GLOBAL ANALYSIS OF SIS EPIDEMIC MODEL WITH A SIMPLE VACCINATION AND MULTIPLE ENDEMIC EQUILIBRIA

An SIS epidemic model with a simple vaccination is investigated in this article. The efficiency of vaccine, the disease-related death rate and population dynamics are also considered in this model. The authors find two threshold R0 and Rc (Rc may not exist). There is a unique endemic equilibrium for R0 > 1 or Rc = R0; there are two endemic equilibria for Rc < R0 < 1; and there is no endemic equilibrium for R0 < Rc < 1. When Rc exists, there is a backward bifurcation from the disease-free equilibrium for R0 = 1. They analyze the stability of equilibria and obtain the globally dynamic behaviors of the model. The results acquired in this article show that an accurate estimation of the efficiency of vaccine is necessary to prevent and controll the spread of disease.     

一类带有种痘年龄的流行病模型

A simple SI epidemic model with age of vaccination is discussed in this paper. Both varing birth rate,the mortality rate caused by disease and vaccine waning rate are considered in this model.We prove that the global dynamics is completely determined by the basic reproductive number R(Ψ)(Ψdenotes per capita vaccination rate).If R(0)＜1, the disease-free equilibrium is a global attractor;If R(Ψ)＜1,the disease-free equilibrium is locally asymptotically stable;If R(Ψ)＞1,an unique endemic equilibrium exists and is locally asymptotically stable under certain condition.     

Two-group SIR epidemic model with stochastic perturbation

A stochastic two-group SIR model is presented in this paper.The existence and uniqueness of its nonnegative solution is obtained,and the solution belongs to a positively invariant set.Furthermore,the globally asymptotical stability of the disease-free equilibrium is deduced by the stochastic Lyapunov functional method if R0 ≤ 1,which means the disease will die out.While if R0 1,we show that the solution is fluctuating around a point which is the endemic equilibrium of the deterministic model in time average.In addition,the intensity of the fluctuation is proportional to the intensity of the white noise.When the white noise is small,we consider the disease will prevail.At last,we illustrate the dynamic behavior of the model and their approximations via a range of numerical experiments.     

具有接种的确定性和随机SIS流行病模型的全局稳定性

We study the stability of endemic equilibriums of the deterministic and stochastic SIS epidemic models with vaccination. The deterministic SIS epidemic model with vaccination was proposed by Li and Ma(2004), for which some sufficient conditions for the global stability of the endemic equilibrium were given in some earlier works. In this paper, we first prove by Lyapunov function method that the endemic equilibrium of the deterministic model is globally asymptotically stable whenever the basic reproduction number is larger than one. For the stochastic version, we obtain some sufficient conditions for the global stability of the endemic equilibrium by constructing a class of different Lyapunov functions.     

SIVS EPIDEMIC MODELS WITH INFECTION AGE AND NONLINEAR VACCINATION RATE

Vaccination is a very important strategy for the elimination of infectious diseaVaccination is a very important strategy for the elimination of infectious diseases. A SIVS epidemic model with infection age and nonlinear vaccination has been formulated in this paper. Using the theory of differential and integral equation, we show the local asymptotic stability of the infection-free equilibrium and the endemic equilibrium under some assumptions.     

STABILITY OF AN SEIS EPIDEMIC MODEL WITH CONSTANT RECRUITMENT AND A VARYING TOTAL POPULATION SIZE

This paper considers an SEIS epidemic model with infectious force in the latent period and a general population-size dependent contact rate. A threshold parameter R is identified. If R≤1, the disease-free equilibrium O is globally stable. If R〉1, there is a unique endemic equilibrium and O is unstable. For two important special cases of bilinear and standard incidence ,sufficient conditions for the global stability of this endemic equilibrium are given. The same qualitative results are obtained provided the threshold is more than unity for the corresponding SEIS model with no infectious force in the latent period. Some existing results are extended and improved.     

TWO DIFFERENTIAL INFECTIVITY EPIDEMIC MODELS WITH NONLINEAR INCIDENCE RATE

This paper considers two differential infectivity（DI） epidemic models with a nonlinear incidence rate and constant or varying population size. The models exhibits two equilibria, namely., a disease-free equilibrium O and a unique endemic equilibrium. If the basic reproductive number σ is below unity,O is globally stable and the disease always dies out. If σ〉1, O is unstable and the sufficient conditions for global stability of endemic equilibrium are derived. Moreover,when σ〈 1 ,the local or global asymptotical stability of endemic equilibrium for DI model with constant population size in n-dimensional or two-dimensional space is obtained.     

Banach空间中复指数函数系的不完备性

In this paper, we obtain a necessary and sufficient condition for the incompleteness of complex exponential system in Cα, where Cα is a weighted Banach space of complex continuous functions f on the real axis R with f(t) exp{-α(t)}vanishing at infinity, in the uniform norm.     

一类具有时滞和治疗的宿主 寄生虫交互动力学模型的研究(英文)

In this paper, the dynamic behaviors of a host-parasite model with intracellular delay and drug treatment is investigated. we obtain the basic reproductive ratio R0 , which determined the behaviors of the system. If R0<1, the infection-free equilibrium is globally asymptotic stability. If 1 3, by choosing the delay τ as a bifurcation parameter, the infection equilibrium bifurcate a family of periodic solution as τ crosses a critical value. Furthermore, we give some examples to verify our theoretical results and discuss the sensitivity of the drug efficacy and effect of the delay. In the finally, we give a brief discussion.     

