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.     

Dynamic behavior of a delay cholera model with constant infectious period

In this paper, a delay cholera model with constant infectious period is investigated. By analyzing the characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium of the model is established. It is proved that if the basic reproductive number $\mathcal{R}_0>1$, the system is permanent. If $\mathcal{R}_0<1$, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the disease-free equilibrium. If $\mathcal{R}_0>1$, also by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators

Let L_2=(-?)~2+ V~2 be the Schr?dinger type operator, where V■0 is a nonnegative potential and belongs to the reverse H?lder class RH_(q1) for q_1 n/2, n ≥5. The higher Riesz transform associated with L_2 is denoted by ■and its dual is denoted by ■. In this paper, we consider the m-order commutators [b~m, R] and [b~m, R*], and establish the(L~p, L~q)-boundedness of these commutators when b belongs to the new Campanato space Λ_β~θ(ρ) and 1/q = 1/p-mβ/n.     

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.     

Boundedness of the Cesàro Operator in Hardy Spaces

The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$.     

Lipschitz Estimates for Commutators of $N$-Dimensional Fractional Hardy Operators

In this paper, it is proved that the commutator$\mathcal{H}_{β,b}$ which is generated by the $n$-dimensional fractional Hardy operator $\mathcal{H}_β$ and $b\in \dot{Λ}_α(\mathbb{R}^n)$ is bounded from $L^P(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$, where $0<α<1,1相似文献   

Regularity in Capacity and the Dirichlet Laplacian

Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set of all open subsets of which are regular in capacity. Then one can define metrics and on only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function . We prove that the spaces and are complete and contain the set of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset.     

有向跳图的生成欧拉子有向图

一个有向多重图D的跳图$J(D)$是一个顶点集为$D$的弧集,其中$(a,b)$是$J(D)$的一条弧当且仅当存在有向多重图$D$中的顶点$u_1$, $v_1$, $u_2$, $v_2$,使得$a=(u_1,v_1)$, $b=(u_2,v_2)$ 并且$v_1\neq u_2$.本文刻画了有向多重图类$\mathcal{H}_1$和$\mathcal{H}_2$,并证明了一个有向多重图$D$的跳图$J(D)$是强连通的当且仅当$D\not\in \mathcal{H}_1$.特别地, $J(D)$是弱连通的当且仅当$D\not\in \mathcal{H}_2$.进一步, 得到以下结果: (i) 存在有向多重图类$\mathcal{D}$使得有向多重图$D$的强连通跳图$J(D)$是强迹连通的当且仅当$D\not\in\mathcal{D}$. (ii) 每一个有向多重图$D$的强连通跳图$J(D)$是弱迹连通的,因此是超欧拉的. (iii) 每一个有向多重图D的弱连通跳图$J(D)$含有生成迹.     

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

In this paper, some improved regularity criteria for the 3D magneto-micropolar fluid equations are established in Morrey–Campanato spaces. It is proved that if the velocity field satisfies

Local André-Oort conjecture for the universal abelian variety

We prove a p-adic version of the André-Oort conjecture for subvarieties of the universal abelian varieties. Let g and n be integers with n≥3 and p a prime number not dividing n. Let R be a finite extension of , the ring of Witt vectors of the algebraic closure of the field of p elements. The moduli space of g-dimensional principally polarized abelian varieties with full level n-structure as well as the universal abelian variety over may be defined over R. We call a point R-special if is a canonical lift and ξ is a torsion point of its fibre. Employing the model theory of difference fields and work of Moonen on special subvarieties of , we show that an irreducible subvariety of containing a dense set of R-special points must be a special subvariety in the sense of mixed Shimura varieties.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.     

