bifurcations in an epidemic model with constant removal rate of the infectives an epidemic model with a constant removal rate of infective individuals is proposed to understand the effect of limited resources for treatment of infectives on the disease spread it is found that it is unnecessary to take such a large treatment capacity that endemic equilibria disappear to eradicate the disease it is shown that the outcome of disease spread may depend on the position of the initial states for certain range of parameters it is also shown that the model undergoes a sequence of bifurcations including saddle node bifurcation subcritical hopf bifurcation and homoclinic bifurcation

stationary distribution of a stochastic delayed sveir epidemic model with vaccination and saturation incidence abstract in this paper we develop and analyze a stochastic delayed sveir epidemic model with vaccination and saturation incidence by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model the existence of a stationary distribution implies stochastic weak stability numerical simulations are introduced to illustrate the analytical result

interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback models of class b lasers subject to either an optoelectronic or an optical feedback are investigated analytically and numerically we derive slow time amplitude equations from the laser delay differential equations and find multiple bifurcating and isolated branches of periodic solutions we then show that secondary bifurcations to tori result from the interaction of pairs of hopf bifurcations the branches emerging from these bifurcations are followed numerically using a continuation method developed for delay differential equations

stability and oscillations of multistage sis models depend on the number of stages abstract in this paper we consider multistage sis models of infectious diseases where infected individuals are passing through infectious stages i 1 i 2 i n and then return to the susceptible compartment first we calculate the basic reproduction number r 0 and prove that the disease dies out for r 0 1 while a unique endemic equilibrium exists for r 0 1 our main result is that the stability of the endemic equilibrium depends on the number of stages the endemic equilibrium is always stable when n 3 while for any n 3 it can be either stable or unstable depending on the particular choice of the parameters we generalize previous stability results for sirs models as well and point out a mistake in the literature for multistage seirs models our results have important implications on the discretization of infectious periods with varying infectivity

nonlinear oscillations in epidemic models constant parameter epidemic models for a closed population described by integral and integrodifferential equations are analyzed to determine whether time delays can give rise to periodic oscillations delays can destabilize the steady state in one finite delay case and in one infinite delay case considered hopf bifurcation techniques are used to show the existence of locally asymptotically stable periodic solutions for certain parameter values inclusion of time delays in another related model does not alter the local asymptotic stability of the endemic steady state

an efficient method for computing invariant manifolds of planar maps abstract an efficient method for computing invariant manifolds of mappings in the plane is described and compared to a simple method in common use the emphasis is on numerically producing smoothly resolved segments of invariant manifolds of arbitrary length while requiring a minimum number of calls to the map itself a brief estimate of errors is given and applications are discussed qualitative and quantitative comparisons between the two methods are made using an example problem the present method is applied to computing unstable manifolds for the well know periodically forced duffing system and the henon map for which the manifold is a strange attractor

a hepatitis b and c virus model with age since infection that exhibits backward bifurcation we propose a mathematical model describing the dynamics of hepatitis b or c virus infection the method of characteristics reduces the age structured model to a system of differential equations with distributed delay the model is rigorously analyzed and a detailed stability analysis is performed the model consisting of two mutually exclusive compartments representing the interaction of the virus in the liver and the blood has a locally asymptotically stable disease free equilibrium dfe whenever a certain epidemiological threshold known as the basic reproduction number r 0 is less than unity a sufficient condition for the global asymptotic stability of the trivial steady state is obtained further under biologically realistic hypotheses the model exhibits the phenomenon of backward bifurcation where the stable dfe coexists with a stable endemic equilibrium this means that reducing the basic reproduction number r 0 below 1 is not always sufficient to control hbv hcv infection when the viru

mixed pulse vaccination strategy in epidemic model with realistically distributed infectious and latent times in this paper we study the use of a pulse vaccination strategy to eradicate infectious diseases modelable by epidemic model having gamma distributed infectious and latent time we demonstrate the global asymptotic stability of the eradication solution for a general model in which co presence of a continuous vaccination strategy the non permanent duration of the immunization and the emerging problem of the vaccine failures are considered