PERMANENCE OF A SIRS EPIDEMIC MODEL WITH IMPULSIVE VACCINATION AND DISTRIBUTED DELAYS

In this paper, we consider a SIRS epidemic model with impulsive vaccination and distributed time delays. By the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution to the system. Further, using the comparison theorem, we prove that the infection-free periodic solution is globally attractive under an assumption. A sufficient condition for the permanence of the model is investigated.     

Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems

This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].     

Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the L p Dirichlet Problem

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L_0 = div A~0(x)? + B~0(x) · ? is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the LpDirichlet problem for the operator L_0 is solvable in the upper half-space Rn+. In this paper we prove that the Lpsolvability is stable under small perturbations of L_0. That is if L_1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L_0 and L_1 are sufficiently close in the sense of Carleson measures, then the LpDirichlet problem for the operator L_1 is solvable for the same value of p. As a corollary we obtain a new result on Lpsolvability of the Dirichlet problem for operators of the form L = div A(x)? + B(x) · ? where the matrix A satisfies weaker Carleson condition(expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiable and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoˇs,Petermichl and Pipher.     

Blowing up of solutions to the Cauchy problem for the generalized Zakharov system with combined power-type nonlinearities

This paper deals with blowing up of solutions to the Cauchy problem for a class of general- ized Zakharov system with combined power-type nonlinearities in two and three space dimensions. On the one hand, for c0 = +∞ we obtain two finite time blow-up results of solutions to the aforementioned system. One is obtained under the condition α≥ 0 and 1 + 4/N ≤ p N +2/N-2 or α 0 and 1 p 1 + 4/N (N = 2, 3); the other is established under the condition N = 3, 1 p N +2/N-2 and α(p-3) ≥ 0. On the other hand, for c0 +∞ and α(p-3) ≥ 0, we prove a blow-up result for solutions with negative energy to the Zakharov system under study.     

GLOBAL DYNAMICS OF AN SEIR EPIDEMIC MODEL WITH IMMIGRATION OF DIFFERENT COMPARTMENTS

The SEIR epidemic model studied here includes constant inflows of new susceptibles, exposeds, infectives, and recovereds. This model also incorporates a population size dependent contact rate and a disease-related death. As the infected fraction cannot be eliminated from the population, this kind of model has only the unique endemic equilibrium that is globally asymptotically stable. Under the special case where the new members of immigration are all susceptible, the model considered here shows a threshold phenomenon and a sharp threshold has been obtained. In order to prove the global asymptotical stability of the endemic equilibrium, the authors introduce the change of variable, which can reduce our four-dimensional system to a three-dimensional asymptotical autonomous system with limit equation.     

On the bifurcations and multiple endemic states of a single strain HIV model

The dynamics of a single strain HIV model is studied. The basic reproduction number R0 used as a bifurcation parameter shows that the system undergoes transcritical and saddle-node bifurcations. The usual threshold unit value of R0 does not completely determine the eradication of the disease in an HIV infected person. In particular, a sub-threshold value Rc is established which determines the system＇s number of endemic states： multiple if Rc 〈 Ro 〈 1, only one if Rc=Ro = 1, and none if R0 〈 Rc 〈 1.     

STABILITY ANALYSIS OF A COMPUTER VIRUS PROPAGATION MODEL WITH ANTIDOTE IN VULNERABLE SYSTEM

We study a proposed model describing the propagation of computer virus in the network with antidote in vulnerable system. Mathematical analysis shows that dynamics of the spread of computer viruses is determined by the threshold R_0. If R_0 ≤ 1, the virusfree equilibrium is globally asymptotically stable, and if R_0 1, the endemic equilibrium is globally asymptotically stable. Lyapunov functional method as well as geometric approach are used for proving the global stability of equilibria. A numerical investigation is carried out to confirm the analytical results. Through parameter analysis, some effective strategies for eliminating viruses are suggested.     

THE MAIN INVARIANTS OF A COMPLEX FINSLER SPACE

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T ′M : two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with K¨ahler spaces, in the two-dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the K¨ahler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0.Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

GLOBAL STABILITY OF SIVS MODEL WITH SATURATION INCIDENCE, INFECTION AGE AND IMPULSIVE VACCINATED RATE

In this paper, an impulsive vaccinated strategy to eradicate SIVS epidemic model is studied. Since infection age is an important factor of epidemic progression, we incorporate the infection age into the model. In this model, we analyze the dynamic behaviors of this model and obtain that there exists an infection-free periodic solution which is globally asymptotically stable under a sufficient condition. Our results indicate that a short period of pulse or a large pulse vaccination rate is the sufficient con...     

On Exact Controllability of Networks of Nonlinear Elastic Strings in 3-Dimensional Space

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings.The authors derive the equations and appropriate nodal conditions,determine equilibrium solutions,and,by using the methods of quasilinear hyperbolic systems,prove that for tree networks the natural initial,boundary value problem has classical solutions existing in neighborhoods of the "stretched" equilibrium solutions.Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved.Finally,it is proved that,given two different equilibrium states satisfying certain conditions,it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a suffciently large time interval.     

An SIS epidemic model with diffusion

The aim of this paper is to study the dynamics of an SIS epidemic model with diffusion. We first study the well-posedness of the model. And then, by using linearization method and constructing suitable Lyapunov function, we establish the local and global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Furthermore, in view of Schauder fixed point theorem, we show that the model admits traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium when R_0 1 and c c~*. And also, by virtue of the two-sided Laplace transform, we prove that the model has no traveling wave solution connecting the two equilibria when R_0 1 and c ∈ [0, c~*).