threshold and stability results for an age structured seir epidemic model abstract in this paper an age structured seir epidemic model is considered the existence and uniqueness of a positive solution for this model is proved threshold results for the existence of endemic states are established under certain conditions the local stability for the steady states is also examined

a branch and cut algorithm for nonconvex quadratically constrained quadratic programming we present a branch and cut algorithm that yields in finite time a globally e optimal solution with respect to feasibility and optimality of the nonconvex quadratically constrained quadratic programming problem the idea is to estimate all quadratic terms by successive linearizations within a branching tree using reformulation linearization techniques rlt to do so four classes of linearizations cuts depending on one to three parameters are detailed for each class we show how to select the best member with respect to a precise criterion the cuts introduced at any node of the tree are valid in the whole tree and not only within the subtree rooted at that node in order to enhance the computational speed the structure created at any node of the tree is flexible enough to be used at other nodes computational results are reported that include standard test problems taken from the literature some of these problems are solved for the first time with a proof of global optimality

dynamical aspects of an age structured sir endemic model the main idea of this work is to present and study the dynamical behavior of an age dependent sir endemic model first the age dependent sir endemic model is formulated from existing sir epidemic models by adding age dependent recruitment rate the behavior of the model is analyzed by using the basic reproductive number r 0 the stability theory is used for the analysis of disease free and endemic equilibrium finally finite difference method of characteristics is used to present numerical results for the suggested age structured endemic model

the characterization of continuous four coefficient scaling functions and wavelets four coefficient dilation equations are examined and results converse to a theorem of daubechies lagarias are given these results complete the characterization of those four coefficient dilation equations having a continuous solution

a sirs epidemic model with infection age dependence based on j mena lorca and h w hethcote s epidemic model a sirs epidemic model with infection age dependent infectivity and general nonlinear contact rate is formulated under general conditions the unique existence of its global positive solutions is obtained moreover under more general assumptions than the existing the existence and asymptotical stability of its equilibria are discussed in the end the condition on the stability of endemic equilibrium is verified by a special model

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

a simplicial branch and bound algorithm for solving quadratically constrained quadratic programs we propose a branch and bound algorithm for solving nonconvex quadratically constrained quadratic programs the algorithm is novel in that branching is done by partitioning the feasible region into the cartesian product of two dimensional triangles and rectangles explicit formulae for the convex and concave envelopes of bilinear functions over triangles and rectangles are derived and shown to be second order cone representable the usefulness of these new relaxations is demonstrated both theoretically and computationally

lyapunov equations and gram matrices if p is a polynomial with all roots inside the unit disc and c its companion matrix then the lyapunov equation x c xc p has a unique solution for every positive semidefinite matrix p we characterize sets of vectors x0 xn 1 and y0 yn 1 such that x g x0 xn 1 g y0 yn 1 1 geometrical connections between such bases and contractions with one dimensional defect spaces are established

contractibility of compact contractions in hilbert space for a finite set of compact contractions in a complex hilbert space h it is shown that r a 1 for all a in the multiplicative semigroup generated by if and only if there exists a positive integer n such that a 1 for all a in the multiplicative semigroup generated by with length greater than n here r a denotes the spectral radius of a as an application an answer is given to an infinite dimensional case of the finiteness conjecture for the generalized spectral radius attributed to j c lagarias and y wang linear algebra appl 214 1995 17

exact quadratic convex reformulations of mixed integer quadratically constrained problems we propose a solution approach for the general problem qp of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints the resolution is based on the reformulation of the original problem qp into an equivalent quadratic problem whose continuous relaxation is convex so that it can be effectively solved by a branch and bound algorithm based on quadratic convex relaxation we concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible furthermore we extend our method to the case of mixed integer quadratic problems with the following restriction all quadratic sub functions of purely continuous variables are already convex finally we illustrate the different results of the article by small examples and we present some computational experiments on pure integer and mixed integer instances of qp most of the considered instances with up to 53 variables can be solved by our approach combined with the use of cplex

global dynamics of a class of seirs epidemic models in a periodic environment abstract in this paper we study a class of periodic seirs epidemic models and it is shown that the global dynamics is determined by the basic reproduction number r 0 which is defined through the spectral radius of a linear integral operator if r 0 1 then the disease free periodic solution is globally asymptotically stable and if r 0 1 then the disease persists our results really improve the results in t zhang z teng on a nonautonomous seirs model in epidemiology bull math biol 69 8 2007 2537 2559 for the periodic case moreover from our results we see that the eradication policy on the basis of the basic reproduction number of the time averaged system may overestimate the infectious risk of the periodic disease numerical simulations which support our theoretical analysis are also given