lyapunov functions and global stability for age structured hiv infection model we study the basic age structured population model describing the hiv infection process which is defined by pdes the model allows the production rate of viral particles and the death rate of productively infected cells to vary and depend on the infection age by using the direct lyapunov method and constructing suitable lyapunov functions dynamical properties of the age structured model without or with drug treatment are established the results show that the global asymptotic stability of the infection free steady state and the infected steady state depends only on the basic reproductive number determined by the burst size further we establish mathematically that the typical ode and dde delay differential equation models of hiv infection are equivalent to two special cases of the above pde models

global dynamics of an seir epidemic model with vertical transmission we study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission the total host population is assumed to have constant density and the incidence term is of the bilinear mass action form we prove that the global dynamics are completely determined by the basic reproduction number r0 p q where p and q are fractions of infected newborns from the exposed and infectious classes respectively if r 0 p q le 1 the disease free equilibrium is globally stable and the disease always dies out if r0 p q 1 a unique endemic equilibrium exists and is globally stable in the interior of the feasible region and the disease persists at an endemic equilibrium state if it initially exists the contribution of the vertical transmission to the basic reproduction number is also analyzed

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

growing 1d and quasi 2d unstable manifolds of maps we present a new 1d algorithm for computing the global one dimensional unstable manifold of a saddle point of a map our method can be generalized to compute two dimensional unstable manifolds of maps with three dimensional state spaces this is shown here with a quasi 2d q2d algorithm for the special case of a quasiperiodically forced map which allows for a substantial simplification of the general case described in krauskopf and osinga 1998 the key idea is to grow the manifold in steps which consist of finding a new point on the manifold at a prescribed distance from the last point the speed of growth is determined only by the curvature of the manifold and not by the dynamics the performance of the 1d algorithm is demonstrated with a constructed test example and it is then used to compute one dimensional manifolds of a map modeling mixing in a stirring tank with the q2d algorithm we compute two dimensional unstable manifolds in the quasiperiodically forced henon map

one dimensional unstable eigenfunction and manifold computations in delay differential equations in this paper we present a new numerical technique for computing the unstable eigenfunctions of a saddle periodic orbit in a delay differential equation this is used to obtain the necessary starting data for an established algorithm for computing one dimensional 1d unstable manifolds of an associated saddle fixed point of a suitable poincare map to illustrate our method we investigate an intermittent transition to chaos in a delay system describing a semiconductor laser subject to phase conjugate feedback

the effects of pulse vaccination on seir model with two time delays abstract in this paper impulsive vaccination has been applied to control the spread and transmission of an infectious disease an seir epidemic model with two time delays and pulse vaccination is formulated the exact infection free periodic solution of the impulsive epidemic system is obtained moreover we show that if the vaccination rate is larger than the infectious population disappear so the disease dies out while if the vaccination rate is less than the infectious population persist our results indicate that a large vaccination rate or a short period of pulsing or a long latent period of the disease is a sufficient condition for the eradication of the disease

vaccination policies and nonlinear force of infection generalization of an observation by alexander and moghadas 2004 it has been recently showed in m e alexander s m moghadas periodicity in an epidemic model with a generalized non linear incidence math biosci 189 2004 75 96 that one of the possible causes of unexpected failures of some vaccination campaigns may be the nonlinearity of the force of infection the aim of this paper is to give a simple and biologically meaningful sufficient condition for the globally stable eradication of diseases whose spreading is described via a generic class of sir and seir epidemic models ruled by nonlinear force of infection we consider a scenario in which there are both traditional and pulse vaccination strategies

on global properties of time delayed feedback control weakly nonlinear analysis we address the problem which type of initial state yields successful stabilisation when time delayed feedback control is applied using well known concepts from bifurcation theory we provide a general mechanism that limits the corresponding basins of attraction application of normal form analysis to general delay differential equations proves the universality of our results the empirical criteria which determine the prospective size of the basin are easily accessible and a demonstration in terms of a simple numerical simulation is provided

global stability of an age structure epidemic model with imperfect vaccination and relapse abstract a new age structured epidemic model with imperfect vaccination and relapse is proposed the total population of our model is partitioned into five subclasses susceptible class s vaccinated class v exposed class e infectious class i and removed class r age structures are equipped with in exposed and recovered classes furthermore imperfect vaccination is also introduced in our model the basic reproduction number r 0 is defined and proved as a threshold parameter of the model asymptotic smoothness of solutions and uniform persistence of the system are showed via reformulating the system as a system of volterra integral equation furthermore by constructing proper volterra type lyapunov functional we get when r 0 1 the disease free equilibrium is globally asymptotically stable when r 0 1 the endemic equilibrium is globally stable our results show that to increase the efficiency of vaccination and reduce influence of relapse are vital essential for controlling epidemic

