了一类具有一般发生率的疟疾传播模型,得到了模型的平衡点和基本再生数R_0.通过构造Lyapunov函数得到当R_0≤1时,无病平衡点是全局渐近稳定的;当R_01时,正平衡点是全局渐近稳定的.通过例子说明所得的理论结果.     

具有种群Logistic增长及饱和传染率的SIS模型的稳定性和Hopf分支

该文研究一类具有种群Logistic增长及饱和传染率的SIS传染病模型,讨论了平衡点的存在性及全局渐近稳定性,得到疾病消除的阈值就是基本再生数$R_{0}=1$. 证明了,当$R_{0}<1$ 时,无病平衡点全局渐近稳定;当$R_{0}>1$ 且$\alpha K\leq 1$ 时,正平衡点全局渐近稳定;当$R_{0}>1$ 且$\Delta ={0}$ 时,系统在正平衡点附近发生Hopf分支;当$R_{0}>1$ 且$\Delta <{0}$ 时,系统在正平衡点外围附近存在唯一稳定的极限环.     

Stability and bifurcation analysis of a viral infection model with delayed immune response

In this paper, we study a viral infection model with an immunity time delay accounting for the time between the immune system touching antigenic stimulation and generating CTLs. By calculation, we derive two thresholds to determine the global dynamics of the model, i.e., the reproduction number for viral infection $R_{0}$ and for CTL immune response $R_{1}$. By analyzing the characteristic equation, the local stability of each feasible equilibrium is discussed. Furthermore, the existence of Hopf bifurcation at the CTL-activated infection equilibrium is also studied. By constructing suitable Lyapunov functionals, we prove that when $R_{0}\leq1$, the infection-free equilibrium is globally asymptotically stable; when $R_{0}>1$ and $R_{1}\leq1$, the CTL-inactivated infection equilibrium is globally asymptotically stable; Numerical simulation is carried out to illustrate the main results in the end.     

The dynamic behavior of deterministic and stochastic delayed SIQS model

In this paper, we present the deterministic and stochastic delayed SIQS epidemic models. For the deterministic model, the basic reproductive number $R_{0}$ is given. Moreover, when $R_{0}<1$, the disease-free equilibrium is globally asymptotical stable. When $R_{0}>1$ and additional conditions hold, the endemic equilibrium is globally asymptotical stable. For the stochastic model, a sharp threshold $\overset{\wedge }{R}_{0}$ which determines the extinction or persistence in the mean of the disease is presented. Sufficient conditions for extinction and persistence in the mean of the epidemic are established. Numerical simulations are also conducted in the analytic results.     

整数集的加权表示的渐进基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. Let Rk1,k2(A, n)be the number of solutions of n = k1a1 + k2a2, where a1, a2 ∈ A. Recently, Xiong proved that there is a set A  Z such that Rk1,k2(A, n) = 1 for all n ∈ Z. Let f : Z-→ N0∪ {∞} be a function such that f-1(0) is finite. In this paper, we generalize Xiong's result and prove that there exist uncountably many sets A  Z such that Rk1,k2(A, n) = f(n) for all n ∈ Z.     

Birds movement impact on the transmission of West Nile virus between patches

Spatial heterogeneity plays an important role in the distribution and persistence of many infectious disease. In the paper, a multi-patch model for the spread of West Nile virus among $n$ discrete geographic regions is presented that incorporates a mobility process. In the mobility process, we assume that the birds can move among regions, but not the mosquitoes based on scale-space. We show that the movement of birds between patches is sufficient to maintain disease persistence in patches. We compute the basic reproduction number $R_{0}$. We prove that if $R_{0}<1$, then the disease-free equilibrium of the model is globally asymptotically stable. When $R_{0}>1$, we prove that there exists a unique endemic equilibrium, which is globally asymptotically stable on the biological domain. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between two regions.     

基于完备BR0-代数的全蕴涵三I算法

研究了基础$BR_0$-代数的性质和基于完备基础$BR_0$-代数的全蕴涵三I算法,对一般蕴涵算子给出了三I算法解存在的一个充分条件,并将结果应用于$R_0$-单位区间$\overline{W}$,不但极大的简化了$R_0$-单位区间$\overline{W}$的$R_0$-型$\alpha$-三I算法结果的证明,而且使其证明过程与相应的模糊命题演算系统结合起来,说明了$R_0$-型三I算法是与$B{\cal L}^*$系统相匹配的模糊推理方法.     

Analysis of an HIV model with post-treatment control

Recent investigation indicated that latent reservoir and immune impairment are responsible for the post-treatment control of HIV infection. In this paper, we simplify the disease model with latent reservoir and immune impairment and perform a series of mathematical analysis. We obtain the basic infection reproductive number $R_{0}$ to characterize the viral dynamics. We prove that when $R_{0}<1$, the uninfected equilibrium of the proposed model is globally asymptotically stable. When $R_{0}>1$, we obtain two thresholds, the post-treatment immune control threshold and the elite control threshold. The model has bistable behaviors in the interval between the two thresholds. If the proliferation rate of CTLs is less than the post-treatment immune control threshold, the model does not have positive equilibria. In this case, the immune free equilibrium is stable and the system will have virus rebound. On the other hand, when the proliferation rate of CTLs is greater than the elite control threshold, the system has stable positive immune equilibrium and unstable immune free equilibrium. Thus, the system is under elite control.     

Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force

This paper formulates a stochastic SIR epidemic model by supposing that the infection force is perturbed by Brown motion and L\''{e}vy jumps. The globally positive and bounded solution is proved firstly by constructing the suitable Lyapunov function. Then, a stochastic basic reproduction number $R_0^{L}$ is derived, which is less than that for the deterministic model and the stochastic model driven by Brown motion. Analytical results show that the disease will die out if $R_0^{L}<1$, and $R_0^{L}>1$ is the necessary and sufficient condition for persistence of the disease. Theoretical results and numerical simulations indicate that the effects of L\''{e}vy jumps may lead to extinction of the disease while the deterministic model and the stochastic model driven by Brown motion both predict persistence. Additionally, the method developed in this paper can be used to investigate a class of related stochastic models driven by L\''{e}vy noise.     

关于杨全会和陈永高提出的一个问题

对任意的正整数与集合,令为解的个数.杨全会和陈永高证明了:若整数且,则不存在集合使得对所有充分大的整数成立,其中.对整数和,定义为满足对所有整数成立的集合的个数.杨全会和陈永高证明了是有限的,且.同时,他们问对任意整数,是否存在使得对所有整数成立.在本文中,我们给出了在时的准确公式.从而推出在时成立.     

Dynamics Analysis of an SIS Epidemic Model with the Effects of Awareness

In this paper, an SIS model incorporating the effects of awareness spreading on epidemic is analyzed. Four kinds of equilibria of the model are given, and a new method is used to prove the stability of the equilibria. The threshold of awareness is $R_{1}^{a}$, which measures whether awareness spreads. When awareness does not spread, the basic reproduction number of disease is $R_{1}^{d}$, it is $R_{2}^{d}$ when awareness spreads. The relationship among the three kinds of thresholds is discussed in details. Specially, the effects of various awareness parameters on epidemic are analyzed. Our theoretical results suggest that raising awareness can effectively reduce the basic reproduction number of disease and reduce the spread of disease. Furthermore, numerical simulations are performed to illustrate our results.