Pulse vaccination on SEIR epidemic model with nonlinear incidence rate

In this paper, we consider an SEIR epidemic model with two time delays and nonlinear incidence rate, and study the dynamical behavior of the model with pulse vaccination. By using the Floquet theorem and comparison theorem, we prove that the infection-free periodic solution is globally attractive when R^*<1, and using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination when R_*>1.     

Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model

A delayed SEIRS epidemic model with pulse vaccination and bilinear incidence rate is investigated. Using Krasnoselskii’s fixed-point theorem, we obtain the existence of disease-free periodic solution (DFPS for short) of the delayed impulsive epidemic system. Further, using the comparison method, we prove that under the condition R^* < 1, the DFPS is globally attractive, and that R_* > 1 implies that the disease is permanent. Theoretical results show that the disease will be extinct if the vaccination rate is larger than θ^* and the disease is uniformly persistent if the vaccination rate is less than θ_*. Our results indicate that a long latent period of the disease or a large pulse vaccination rate will lead to eradication of the disease.     

Impulsive epidemic model with differential susceptibility and stage structure

A stage-structured epidemic model with two differential susceptible of susceptibility and pulse vaccination is proposed. We introduce two thresholds R^∗ and R_∗, and further prove that a disease cannot invade the disease-free state if R^∗ is less than unit and can invade if R^∗ is larger than unit. Two numerical simulations are also given to illustrate our main results.     

Dynamic behavior for a nonautonomous SIRS epidemic model with distributed delays

A nonautonomous SIRS epidemic model with distributed delay is investigated. Two new threshold values, R_∗ and R^∗ are derived. The model is permanent as R_∗>1 and R^∗<1 implies the extinction of the disease. Using the Liapunov functional method, global behavior of the model is studied.     

Renormalized coupling constants for the three-dimensional scalar <Emphasis Type="Italic">λ</Emphasis><Emphasis Type="Italic">ϕ</Emphasis><Superscript>4</Superscript> field theory and pseudo-<Emphasis Type="Italic">ε</Emphasis>-expansion

The renormalized coupling constants g _2k that enter the equation of state and determine nonlinear susceptibilities of the system have universal values g _2k ^* at the Curie point. We use the pseudo-ε-expansion approach to calculate them together with the ratios R _2k = g _2k /g ₄ ^k-1 for the three-dimensional scalar λ ? ⁴ field theory. We derive pseudo-ε-expansions for g ₆ ^* , g ₈ ^* , R ₆ ^* , and R ₈ ^* in the five-loop approximation and present numerical estimates for R ₆ ^* and R ₈ ^* . The higher-order coefficients of the pseudo-ε-expansions for g ₆ ^* and R ₆ ^* are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives R ₆ ^* = 1.650, while the recent lattice calculation gave R ₆ ^* = 1.649(2). The pseudo-ε-expansions of g ₈ ^* and R ₈ ^* are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for R ₈ ^* gives the estimate R ₈ ^* = 0.890, differing only slightly from the values R ₈ ^* = 0.871 and R ₈ ^* = 0.857 extracted from the results of lattice and field theory calculations.     

Analysis of SE τ IR ω S epidemic disease models with vertical transmission in complex networks

When the role of network topology is taken into consideration, one of the objectives is to understand the possible implications of topological structure on epidemic models. As most real networks can be viewed as complex networks, we propose a new delayed SEτ IRωS epidemic disease model with vertical transmission in complex networks. By using a delayed ODE system, in a small-world (SW) network we prove that, under the condition R0 ≤ 1, the disease-free equilibrium (DFE) is globally stable. When R0 > 1, the endemic equilibrium is unique and the disease is uniformly persistent. We further obtain the condition of local stability of endemic equilibrium for R0 > 1. In a scale-free (SF) network we obtain the condition R1 > 1 under which the system will be of non-zero stationary prevalence.     

极大高阶交换子的端点估计

该文主要确立了当b∈BMO 时, 极大高阶奇异积分算子交换子T_{b, m}* 满足如下不等式 |{y∈Rⁿ:T_{b, m}*f(y)>λ}|≤C||b||^m_BMO∫_Rⁿ|f(y)|/λ (1+log⁺|f(y)|/λ)^mdy 且T_{b, m}* 在L^p(Rⁿ)(1 < p <∞上有界.     

A note on rings with involution

LetR be a ring with involution_* and π a right ideal ofR. It is shown that if every symmetric elements∈p satisfiess ⁿ=0 wheren is a fixed positive integer then ?^*?(L(R), the lower nil radical ofR.     

On a parabolic equation in MEMS with fringing field

We consider the asymptotic behavior of the solutions to the equation ${u_{t}-u_{xx} = \lambda(1 + {\delta}u_{x}^{2})(1 - u)^{-2}}$ , which comes from Micro-Electromechanical Systems (MEMS) devices modeling. It is shown that when the fringing field exists (i.e., δ?> 0), there is a critical value λ _δ ^* > 0 such that if 0 < λ < λ _δ ^* , the equation has a global solution for some initial data; while for λ > λ _δ ^* , all solutions to the equation will quench at finite time. When the quenching happens, u has only finitely many quenching points for particular initial data. A one-side estimate is deduced for the quenching rate of u.     