a behavioral approach to delay differential systems we will study linear time invariant delay differential systems from the behavioral point of view as it was introduced for dynamical systems by willems dynam report 2 1989 pp 171 269 a ring cal h which lies between bbb r s z z 1 and bbb r s z z 1 will be presented whose elements can be interpreted as a generalized version of delay differential operators on cal c infty bbb r bbb r in this framework a behavior is the kernel of such an operator using the ring cal h an algebraic characterization of inclusion respectively equality of the behaviors under consideration is given finally controllability of the behaviors is characterized in terms of the rank of the associated matrices in the case of time delay state space systems this criterion becomes the known hautus criterion for spectral controllability

inverse problems for delay differential equations using the collage theorem abstract we present a theoretical framework to solve inverse problems for systems of delayed ordinary differential equations delay odes that allows us to estimate the values of unknown parameters such as coefficients and time delays using available time series data this work builds on similar results for non delayed odes inspired by the collage theorem we discuss technical details related to the implementation of the method including the use of non convex optimization to recover unknown delay values the performance of the method is demonstrated using simulated and noised datasets to recover parameters in models applied to human health these include an additive delay model for the population dynamics of malaria transmission and a distributed delay model for the homeostasis of glucose and insulin in the bloodstream

a mathematical study of a model for childhood diseases with non permanent immunity protecting children from diseases that can be prevented by vaccination is a primary goal of health administrators since vaccination is considered to be the most effective strategy against childhood diseases the development of a framework that would predict the optimal vaccine coverage level needed to prevent the spread of these diseases is crucial this paper provides this framework via qualitative and quantitative analysis of a deterministic mathematical model for the transmission dynamics of a childhood disease in the presence of a preventive vaccine that may wane over time using global stability analysis of the model based on constructing a lyapunov function it is shown that the disease can be eradicated from the population if the vaccination coverage level exceeds a certain threshold value it is also shown that the disease will persist within the population if the coverage level is below this threshold these results are verified numerically by constructing and then simulating a robust semi explicit second order finite difference method

some basic remarks on eigenmode expansions of time delay dynamics abstract we describe in elementary terms how eigenmode expansions can be used to deal with differential difference equations as particular applications we present the full analytical solution of linear stochastic time delay systems and the weakly nonlinear analysis of nonlinear differential difference equations in the limit of large time delay our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the lambert w function and on the explicit formula for the eigenfunctions of the adjoint equation

global stability of infectious disease models using lyapunov functions two systematic methods are presented to guide the construction of lyapunov functions for general infectious disease models and are thus applicable to establish their global dynamics specifically a matrix theoretic method using the perron eigenvector is applied to prove the global stability of the disease free equilibrium while a graph theoretic method based on kirchhoff s matrix tree theorem and two new combinatorial identities are used to prove the global stability of the endemic equilibrium several disease models in the literature and two new cholera models are used to demonstrate the applications of these methods

semiconductor laser with phase conjugate feedback dynamics and bifurcations this paper presents the dynamics and bifurcations of a semiconductor laser subject to instantaneous phase conjugate feedback recently the behavior of such a laser has been explored by means of bifurcation diagrams however the exact nature of the involved dynamics and bifurcations remained unclear here we present a detailed study of the changes of the dynamics as the feedback strength is varied most prominent are symmetry breaking and restoring bifurcations tori and their bifurcations and a sudden transition between chaos and a stable limit cycle due to a saddle node bifurcation of limit cycles

a class of nonlinear sturm liouville problems with infinitely many solutions on etudie l existence des solutions des problemes de sturm liouville non lineaires dont la partie lineaire est a la resonance on montre que de tels problemes peuvent avoir une infinite de solutions si les perturbations non lineaires sont periodiques