global optimization algorithm for mixed integer quadratically constrained quadratic program abstract mixed integer quadratic programs with quadratic constraints miqqp occur frequently in various areas of engineering practice and management science but most solution methods for this kind of problems are often designed for its special cases in this paper we present a simple global optimization algorithm for solving problem miqqp we first convert problem miqqp into an equivalent generalized bilinear programming problem with integer variables eiqqp we next show that replacing the quadratic objective and constraint functions with their convex envelopes is dominated by an alternative methodology based on convexifying the range of the bilinear terms on the feasible region finally by incorporating the reduction correction techniques and sampling strategies into the branch and bound scheme the proposed algorithm is developed for solving miqqp convergence and optimality of the algorithm are presented and numerical examples taken from some recent literature and minlplib2 are carried out to validate the performance of the proposed algorithm

global optimization of mixed integer quadratically constrained quadratic programs miqcqp through piecewise linear and edge concave relaxations we propose a deterministic global optimization approach whose novel contributions are rooted in the edge concave and piecewise linear underestimators to address nonconvex mixed integer quadratically constrained quadratic programs miqcqp to epsilon global optimality the facets of low dimensional n 3 edge concave aggregations dominating the termwise relaxation of miqcqp are introduced at every node of a branch and bound tree concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge concave aggregations are addressed through piecewise linear relaxations extensive computational studies are presented for point packing problems standard and generalized pooling problems and examples from globallib meeraus globallib http www gamsworld org global globallib htm

general four step discrete time zeroing and derivative dynamics applied to time varying nonlinear optimization abstract time varying nonlinear optimization tvno problems are considered as important issues in various scientific disciplines and industrial applications in this paper the continuous time derivative dynamics ctdd model is developed for obtaining the real time solutions of tvno problems furthermore aiming to remedy the weaknesses of ctdd model a continuous time zeroing dynamics ctzd model is presented and investigated for potential digital hardware realization by using bilinear transform a general four step zhang et al discretization zead formula is proposed and applied to the discretization of both ctdd and ctzd models a general four step discrete time derivative dynamics general four step dtdd model and a general four step discrete time zeroing dynamics general four step dtzd model are proposed on the basis of this general four step zead formula further theoretical analyses indicate that the general four step dtzd model is zero stable consistent and convergent with the truncation error of o g 4 which denotes a vector with every entries being o g 4 with g denoting the sampling period theoretical analyses also indicate that the maximal steady state residual error mssre has an o g 4 pattern confirmedly the efficacy and accuracy of the general four step dtdd and dtzd models are further illustrated by numerical examples

unitary part of a contraction for a contraction a on a hilbert space h we define the index j a resp k a as the smallest nonnegative integer j resp k such that ker i aj aj resp ker i ak ak ker i akak equals the subspace of h on which the unitary part of a acts we show that if n dimh then j a n resp k a n 2 and the equality holds if and only if a is of class sn resp one of the three conditions is true 1 a is of class sn 2 n is even and a is completely nonunitary with an 2 1 and an 1 1 and 3 n is even and a u a where u is unitary on a one dimensional space and a is of class sn 1

analysis on the existence of the steady states for an ecological mathematical model with predator prey dependent functional response abstract we establish a reaction diffusion ecological mathematical model with predator prey dependent functional response and the neumann boundary conditions first we study the bifurcation solution emitting from the unique positive constant solution by considering the local bifurcation finally we mainly analyze the coexistence of the prey and predator with the help of the fixed point index theory where we use a different approach in calculating one of the fixed point indices

global dynamics of vector borne diseases with horizontal transmission in host population the paper presents the dynamical features of a vector host epidemic model with direct transmission first we extended the model by taking into account the exposed individuals in both human and vector population with the impact of disease related deaths and total time dependent population size using lyapunov function theory some sufficient conditions for global stability of both the disease free equilibrium and the endemic equilibrium are obtained for the basic reproductive number r 0 1 a unique endemic equilibrium exists and is globally asymptotically stable furthermore it is found that the model exhibits the phenomenon of backward bifurcation where the stable disease free equilibria coexists with a stable endemic equilibrium finally numerical simulations are carried out to investigate the influence of the key parameters on the spread of the vector borne disease to support the analytical conclusion and illustrate possible behavioral scenarios of the model

