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 Abdelilah Kaddar Electronic Journal of Differential Equations , 2009, Abstract: In this paper, a delayed SIR epidemic model with modified saturated incidence rate is proposed. The local stability and the existence of Hopf bifurcation are established. Also some numerical simulations are given to illustrate the theoretical analysis.
 International Journal of Differential Equations , 2012, DOI: 10.1155/2012/408637 Abstract: We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. We investigate the qualitative behaviour of the model and find the conditions that guarantee the asymptotic stability of corresponding steady states. We present the conditions in the time lag in which the DDE model is stable. Hopf bifurcation analysis is also addressed. Numerical simulations are provided in order to illustrate the theoretical results and gain further insight into the behaviour of this system. 1. Introduction Epidemics have ever been a great concern of human kind, because the impact of infectious diseases on human and animal is enormous, both in terms of suffering and social and economic consequences. Mathematical modeling is an essential tool in studying a diverse range of such diseases to gain a better understanding of transmission mechanisms, and make predictions; determine and evaluate control strategies. Many authors have proposed various kinds of epidemic models to understand the mechanism of disease transmission (see [1–10] and references therein). The basic elements for the description of infectious diseases have been considered by three epidemiological classes: that measures the susceptible portion of population, the infected, and the removed ones. Kermack and McKendrick  described the simplest SIR model which computes the theoretical number of people infected with a contagious illness in a closed population over time. Transmission of a disease is a dynamical process driven by the interaction between susceptible and infective. The behaviour of the SIR models are greatly affected by the way in which transmission between infected and susceptible individuals are modelled. The simplest model in which recovery does not give immunity is the SIS model, since individuals move from the susceptible class to the infective class and then back to the susceptible class upon recovery. If individuals recover with permanent immunity, then the simplest model is an SIR model. If individuals recover with temporary immunity so that they eventually become susceptible again, then the simplest model is an SIRS model. If individuals do not recover, then the simplest model is an SI model. In general, SIR (epidemic and endemic) models are appropriate for viral agent diseases such as measles, mumps, and smallpox, while SIS models are appropriate for some bacterial agent diseases such as meningitis, plague, and sexually transmitted diseases, and for protozoan agent diseases such
 卢加荣, 张建刚, 罗宏伟, 殷俊, 庞琴 Advances in Applied Mathematics (AAM) , 2016, DOI: 10.12677/AAM.2016.54070 Abstract: 本文主要分析带有时滞影响下的双传染病假设模型的稳定性及Hopf分岔，研究了不同情况下的系统唯一的正平衡点的稳定性，并通过分析相应线性系统的特征根的分布，得到了使得系统平衡的条件。然而，当时滞的值通过一个临界值的时候，Hopf分岔就会发生。最后, 运用MATLAB数值模拟去验证得到的理论结果。 This paper mainly investigates the stability and Hopf bifurcation in a delayed epidemic model system with double epidemic hypothesis. We study the stability of the unique positive equilibrium for the system under different conditions. By analyzing the distribution of characteristic roots of corresponding linearized system, we obtain the conditions for keeping the system to be stable. Moreover, it is illustrated that the Hopf bifurcation will occur when the delay passes through a criti-cal value. Then we use the MATLAB numerical simulations for justifying the theoretical results.
