Nonlinear Analysis: Real World Applications 11 (2010) 88–98 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa A delay SIR epidemic model with pulse vaccination and incubation times ✩ Xinzhu Meng a,∗ , Lansun Chen b , Bo Wu a a College of Science, Shandong University of Science and Technology, Qingdao 266510, PR China b Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, PR China article info Article history: Received 18 April 2008 Accepted 10 October 2008 Keywords: Permanence Pulse vaccination Nonlinear incidence Time delay SIR epidemic model abstract In this paper, a new delay SIR epidemic model with pulse vaccination and incubation times is considered. We obtain an infection-free semi-trivial periodic solution and establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination and nonlinear incidence have significant effects on the dynamics behaviors of the model. Our results are illustrated and corroborated with some numerical experiments. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction The SIR infections disease model is an important biologic model and has been studied by many authors [1–16]. It is well-known that one of strategies to control infectious diseases is vaccination. Then a number of epidemic models in ecology can be formulated as dynamical systems of differential equations with vaccination [17–19]. Systems with sudden perturbations lead to impulsive differential equations, which have been studied intensively and systematically in [20–27]. It is very important that one investigates under what conditions a given agent can invade a partially vaccinated population, i.e., how large a fraction of the population do we have to keep vaccinated in order to prevent the agent from establishing. Pulse vaccination seems more reasonable than traditional continuous constant vaccination in the real world. Recently, pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as a strategy for the elimination of childhood viral infectious such as measles hepatitis, parotitis, smallpox and phthisis. Pulse vaccination strategy (PVS) [11–16,23], consists of periodical repetitions of impulsive vaccinations in a population, on all the age cohorts, differently from the traditional constant vaccination. A model for the spread of an infectious disease (involving only susceptibles and infective individuals) transmitted by a vector (e.g. mosquitoes) after an incubation time, was proposed by Cooke [28]. This is called the phenomena of ‘time delay’ which has very important biologic meaning in epidemic models. Many authors have directly incorporated time delays in the modeling of equations and, as a result, the models take the form of delay differential equations [1–11,28–37]. In recent years, the research on delay SIR epidemic models with impulsive perturbations is a relevant subject in mathematical biology, but not totally developed. See [11,35,36] and the references therein. However, this is an interesting problem in mathematical biology. Since an adopted incidence form like the β e −μω S q (t )I (t − ω) term with time delay in this paper is different from those incidences of the form in [11,34–36], by use of new computational techniques for impulsive ✩ This work is supported by National Natural Science Foundation of China (No. 10771179) and Science and Development Foundation of SDUST(05g016). ∗ Corresponding author. E-mail address: mxz721106@sdust.edu.cn (X. Meng). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.10.041