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t ?1/ω ?1(t ?1) for ${t\in (0,\infty).}Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for t ? (0,￥).{t\in (0,\infty).} In this paper, the authors introduce the generalized VMO spaces VMOr, L(\mathbb Rn){{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}} associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOr,L (\mathbb Rn))*=Bw,L*(\mathbb Rn){({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}, where L * denotes the adjoint operator of L in L2(\mathbb Rn){L^2({\mathbb R}^n)} and Bw,L*(\mathbb Rn){B_{\omega,L^\ast}({\mathbb R}^n)} the Banach completion of the Orlicz–Hardy space Hw,L*(\mathbb Rn){H_{\omega,L^\ast}({\mathbb R}^n)}. Notice that ω(t) = t p for all t ? (0,￥){t\in (0,\infty)} and p ? (0,1]{p\in (0,1]} is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1, L(\mathbb Rn))*=HL*1(\mathbb Rn){({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}, where HL*1(\mathbb Rn){H_{L^\ast}^1({\mathbb R}^n)} was the Hardy space introduced by Hofmann and Mayboroda.     

Permanence and extinction for a nonautonomous SIRS epidemic model with time delay

In this paper, a nonautonomous SIRS epidemic model with time delay is studied. We introduce some new threshold values _R∗$R_{\ast }$ and R^∗$R^{\ast }$ and further obtain the disease will be permanent when _R∗>1$R_{\ast }>1$ and the disease extinct when R^∗<1$R^{\ast }<1$. Using the method of Liapunov functional, some sufficient conditions are derived for the global attractivity of the system. The known results are extended.     

About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ?(d(0, R^?)) be the field of meromorphic functions inside the disk d(0,R⁻) = {x ∈ K ∣ ∣x∣ < R}. Let ?_b(d(0, R^?)) be the subfield of bounded meromorphic functions inside d(0,R⁻) and let ?_u(d(0, R^?)) = ?(d(0, R^?)) ? ?_b(d(0, R^?)) be the subset of unbounded meromorphic functions inside d(0,R⁻). Initially, we consider the Yosida Equation: , where m ∈ ?^* and F(X) is a rational function of degree d with coefficients in ?_b(d(0, R^?)). We show that, if d ≥ 2m + 1, this equation has no solution in ?_u(d(0, R^?)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R⁻). Further, we list the only cases when the equation may eventually admit solutions in ?_u(d(0, R^?)). Particularly, the elliptic equation may not.     

HIV/AIDS epidemic fractional-order model

In this paper a non-linear mathematical model with fractional order ?, 0 < ? ≤ 1 is presented for analyzing and controlling the spread of HIV/AIDS. Both the disease-free equilibrium E₀ and the endemic equilibrium E^* are found and their stability is discussed using the stability theorem of fractional order differential equations. The basic reproduction number R₀ plays an essential role in the stability properties of our system. When R₀ < 1 the disease-free equilibrium E₀ is attractor, but when R₀ > 1, E₀ is unstable and the endemic equilibrium (EE) E^* exists and it is an attractor. Finally numerical Simulations are also established to investigate the influence of the system parameter on the spread of the disease.     

A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

We establish that a pair A, B, of nonsingular matrices over a commutative domain R of principal ideals can be reduced to their canonical diagonal forms D ^A and D ^B by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices U, V _A, and V _B over R such that UAV _a=D^A and UAV _B=D^B if and only if the matrices B _*A and D _* ^B D^A where B _* 0 is the matrix adjoint to B, are equivalent.     

Global stability for an HIV/AIDS epidemic model with different latent stages and treatment

An HIV/AIDS epidemic model with different latent stages and treatment is constructed. The model allows for the latent individuals to have the slow and fast latent compartments. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are determined by the basic reproduction number under some conditions. If R₀ < 1, the disease free equilibrium is globally asymptotically stable, and if R₀ > 1, the endemic equilibrium is globally asymptotically stable for a special case. Some numerical simulations are also carried out to confirm the analytical results.     

Moser's theorem for lower dimensional tori

Moser's C^?-version of Kolmogorov's theorem on the persistence of maximal quasi-periodic solutions for nearly-integrable Hamiltonian system is extended to the persistence of non-maximal quasi-periodic solutions corresponding to lower-dimensional elliptic tori of any dimension n between one and the number of degrees of freedom. The theorem is proved for Hamiltonian functions of class C^? for any ?>6n+5 and the quasi-periodic solutions are proved to be of class C^p for any p with 2<p<p_* for a suitable p_*=p_*(n,?)>2 (which tends to infinity when ?→∞).     

Commutative rings whose zero-divisor graph is a proper refinement of a star graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G _c ^* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G _c ^* has at least two connected components. We prove that the diameter of the induced graph G _c ^* is two if Z(R)² ≠ {0}, Z(R)³ = {0} and G _c ^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs K _n with some end vertices adjacent to a single vertex of K _n .     

Qualitative analysis of a SIR epidemic model with saturated treatment rate

On account of the effect of limited treatment resources on the control of epidemic disease, a saturated removal rate is incorporated into Hethcote’s SIR epidemiological model (Hethcote, SIAM Rev. 42:599–653, 2000). Unlike the original model, the model has two endemic equilibria when R ₀<1. Furthermore, under some conditions, both the disease free equilibrium and one of the two endemic equilibria are asymptotically stable, i.e., the model has bistable equilibria. Therefore, disease eradication not only depends on R ₀ but also on the initial sizes of all sub-populations. By the Poincaré-Bendixson theorem, Poincaré index, center manifold theorem, Hopf bifurcation theorem and Lyapunov-Lasalle theorem, etc., the existence and asymptotical stability of the equilibria, the existence, stability and direction of Hopf bifurcation are discussed, respectively.     

Functional properties of rearrangement invariant spaces defined in terms of oscillations

Function spaces whose definition involves the quantity f^**-f^*, which measures the oscillation of f^*, have recently attracted plenty of interest and proved to have many applications in various, quite diverse fields. Primary role is played by the spaces S_p(w), with 0<p<∞ and w a weight function on (0,∞), defined as the set of Lebesgue-measurable functions on R such that f^*(∞)=0 and