global stability of an sveir epidemic model with ages of vaccination and latency the vaccination period i e the time period retaining acquired immunity following vaccination and the latent period are two important factors affecting disease dynamics classical compartmental disease models unrealistically assume that these periods are exponentially distributed in this paper these two ages are assumed to have arbitrary distributions that are represented by age specific rates leaving the vaccinated and the exposed classes resulting in an sveir model with population dynamics we investigate the global behavior of this model and derive its basic reproduction number the basic reproduction number completely determines the global dynamics of the model i e the disease free equilibrium is globally asymptotically stable and the disease always dies out when the basic reproduction number is less than unity whereas when the basic reproduction number is larger than unity there exists a unique endemic equilibrium which is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists the contributions of the vaccine wane rate to the basic reproduction number and the level of the endemic equilibrium are displayed

defect indices of powers of a contraction abstract let a be a contraction on a hilbert space h the defect index d a of a is by definition the dimension of the closure of the range of i a a we prove that 1 d a n nd a for all n 0 2 if in addition a n converges to 0 in the strong operator topology and d a 1 then d a n n for all finite n 0 n dim h and 3 d a d a implies d a n d a n for all n 0 the norm one index k a of a is defined as sup n 0 a n 1 when dim h m a lower bound for k a was obtained before k a m d a 1 we show that the equality holds if and only if either a is unitary or the eigenvalues of a are all in the open unit disc d a divides m and d a n nd a for all n 1 n m d a we also consider the defect index of f a for a finite blaschke product f and show that d f a d a n where n is the number of zeros of f

a finite difference method for a two sex model of population dynamics an explicit finite difference algorithm is developed to approximate the solution of a nonlinear and nonlocal system of integro differential equations that models the dynamics of a two sex population the algorithm is unconditionally stable the optimal rate of convergence of the algorithm is demonstrated for the maximum norm results from a numerical simulation of u s population growth from 1970 to 1980 are presented these compare favorably with the actual data excerpt

a note on the unitary part of a contraction short and independent proofs are given to two recent results of gau and wu on the unitary part of a contraction

finite blaschke products of contractions let a be a contraction on hilbert space h and a finite blaschke product in this paper we consider the problem when the norm of a is equal to 1 we show that 1 a 1 if and only if ak 1 where k is the number of zeros of counting multiplicity and 2 if h is finite dimensional and a has no eigenvalue of modulus 1 then the largest integer l for which al 1 is at least m n m where n dimh and m dimker i a a and moreover l n 1 if and only if m n 1

extremality of numerical radii of tensor products of matrices abstract for n by n and m by m complex matrices a and b respectively it is known that the inequality w a b a w b holds where w and denote respectively the numerical radius and the operator norm of a matrix in this paper we consider when this becomes an equality we show that the equality w a b a w b holds if and only if a and b have k by k compressions a 1 and b 1 respectively such that rank a 2 i k a 1 a 1 min r dim ker w b i k re e i b 1 we also give some consequences of this result in particular we show that if rank b sup k n a k a k then w a b a w b

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

epidemiological models for mutating pathogens we formulate epidemiological models for the transmission of a pathogen that can mutate in the host to create a second infectious mutant strain the models account for mutation rates that depend on how long the host has been infected we derive explicit formulas for the reproductive number of the epidemic based on the local stability of the infection free equilibrium we analyze the existence and stability of the boundary equilibrium whose infection components are zero and positive respectively and the endemic equilibrium whose components are all positive we establish the conditions for global stability of the infection free and boundary equilibria and local stability of the endemic equilibrium for the case where there is no age structure for the pathogen in the infected population we show that under certain circumstances there is a hopf bifurcation where the endemic equilibrium loses its stability and periodic solutions appear we provide examples and numerical simulations to illustrate the hopf b

endemic threshold results for an age structured sis epidemic model with periodic parameters abstract the main contribution of this paper is to obtain a threshold value for the existence and uniqueness of a nontrivial endemic periodic solution of an age structured sis epidemic model with periodic parameters under the assumption of the weak ergodicity of a non autonomous lotka mckendrick system we formulate a normalized system for an infected population as an initial boundary value problem of a partial differential equation the existence problem for endemic periodic solutions is reduced to a fixed point problem of a nonlinear integral operator acting on a banach space of locally integrable periodic l 1 valued functions we prove that the spectral radius of the frechet derivative of the integral operator at zero plays the role of a threshold for the existence and uniqueness of a nontrivial fixed point of the operator corresponding to a nontrivial periodic solution of the original differential equation in a weak sense if the malthusian parameter of the host population is equal to zero our threshold value is equal to the well known epidemiological threshold value the basic reproduction number r 0 however if it is not the case then two threshold values are different from each other and we have to pay attention on their actual biological implications

asymptotic stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem