有Logistic增长的媒介传播模型分析

建立了贮存宿主鸟类与传染宿主蚊子都具有Logistic增长的西尼罗河病毒传播模型,获得了基本再生数R_0.当R_0<1时,通过构造Lyapunov函数,证明了无病平衡点的全局渐近稳定性.当R_0>1,且满足不同条件时,得到了正平衡点的存在性,数值模拟验证了理论结果的正确性.     

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.     

Mathematical Analysis of West Nile Virus Model with Discrete Delays

The paper presents the basic model for the transmission dynamics of West Nile virus (WNV). The model, which consists of seven mutually-exclusive compartments representing the birds and vector dynamics, has a locally-asymptotically stable disease-free equilibrium whenever the associated reproduction number (?₀) is less than unity. As reveal in [3, 20], the analyses of the model show the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity). It is shown, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. Analysis of the reproduction number of the model shows that, the disease will persist, whenever ?₀ > 1, and increase in the length of incubation period can help reduce WNV burden in the community if a certain threshold quantities, denoted by Δ_b and Δ_v are negative. On the other hand, increasing the length of the incubation period increases disease burden if Δ_b > 0 and Δ_v > 0. Furthermore, it is shown that adding time delay to the corresponding autonomous model with standard incidence (considered in [2]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease).     

On the structure of partition which the difference of their representation function is a constant

Periodica Mathematica Hungarica - Let $$\mathbb {N}$$ be the set of nonnegative integers. For any set $$A \subset \mathbb {N}$$ , let $$R_1(A, n)$$ , $$R_2(A, n)$$ and $$R_3(A, n)$$ be the number...     

Parabolic Raynaud bundles

Let X be an irreducible smooth projective curve defined over the field of complex numbers, a finite set of closed points and N ≥ 2 a fixed integer. For any pair , there exists a parabolic vector bundle on X, with parabolic structure over S and all parabolic weights in , that has the following property: Take any parabolic vector bundle of rank r on X whose parabolic points are contained in S, all the parabolic weights are in and the parabolic degree is d. Then is parabolically semistable if and only if there is no nonzero parabolic homomorphism from to .     

Modeling the Transmission of West Nile Virus with {\it Wolbachia} in a Heterogeneous Environment

{\it Wolbachia} are maternally transmitted endosymbiotic bacteria. To investigate the effect of {\it Wolbachia} on the spreading and vanishing of West Nile virus, we construct a reaction-diffusion model associated with the {\it Wolbachia} parameter in a heterogeneous environment, which has nonlinear infectious disease parameters. Based on the spectral radius of next infection operator and the related eigenvalue problem, we present a corresponding explicit expression describing the basic reproduction number. Furthermore, utilizing this number, we not only give out the stability of disease-free equilibrium, but also analyze the uniqueness and globally asymptotic behavior of endemic equilibrium. Our theoretical results and numerical simulations indicate that only if {\it Wolbachia} reach a certain magnitude in mosquitoes, it can be effective in the control of West Nile virus.     

On Semimonotone Matrices, $$R_0$$ -Matrices and Q-Matrices

Journal of Optimization Theory and Applications - In 1979, Pang proved that within the class of semimonotone matrices, $$R_0$$ -matrices are Q-matrices and conjectured that the converse is also...     

On Some Rings of Arithmetical Functions

In this paper, we consider several constructions which from a given B-product * _B lead to another one We shall be interested in finding what algebraic properties of the ring are shared also by the ring . In particular, for some constructions the rings R _B and will be isomorphic and therefore have the same algebraic properties.     

Global properties for virus dynamics model with mitotic transmission and intracellular delay

In this paper we study the basic model of viral infections with mitotic transmission and intracellular delay discrete. The delay corresponds to the time between infection of uninfected cells and the emission of virus on a cellular level. By means of Volterra-type Lyapunov functionals, we provide the global stability for this model. Let η be the number of virus produced per infected cell. If ηcrit, the critical number, satisfies η?ηcrit, then the virus-free steady state is globally asymptotically stable. On the contrary if η>ηcrit, then the infected steady state is globally asymptotically stable if a sufficient condition is satisfied.     

On weakening tightness to weak tightness

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property $$wt(X)\le t(X)$$. We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $$wt(X)=\aleph _0<t(X)$$ under $$2^{\aleph _0}=2^{\aleph _1}$$. In particular, this demonstrates the celebrated Balogh’s (Proc Am Math Soc 105(3):755–764, 1989) Theorem does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then $$|X|\le 2^{wt(X)\pi \chi (X)}$$. This refines a theorem of de la Vega (Topol Appl 153:2118–2123, 2006). In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhász and van Mill (Proc Am Math Soc 146(1):429–437, 2018). In this connection we also show $$w(X)\le 2^{wt(X)}$$ for a homogeneous compactum. Third, we show that if X is a $$T_1$$ space, $$wt(X)\le \kappa $$, X is $$\kappa ^+$$-compact, and $$\psi (\overline{D},X)\le 2^\kappa $$ for any $$D\subseteq X$$ satisfying $$|D|\le 2^\kappa $$, then (a) $$d(X)\le 2^\kappa $$ and (b) X has at most $$2^\kappa $$-many $$G_\kappa $$-points. This is a variation of another theorem of Balogh (Topol Proc 27:9–14, 2003). Finally, we show that if X is a regular space, $$\kappa =L(X)wt(X)$$, and $$\lambda $$ is a caliber of X satisfying $$\kappa <\lambda \le \left( 2^{\kappa }\right) ^+$$, then $$d(X)\le 2^{\kappa }$$. This extends of theorem of Arhangel$$'$$skiĭ (Topol Appl 104:13–26, 2000).     

Global analysis of virus dynamics model with logistic mitosis,cure rate and delay in virus production

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra—type functions, composite quadratic functions and Volterra—type functionals, we provide the global stability for this model. If R₀, the basic reproductive number, satisfies R₀ ≤ 1, then the infection‐free equilibrium state is globally asymptotically stable. Our system is persistent if R₀ > 1. On the other hand, if R₀ > 1, then infection‐free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if R₀ > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

四阶微分方程的迭代解

利用一个构造性的方法,在假设边值问题存在上解α和下解β,满足β≤α的前提下,给出了两个单调序列它们一致收敛于如下两类边值问题的极值解u(4)(x)-Mu″(x)=f(x,u(x),u'(x),u″(x),u″'(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0;u(4)(x)-Mu″(x)=g(x,u(x),u'(x),u″(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0.     

A delayed computer virus propagation model and its dynamics

In this paper, we propose a delayed computer virus propagation model and study its dynamic behaviors. First, we give the threshold value R₀ determining whether the virus dies out completely. Second, we study the local asymptotic stability of the equilibria of this model and it is found that, depending on the time delays, a Hopf bifurcation may occur in the model. Next, we prove that, if R₀ = 1, the virus-free equilibrium is globally attractive; and when R₀ < 1, it is globally asymptotically stable. Finally, a sufficient criterion for the global stability of the virus equilibrium is obtained.     

Fractional Gaussian estimates and holomorphy of semigroups

Let $$\Omega \subset {\mathbb {R}}^N$$ be an arbitrary open set, $$0<s<1$$ and denote by $$(e^{-t(-\Delta )_{{{\mathbb {R}}}^N}^s})_{t\ge 0}$$ the semigroup on $$L^2({{\mathbb {R}}}^N)$$ generated by the fractional Laplace operator. In the first part of the paper, we show that if T is a self-adjoint semigroup on $$L^2(\Omega )$$ satisfying a fractional Gaussian estimate in the sense that $$|T(t)f|\le Me^{-bt(-\Delta )_{{{\mathbb {R}}}^N}^s}|f|$$, $$0\le t \le 1$$, $$f\in L^2(\Omega )$$, for some constants $$M\ge 1$$ and $$b\ge 0$$, then T defines a bounded holomorphic semigroup of angle $$\frac{\pi }{2}$$ that interpolates on $$L^p(\Omega )$$, $$1\le p<\infty $$. Using a duality argument, we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.     

Mathematical Modeling and Analysis of an Epidemic Model with Quarantine, Latent and Media Coverage

Epidemic models are very important in today''s analysis of diseases. In this paper, we propose and analyze an epidemic model incorporating quarantine, latent, media coverage and time delay. We analyze the local stability of either the disease-free and endemic equilibrium in terms of the basic reproduction number $\mathcal{R}_{0}$ as a threshold parameter. We prove that if $\mathcal{R}_{0}<1,$ the time delay in media coverage can not affect the stability of the disease-free equilibrium and if $\mathcal{R}_{0}>1$, the model has at least one positive endemic equilibrium, the stability will be affected by the time delay and some conditions for Hopf bifurcation around infected equilibrium to occur are obtained by using the time delay as a bifurcation parameter. We illustrate our results by some numerical simulations such that we show that a proper application of quarantine plays a critical role in the clearance of the disease, and therefore a direct contact between people plays a critical role in the transmission of the disease.     

Global analysis of a multi-group animal epidemic model with indirect infection and time delay

The transmission mechanism of some animal diseases is complex because of the multiple transmission pathways and multiple-group interactions, which lead to the limited understanding of the dynamics of these diseases transmission. In this paper, a delay multi-group dynamic model is proposed in which time delay is caused by the latency of infection. Under the biologically motivated assumptions, the basic reproduction number $R_0$ is derived and then the global stability of the disease-free equilibrium and the endemic equilibrium is analyzed by Lyapunov functionals and a graph-theoretic approach as for time delay. The results show the global properties of equilibria only depend on the basic reproductive number $R_0$: the disease-free equilibrium is globally asymptotically stable if $R_0\leq 1$; if $R_0>1$, the endemic equilibrium exists and is globally asymptotically stable, which implies time delay span has no effect on the stability of equilibria. Finally, some specific examples are taken to illustrate the utilization of the results and then numerical simulations are used for further discussion. The numerical results show time delay model may experience periodic oscillation behaviors, implying that the spread of animal diseases depends largely on the prevention and control strategies of all sub-populations.     

On the Cubicity of Interval Graphs

A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product where R _i (for 1 ≤ i ≤ k) is an interval of the form [a _i , a _i  + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, . This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to .     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.