 Abstract and Applied Analysis , 2014, DOI: 10.1155/2014/930541 Abstract: We consider an SIR endemic model in which the contact transmission function is related to the number of infected population. By theoretical analysis, it is shown that the model exhibits the bistability and undergoes saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. Furthermore, we find that the threshold value of disease spreading will be increased, when the half-saturation coefficient is more than zero, which means that it is an effective intervention policy adopted for disease spreading. However, when the endemic equilibria exist, we find that the disease can be controlled as long as we let the initial values lie in the certain range by intervention policy. This will provide a theoretical basis for the prevention and control of disease. 1. Introduction The classical SIR model for disease transmission has been widely studied. It is one of the most important issues that the dynamical behaviors are changed by the different incidence rate in epidemic system. For the incidence rate, we divided into two categories: one is that Capasso and Serio  proposed the infection force which is a saturated curve, described “crowding effect” or “protection measures;” the other is the infection force that describes the effect of “intervention policy,” for example, closing schools and restaurants and postponing conferences (see Figure 1). For the model with the saturated infection force, , which is one of the typical infection forces, the rich dynamical behaviors were founded by Ruan and Wang  and Tang et al. . The model with the incidence rate can be suited for many infectious diseases, including measles, mumps, rubella, chickenpox, and influenza. For more research literatures about nonlinear infection rate see [4–9]. However, for some parasite-host models, by observing macro- and microparasitic infections, one finds that the infection rate is an increasing function of the parasite dose, usually sigmoidal in shape [10, 11]. So we will build a model with sigmoidal incidence rate which is taken into account “crowding effect” and “saturated effect.” According to the parasite-host model which is proposed by Anderson and May (1979) [12, 13], the model is as follows: where are susceptible hosts, infected hosts, and removed hosts, respectively. is the birth rate of susceptible host, is the natural death rate of a population, is the removal rate, and is the per capita infection-related death rate. If we denote infection force , can be explained as the rate of valid contact. At the beginning of disease, most people have poor awareness of
 Fatihcan M. Atay Mathematics , 2008, DOI: 10.1137/060673813 Abstract: The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean.
 Mathematics , 2015, Abstract: We study the dynamics of a SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment, that takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions we prove the existence of backward bifurcation, the stability and the direction of Hopf bifurcation. We also explore how the mechanism of backward bifurcation affects the control of the infectious disease. Numerical simulations are presented to illustrate the theoretical findings.
 Quantitative Biology , 2009, DOI: 10.1103/PhysRevE.80.061920 Abstract: The bifurcation diagram of a model stochastic differential equation with delayed feedback is presented. We are motivated by recent research on stochastic effects in models of transcriptional gene regulation. We start from the normal form for a pitchfork bifurcation, and add multiplicative or parametric noise and linear delayed feedback. The latter is sufficient to originate a Hopf bifurcation in that region of parameters in which there is a sufficiently strong negative feedback. We find a sharp bifurcation in parameter space, and define the threshold as the point in which the stationary distribution function p(x) changes from a delta function at the trivial state x=0 to p(x) ~ x^alpha at small x (with alpha = -1 exactly at threshold). We find that the bifurcation threshold is shifted by fluctuations relative to the deterministic limit by an amount that scales linearly with the noise intensity. Analytic calculations of the bifurcation threshold are also presented in the limit of small delay tau -> 0 that compare quite favorably with the numerical solutions even for tau = 1.
 物理学报 , 2011, Abstract: In this paper, a resistive-capacitive-shunted Josephson junction with linear delayed feedback is considered. The stability of trivial solution of the controlled system is analyzed using nonlinear dynamics theory, and the theoretical results show that the stable trivial solution of the system will lose its stability via Hopf bifurcation as control parameter varies. The critical parameter condition of Hopf bifurcation is also derived. Numerical analysis of the controlled system is carried out under different parameter conditions, and the results show that the stable periodic solution generated by supercritical Hopf bifurcation may transit to chaos gradually through a process of symmetry-breaking bifurcation and period-doubling bifurcation.
 Mathematical Problems in Engineering , 2009, DOI: 10.1155/2009/759363 Abstract: An efficient method is proposed to study delay-induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension-three and codimension-two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.
 Electronic Journal of Differential Equations , 2007, Abstract: In this paper, a delayed SIS (Susceptible Infectious Susceptible) model with stage structure is investigated. We study the Hopf bifurcations and stability of the model. Applying the normal form theory and the center manifold argument, we derive the explicit formulas determining the properties of the bifurcating periodic solutions. The conditions to guarantee the global existence of periodic solutions are established. Also some numerical simulations for supporting the theoretical are given.
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