%0 Journal Article
%T Disease Control of Delay SEIR Model with Nonlinear Incidence Rate and Vertical Transmission
%A Yan Cheng
%A Qiuhui Pan
%A Mingfeng He
%J Computational and Mathematical Methods in Medicine
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/830237
%X The aim of this paper is to develop two delayed SEIR epidemic models with nonlinear incidence rate, continuous treatment, and impulsive vaccination for a class of epidemic with latent period and vertical transition. For continuous treatment, we obtain a basic reproductive number and prove the global stability by using the Lyapunov functional method. We obtain two thresholds and for impulsive vaccination and prove that if then the disease-free periodic solution is globally attractive and if then the disease is permanent by using the comparison theorem of impulsive differential equation. Numerical simulations indicate that pulse vaccination strategy or a longer latent period will make the population size infected by a disease decrease. 1. Introduction Mathematical models describing the population dynamics of infectious diseases have been playing an important role in understanding epidemiological patterns and disease control. Researchers have studied the epidemic models by ordinary differential equations [1¨C3] and the references cited therein. A customarily epidemic model is susceptible, infectious, and recovered model (SIR for short) [4¨C7]. But in real life, many diseases have a period of incubation time inside the hosts before the hosts become infectious; if we include incubation period of the hosts, the model is described as SEIR model. As tuberculosis (TB), measles and so on, a susceptible individual becomes exposed (infected but not infective) by adequate contact with an infectious individual. SEIR infections disease model has been studied by many authors for its important biological meaning [8¨C13]. In [13], the authors considered the following delayed SEIR epidemic model: where , , , and represent the number of individuals who are susceptible, exposed, infected, and removed, respectively. The parameters , , , and are positive constants, and here is the constant recruitment rate into the population, is the contact rate, is the birth and death rate, is the removal rate. represents a time delay describing the latent period of the disease and the term represents the individuals surviving in the latent period and becoming infective at time . The sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. In the study of epidemic model, the spread of an infectious disease is a crucial issue, which depends on both the population behavior and the infectivity of the disease. These two aspects are captured in the incidence rate of a disease. In many epidemiological models, the incidence rate is described as mass action
%U http://www.hindawi.com/journals/cmmm/2013/830237/