numerical bifurcation analysis of the traveling wave model of multisection semiconductor lasers abstract the traveling wave model consisting of a hyperbolic system of linear first order pde s nonlinearly coupled with a system of ode s is used to model the dynamics of multisection semiconductor lasers it is shown how low dimensional systems of ode s based on the projection of the original system into the subspace spanned only by a few instantaneously changing spectral elements allow a precise approximation of the original model the constructed low dimensional mode approximation systems are accessible to classical bifurcation analysis and numerical tools the presented method is applied to characterise quasi periodic orbits arising in a three section distributed feedback laser the obtained results are highly important for tailoring such pulsating lasers to be applied in optical communication systems

global attractivity and permanence of a delayed sveir epidemic model with pulse vaccination and saturation incidence in this study we formulate and analyze a new sveir epidemic disease model with time delay and saturation incidence and analyze the dynamic behavior of the model under pulse vaccination using the discrete dynamical system determined by the stroboscopic map we obtain an infection free periodic solution further show that the infection free periodic solution is globally attractive for some parameters of the model under appropriate conditions the permanence of the model is investigated analytically by computer simulation it is concluded that a large vaccination rate or a short pulse of vaccination or a long latent period are each a sufficient condition for the extinction of the disease

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

global results for an epidemic model with vaccination that exhibits backward bifurcation vaccination of both newborns and susceptibles is included in a transmission model for a disease that confers immunity the interplay of the vaccination strategy together with the vaccine efficacy and waning is studied in particular it is shown that a backward bifurcation leading to bistability can occur under mild parameter constraints compound matrices are used to show that each orbit limits to an equilibrium in the case of bistability this global result requires a novel approach since there is no compact absorbing set

multiple solutions of the periodic boundary value problem for some forced pendulum type equations if f x c 0 and e is a constant t then integrating 0 1 over 0 272 after multiplication of both members of the equation by x shows that the only possible solutions are constant and hence satisfy a sin x c consequently if we exclude the trivial translation by a multiple of 275 0 1 will have no solution if dj a one solution if lp a and two solutions if lc l a in 0th er t erms the intersection of the range of the operator d2 dt2 c d dt f u sin acting on zn periodic functions of class g with the subspace of constant functions in the space c o 2711 of real continuous functions on 0 27r is the closed interval a a whose interior points are images of two distinct solutions and boundary points of one we try to obtain similar results for the case where e is no more constant and

dynamics of a discretized sir epidemic model with pulse vaccination and time delay we derive a discretized sir epidemic model with pulse vaccination and time delay from the original continuous model the sufficient conditions for global attractivity of an infection free periodic solution and permanence of our model are obtained improving discretization our results are corresponding to those in the original continuous model

stochastic bifurcation in a driven laser system experiment and theory we analyze the effects of stochastic perturbations in a physical example occurring as a higher dimensional dynamical system the physical model is that of a class b laser which is perturbed stochastically with finite noise the effect of the noise perturbations on the dynamics is shown to change the qualitative nature of the dynamics experimentally from a stochastic periodic attractor to one of chaoslike behavior or noise induced chaos to analyze the qualitative change we apply the technique of the stochastic frobenius perron operator l billings et al phys rev lett 88 234101 2002 to a model of the experimental system our main result is the identification of a global mechanism to induce chaoslike behavior by adding stochastic perturbations in a realistic model system of an optics experiment in quantifying the stochastic bifurcation we have computed a transition matrix describing the probability of transport from one region of phase space to another which approximates the stochastic frobenius perron operator this mechanism depends on both the standard deviation of the noise and the global topology of the system our result pinpoints regions of stochastic transport whereby topological deterministic dynamics subjected to sufficient noise results in noise induced chaos in both theory and experiment

a boundary value problem with a periodic nonlinearity soit g r r une fonction continue et periodique de periode t 0 et soit h l 1 0 on considere le probleme d existence pour le probleme aux limites de picard u u g u h u 0 0 u

on periodic solutions of forced pendulum like equations the existence and multiplicity of t periodic solutions of the forced pen dulum equation y t q t a sin j t e t 1 have been considered recently by a number of authors and we refer to 2 3 for the history and a survey of this problem the main techniques used up to now have been upper and lower solutions topological degree and variational methods a precise and explicit characterizarion of the set of e e

a note on a resonance problem in this paper we prove the existence of at least one solution to the problem where k is an eigenvalue of the linear part h is orthogonal to the eigenspace corresponding to r and g is a nonlinear perturbation which can be for instance a continuous periodic real function with mean value zero we employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which r is assumed to be simple the final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of p rabinowitz