三角Banach代数的Lie弱顺从性

设$\mathcal {A,\ B}$ 是含单位元的Banach代数, $\mathcal M$ 是一个Banach $\mathcal {A,\ B}$-双模. $\mathcal {T}=\left ( \begin{array}{cc} \mathcal {A} & \mathcal M \\ & \mathcal {B} \\ \end{array} \right )$按照通常矩阵加法和乘法,范数定义为$\|\left( \begin{array}{cc} a & m \\ & b\\ \end{array} \right)\|=\|a\|_{\mathcal A}+\|m\|_{\mathcal M}+\|b\|_{\mathcal B}$,构成三角Banach 代数.如果从$\mathcal T$到其$n$次对偶空间$\mathcal T^{n}$上的Lie导子都是标准的,则称$\mathcal T$是Lie $n$弱顺从的.本文研究了三角Banach代数$\mathcal T$上的Lie $n$弱顺从性,证明了有限维套代数是Lie $n$弱顺从的.     

Conclaves of planes in PG (4,2) and certain planes external to the Grassmannian ${\cal G}_{1,4,2} \subset$ PG(9,2)

We show that in $\operatorname{PG}(4,2)$ there exist octets $\mathcal{P} _{8}=\{\pi_{1},\,\ldots\,,\pi_{8}\}$ of planes such that the 28 intersections $\pi_{i}\cap\pi_{j}$ are distinct points. Such conclaves (see [6]) $\mathcal{P}_{8}$ of planes in $\operatorname{PG}(4,2)$ are shown to be in bijective correspondence with those planes $P$ in $\operatorname{PG}(9,2)$ which are external to the Grassmannian $\mathcal{G}_{1,4,2}$ and which belong to the orbit $\operatorname{orb}(2\gamma)$ (see [4]). The fact that, under the action of $\operatorname{GL}(5,2),$ the stabilizer groups $\mathcal{G}_{\mathcal{P}_{8}}$ and $\mathcal{G}_{P}$ both have the structure $2^{3}:(7:3)$ is thus illuminated. Starting out from a regulus-free partial spread $\mathcal{S}_{8}$ in $\operatorname{PG}(4,2)$ we also give a construction of a conclave of planes $P\in\operatorname{orb}(2\gamma)\subset\operatorname{PG}(9,2).$     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

Syzygies of torsion bundles and the geometry of the level ℓ modular variety over \overline{\mathcal{M}}_{g}

We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of ? in its Jacobian. These statements can be viewed an analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding non-vanishing locus in the moduli space $\mathcal{R}_{g,\ell}$ of twisted level ? curves of genus g and use this to derive results about the birational geometry of $\mathcal{R}_{g, \ell}$ . For instance, we prove that $\mathcal{R}_{g,3}$ is a variety of general type when g>11 and the Kodaira dimension of $\mathcal{R}_{11,3}$ is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.     

Computing Minimum-Volume Enclosing Axis-Aligned Ellipsoids

Given a set of points and ε>0, we propose and analyze an algorithm for the problem of computing a (1+ε)-approximation to the minimum-volume axis-aligned ellipsoid enclosing . We establish that our algorithm is polynomial for fixed ε. In addition, the algorithm returns a small core set , whose size is independent of the number of points m, with the property that the minimum-volume axis-aligned ellipsoid enclosing is a good approximation of the minimum-volume axis-aligned ellipsoid enclosing . Our computational results indicate that the algorithm exhibits significantly better performance than the theoretical worst-case complexity estimate. This work was supported in part by the National Science Foundation through CAREER Grants CCF-0643593 and DMI-0237415.     

Modelling the $Wolbachia$ Strains for Dengue Fever Virus Control in the Presence of Seasonal Fluctuation

Consider that infection with $Wolbachiacan$ limit a mosquito''s ability to transmit Dengue fever virus through its saliva, a mathematical model describing the transmission of Dengue fever between vector mosquitoes and human, incorporating $Wolbachia$-carrying mosquito population and seasonal fluctuation, is proposed. Firstly, the stability and bifurcation of this model are investigated exactly in the case where seasonality can be neglected. Further, the basic reproductive number $\mathcal{R}_0^s$ for this model with seasonal variation is obtained, that is, if $\mathcal{R}_0^s$ is less than unity the disease is extinct and $\mathcal{R}_0^s$ is greater than unity the disease is uniformly persistent. Finally, numerical simulations verify the theoretical results. Theoretical results suggest that, compared with the mosquito reduction strategies (such as the elimination of mosquito breeding sites, killing of adult mosquitoes by spraying), introducing $Wolbachia$ strains is as effectual to fight against the transmission of Dengue virus.     

