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A delayed SEIRS epidemic model with pulse vaccination
and nonlinear incidence rate is proposed. We analyze the dynamical behaviors of this
model and point out that there exists an infection-free periodic solution which is globally
attractive if

Infectious diseases are usually caused by pathogenic microorganisms, such as bacteria, viruses, parasites, or fungi; the diseases can be spread directly or indirectly. The severe and sudden epidemics of infectious diseases have a great influence on the human life and socioeconomy, which compel scientists to design and implement more effective control and preparedness pro- grams. Pulse vaccination is an effective method to use in attempts to control infectious diseases.

In recent years, epidemic mathematical models of
ordinary differential equations have been studied by many authors (e.g.,
[

Bilinear and standard incidence rates have been
frequently used in classical epidemic models [

On the one hand, the newborns of the infectious may
already be infected with the disease at birth such as hepatitis and phthisis, and so forth.
This is called vertical transmission. On the other hand, some diseases may be
spread from one individual to another via horizontal contacting transmission.
Some epidemic models with vertical transmission were studied by many authors.
However, only a few literatures [

Most of the research literature on these epidemic
models are established by ODE, delayed ODE or impulsive ODE. However, impulsive
equations with time delay are not many [

The organization of this paper is as follows. In the
next section, we introduce the delayed SEIRS model with pulse vaccination. To
prove our main results, we also give several definitions, notations, and
lemmas. In Section

In the following model, we study a population that is
partitioned into four classes, the susceptible, exposed, infectious, and
recovered, with sizes denoted by S, E, I, and R, respectively, and we consider
pulse vaccination strategy in the delayed SEIRS epidemic model with nonlinear
incidence rate

Here, all coefficients are
positive constants,

The total population size

Before going to any detail, we simplify model (

Before starting our main results, we give the
following lemmas.

Consider the following delay differential equation:

if

if

Consider the following impulsive
differential equations:

In this
section, we study the existence of the infection-free periodic solution of
system (

Denote

If

Since

Set

It is clear that

In system (

Theorem

In this section, it is noted that the disease is
endemic if the infectious population persists above a certain threshold for
sufficiently large time. The endemicity of the disease can be well captured and
studied through the notion of uniform persistence and permanence.

System (

System (

Denote

If

Note that the
second equation of system (

By the comparison theorem for impulsive differential
equation [

Let

If

Suppose that

Set

The following results are true.

In this paper,
we introduce the delayed SEIRS epidemic model with pulse vaccination and
nonlinear incidence rate of the form

This work was supported by National Natural Science Foundation of China (10771179).