global attractivity and permanence of a sveir epidemic model with pulse vaccination and time delay in this study we propose a new sveir epidemic disease model with time delay and analyze the dynamic behavior of the model under pulse vaccination pulse vaccination is an effective strategy for the elimination of infectious disease using the discrete dynamical system determined by the stroboscopic map we obtain an infection free periodic solution we also show that the infection free periodic solution is globally attractive when some parameters of the model under appropriate conditions the permanence of the model is investigated analytically our results indicate that a large vaccination rate or a short pulse of vaccination or a long latent period is a sufficient condition for the extinction of the disease

asymptotic analysis of oscillating parametric integrals and ordinary boundary value problems at resonance abstract this paper deals with boundary value problems whose nonlinear part involves periodic functions and such that the linear part has a one dimensional solution space we shall study the existence and multiplicity of solutions using various methods of nonlinear analysis such as the lyapunov schmidt reduction and methods of critical point theory the proofs are based on some general results on the oscillation and asymptotic behavior of certain parametric integrals

an epidemic model of a vector borne disease with direct transmission and time delay abstract this paper considers an epidemic model of a vector borne disease which has direct mode of transmission in addition to the vector mediated transmission the incidence term is assumed to be of the bilinear mass action form we include both a baseline ode version of the model and a differential delay model with a discrete time delay the ode model shows that the dynamics is completely determined by the basic reproduction number r 0 if r 0 1 the disease free equilibrium is globally stable and the disease dies out if r 0 1 a unique endemic equilibrium exists and is locally asymptotically stable in the interior of the feasible region the delay in the differential delay model accounts for the incubation time the vectors need to become infectious we study the effect of that delay on the stability of the equilibria we show that the introduction of a time delay in the host to vector transmission term can destabilize the system and periodic solutions can arise through hopf bifurcation

almost periodic solutions for an impulsive delay nicholson s blowflies model by means of the contraction mapping principle and gronwall bellman s inequality we prove the existence and exponential stability of positive almost periodic solution for an impulsive delay nicholson s blowflies model the main results are illustrated by an example

some new observations on the classical logistic equation with heredity abstract several new and significant observations are presented pertaining to the classical problem of single population growth with hereditary influences in its conventional form the resulting equation with heredity is mathematically represented by a nonlinear volterra integro differential equation in this paper we propose a new differential formulation where the dependent variable is now defined in terms of the integral of the unknown population this formulation allows us to develop novel analyses leading to enlightening results some particular findings include the development and analysis of an integrated phase plane the elucidation of the exact value for the extremum of the population and several other important functional relations at that corresponding time the development of two analytic expressions for determining the time at which the population peaks the determination of the upper asymptote for the cumulative population and the development of an accurate early time solution as obtained from a riccati equation additionally we illustrate that an analytical solution based on taylor series expansions can be developed with the aid of mathematica tm a pure numerical solution is offered for comparison with the analytic solution

sign patterns that require or allow particular refined inertias abstract the refined inertia ri a of a real n n matrix a is the ordered 4 tuple n n n z 2 n p where n resp n n z 2 n p is the number of positive resp negative zero nonzero pure imaginary eigenvalues of a let h n 0 n 0 0 0 n 2 0 2 2 n 2 0 0 an n n sign pattern s n requires h n if h n ri a a has sign pattern s n and allows h n if h n ri a a has sign pattern s n sign patterns that require or allow h n are investigated for small values of n and for patterns with all diagonal entries negative examples are given relating these concepts to hopf bifurcation in dynamical systems

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

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

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

computing unstable manifolds of periodic orbits in delay differential equations we present the first algorithm for computing unstable manifolds of saddle type periodic orbits with one unstable floquet multiplier in systems of autonomous delay differential equations ddes with one fixed delay specifically we grow the one dimensional unstable manifold wu q of an associated saddle fixed point q of a poincare map defined by a suitable poincare section starting close to q along the linear approximation to wu q given by the associated eigenfunction our algorithm grows the manifold as a sequence of points where the distance between points is governed by the curvature of the one dimensional intersection curve wu q of wu q with our algorithm makes it possible to study global bifurcations in ddes we illustrate this with the break up of an invariant torus and a subsequent crisis bifurcation to chaos in a dde model of a semiconductor laser with phase conjugate feedback