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 Journal of Biomedicine and Biotechnology , 2007, DOI: 10.1155/2007/64870 Abstract: Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as an effective strategy for the elimination of infectious diseases. An SIR epidemic model with pulse vaccination and distributed time delay is proposed in this paper. Using the discrete dynamical system determined by the stroboscopic map, we obtain the exact infection-free periodic solution of the impulsive epidemic system and prove that the infection-free periodic solution is globally attractive if the vaccination rate is larger enough. Moreover, we show that the disease is uniformly persistent if the vaccination rate is less than some critical value. The permanence of the model is investigated analytically. Our results indicate that a large pulse vaccination rate is sufficient for the eradication of the disease.
 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 [11] 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
 Abstract and Applied Analysis , 2013, DOI: 10.1155/2013/670263 Abstract: Mathematical models can assist in the design and understanding of vaccination strategies when resources are limited. Here we propose and analyse an SIR epidemic model with a nonlinear pulse vaccination to examine how a limited vaccine resource affects the transmission and control of infectious diseases, in particular emerging infectious diseases. The threshold condition for the stability of the disease free steady state is given. Latin Hypercube Sampling/Partial Rank Correlation Coefficient uncertainty and sensitivity analysis techniques were employed to determine the key factors which are most significantly related to the threshold value. Comparing this threshold value with that without resource limitation, our results indicate that if resources become limited pulse vaccination should be carried out more frequently than when sufficient resources are available to eradicate an infectious disease. Once the threshold value exceeds a critical level, both susceptible and infected populations can oscillate periodically. Furthermore, when the pulse vaccination period is chosen as a bifurcation parameter, the SIR model with nonlinear pulse vaccination reveals complex dynamics including period doubling, chaotic solutions, and coexistence of multiple attractors. The implications of our findings with respect to disease control are discussed. 1. Introduction Epidemiology is the study of the spread of diseases with the objective of tracing factors that are responsible for or contribute to their occurrence and serves as the foundation and logic of interventions made in the interest of public health and preventive medicine. Mathematical models describing the population dynamics of infectious diseases have played an important role in better understanding epidemiological patterns and disease control for a long time. Various epidemic models have been proposed and explored extensively and considerable progress has been achieved in the studies of disease control and prevention (see [1–3] and the references therein). Outbreaks of infectious diseases have not only caused the loss of billions of lives but have often also rapidly damaged social economic systems, bringing about much human misery. Consequently, the focus of our research has been on how to prevent and cure infectious diseases effectively. It is well known that one of the most important concerns in the analysis of epidemic logical models is the efficacy of vaccination programmes. This subject gained prominence as a result of highly successful application of vaccinations for the worldwide eradication of smallpox
 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.
 Electronic Journal of Differential Equations , 2012, Abstract: In this article we propose a comparison of a delayed SIR model and its corresponding SEIR model in terms of global stability. We consider a saturated incidence rate and we determine, using Lyapunov functionals, conditions by which the disease-free equilibrium and the endemic equilibrium are globally asymptotically stable. Also some numerical simulations are given to compare a global behaviour of a delayed SIR model and its corresponding SEIR model.
 Physics , 2006, DOI: 10.1142/S1793524508000370 Abstract: A pulse vaccination SIR model with periodic infection rate $\beta (t)$ have been proposed and studied. The basic reproductive number $R_0$ is defined. The dynamical behaviors of the model are analyzed with the help of persistence, bifurcation and global stability. It has been shown that the infection-free periodic solution is globally stable provided $R_0 < 1$ and is unstable if $R_0>1$. Standard bifurcation theory have been used to show the existence of the positive periodic solution for the case of $R_0 \to1^+$. Finally, the numerical simulations have been performed to show the uniqueness and the global stability of the positive periodic solution of the system.
 Open Journal of Modelling and Simulation (OJMSi) , 2015, DOI: 10.4236/ojmsi.2015.33011 Abstract: We consider a SIR epidemic model with saturated incidence rate and treatment. We show that if the basic reproduction number, R0 is less than unity and the disease free equilibrium is locally asymptotically stable. Moreover, we show that if R0 > 1, the endemic equilibrium is locally asymptotically stable. In the end, we give some numerical results to compare our model with existing model and to show the effect of the treatment term on the model.