Height of a Superposition

We observe that, given a poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ and a finite covering ${\user1{\mathcal{R}}} = {\user1{\mathcal{R}}}_{1} \cup \cdots \cup {\user1{\mathcal{R}}}_{n} $ of its ordering, the height of the poset does not exceed the natural product of the heights of the corresponding sub-relations: $$\mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}} \right)} \leqslant \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{1} } \right)} \otimes \cdots \otimes \mathfrak{h}{\left( {E,{\user1{\mathcal{R}}}_{n} } \right)}.$$ Conversely for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, every poset ${\left( {E,{\user1{\mathcal{R}}}} \right)}$ of height at most $\xi_1\otimes\cdots\otimes\xi_n$ admits a partition ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ such that each ${\left( {E,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at most $\xi_k$ . In particular for every finite sequence $(\xi_1,\cdots,\xi_n)$ of ordinals, the ordinal $$\xi _{1} \underline{ \otimes } \cdots \underline{ \otimes } \xi _{n} : = \sup {\left\{ {{\left( {\xi ^{\prime }_{1} \otimes \cdots \otimes \xi ^{\prime }_{n} } \right)} + 1:\xi ^{\prime }_{1} < \xi _{1} , \cdots ,\xi ^{\prime }_{n} < \xi _{n} } \right\}}$$ is the least $\xi$ for which the following partition relation holds $$\mathfrak{H}_{\xi } \to {\left( {\mathfrak{H}_{{\xi _{1} }} , \cdots ,\mathfrak{H}_{{\xi _{n} }} } \right)}^{2} $$ meaning: for every poset ${\left( {A,{\user1{\mathcal{R}}}} \right)}$ of height at least $\xi$ and every finite covering ${\left( {{\user1{\mathcal{R}}}_{1} , \cdots ,{\user1{\mathcal{R}}}_{n} } \right)}$ of its ordering ${\user1{\mathcal{R}}}$ , there is a $k$ for which the relation ${\left( {A,{\user1{\mathcal{R}}}_{k} } \right)}$ has height at least $\xi_k$ . The proof will rely on analogue properties of vertex coverings w.r.t. the natural sum.     

Split Partial Isometries

A partial isometry V is said to be a split partial isometry if ${\mathcal{H}=R(V) + N(V)}$ , with R(V) ∩ N(V) = {0} (R(V) = range of V, N(V) = null-space of V). We study the topological properties of the set ${\mathcal{I}_0}$ of such partial isometries. Denote by ${\mathcal{I}}$ the set of all partial isometries of ${\mathcal{B}(\mathcal{H})}$ , and by ${\mathcal{I}_N}$ the set of normal partial isometries. Then $$\mathcal{I}_N\subset \mathcal{I}_0\subset \mathcal{I}, $$ and the inclusions are proper. It is known that ${\mathcal{I}}$ is a C ^∞-submanifold of ${\mathcal{B}(\mathcal{H})}$ . It is shown here that ${\mathcal{I}_0}$ is open in ${\mathcal{I}}$ , therefore is has also C ^∞-local structure. We characterize the set ${\mathcal{I}_0}$ , in terms of metric properties, existence of special pseudo-inverses, and a property of the spectrum and the resolvent of V. The connected components of ${\mathcal{I}_0}$ are characterized: ${V_0,V_1\in \mathcal{I}_0}$ lie in the same connected component if and only if $${\rm dim}\, R(V_0)= {\rm dim}\, R(V_1) \,\,{\rm and}\,\,\, {\rm dim}\, R(V_0)^\perp = {\rm dim}\, R(V_1)^\perp.$$ This result is known for normal partial isometries.