 International Journal of Stochastic Analysis , 2013, DOI: 10.1155/2013/431257 Abstract: We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results. 1. Introduction Many mathematical models have been developed in order to understand disease transmissions and behavior of epidemics. One of the earliest of these models was used by Kermack and Mckendrick [1], by considering the total population into three classes, namely, susceptible individuals, infected individuals, and recovered individuals which is known to us as SIR epidemic model. This SIR epidemic model is very important in today's analysis of diseases. The disease transmission process is unknown in detail. However, several authors proposed different forms of incidences rate in order to model this disease transmission process. In this paper, we consider the following model with specific nonlinear incidence rate: where is the recruitment rate of the population, is the natural death rate of the population, is the death rate due to disease, is the recovery rate of the infective individuals, is the infection coefficient, and is the incidence rate, where are constants. It is very important to note that this incidence rate becomes the bilinear incidence rate if , the saturated incidence rate if or , the modified saturated incidence rate proposed in [2, 3] when , and Crowley-Martin functional response presented in [4–6] if . On the other hand, environmental fluctuations have great influence on all aspects of real life. The aim of this work is to study the effect of these environmental fluctuations on the model (1). We assume that the stochastic perturbations are of white noise type and that they are proportional to the distances of and , respectively. Then, the system (1) will be extended to the following system of stochastic differential equation: where , are the positive points of equilibrium for the corresponding deterministic system (1), are independent standard Brownian motions, and represent the intensities of , respectively. The rest of paper is organized as follows. In the next section, we present the stability analysis of our stochastic model (2). In Section 3, we present the numerical simulation to illustrate our result. The conclusion of our paper is in Section 4. 2. Stability Analysis of Stochastic Model Clearly, the system (1) has a
 Discrete Dynamics in Nature and Society , 2011, DOI: 10.1155/2011/350892 Abstract: Based on some well-known SIR models, a revised nonautonomous SIR epidemic model with distributed delay and density-dependent birth rate was considered. Applying some classical analysis techniques for ordinary differential equations and the method proposed by Wang (2002), the threshold value for the permanence and extinction of the model was obtained. 1. Introduction For understanding the spread of infectious diseases in population, mathematical models that use the theories of ordinary differential equations in epidemiology have been developed rapidly. Epidemic models with delay, including either the autonomous continuous systems or the discrete ones, were discussed by many authors [1–13]. The important subjects for this models are looking for the threshold value that determines whether the infectious disease will be permanent or extinct. Hence, permanence of disease plays an important role in epidemiology. Furthermore, it is well known that models with distributed delay are more appropriate than the discrete ones because it is considered more realistic to assume the infectivity to be a function of the duration since infection and up to some maximum duration. Recently, Song and Ma in [4] and Song et al. in [5] discussed the permanence of disease in a generalized autonomous SIR epidemic model with density dependent birth rate. On the other hand, for the nonautonomous systems [1] (and the reference therein), the literature is still very inadequate. Motivated by the works in [1, 2, 4, 5, 9, 10], in this paper, we will consider the following nonautonomous delayed systems with density-dependent birth rate: where denote the total number of a population at time , is the susceptible population, is the infective population and is the removed population. It is assumed that all newborns are susceptible. Functions and are instantaneous per capita mortality rates of susceptible, infective, and recovered population, respectively; functions and represent the birth rate of the population and the recovery rate of infectives, respectively; function reflects the relation between the birth rate and the density of population. The nonnegative constant is the time delay. The function is nondecreasing and has hounded variation such that 2. Preliminaries Firstly, we give some notations for convenience: if is a continuous bounded function defined on , then we set Secondly, for system (1.1), we introduce the following assumption: functions are nonnegative continuous bounded functions and have positive lower bounds. It is biologically natural to assume that (i.e., epidemics will
 Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.410A2006 Abstract: In this paper, a delayed SIR model with exponential demographic structure and the saturated incidence rate is formulated. The stability of the equilibria is analyzed with delay: the endemic equilibrium is locally stable without delay; and the endemic equilibrium is stable if the delay is under some condition. Moreover the dynamical behaviors from stability to instability will change with an appropriate critical value. At last, some numerical simulations of the model are given to illustrate the main theoretical results.
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