This paper aims to discuss the delay epidemic model with vertical transmission, constant input, and nonlinear incidence. Some sufficient conditions are given to guarantee the existence and global attractiveness of the infection-free periodic solution and the uniform persistence of the addressed model with time delay. Finally, a numerical example is given to demonstrate the effectiveness of the proposed results.

Vaccination has been widely used as a method of disease control; inoculate is an effective approach according to the characteristics of the disease which takes the defense in advance. The implementation of inoculate is not continuous but cyclical. As early as in the 1960s, the principle of the stability has been given in [

On the other hand, the pulse SEIR epidemic model with the incubation period has been established in [

An impulsive vaccination SEIR epidemic model with saturation infectious and constant input has been studied in [

In this paper, the cases of constant population input and vertical transmission are considered. Patients who contact susceptible people with saturated incidence way are taken into account. A new impulsive vaccination SEIR epidemic model with time delay and nonlinear incidence rate is established. The sufficient condition is given to guarantee the stability of the disease-free periodic solution and the persistence of the model. Compared to existing results, the main contributions lie in the following aspects: (i) the nonlinear incidence rate is considered in the model to describe the spread of the disease which is more close to reality; (ii) all kinds of infectious diseases have the incubation period, and therefore it is necessary to deal with the phenomenon of time delay; (iii) a unified pulse SEIR epidemic model including the nonlinear contract rate, vertical transmission, and time delays is established. As discussed in [

We consider an SEIR model by assuming that the input term has a constant population, a nonlinear occurrence rate as

Letting

Noticing

To proceed, we introduce the following two lemmas which will play important roles in the remaining parts of this paper.

Consider the following differential equations with delay:

the sequence

Considering the following differential equations with delay:

if

if

In this section, for model (

Firstly, the analysis result is given to ensure the existence of the disease-free periodic solutions.

If

The existence of the disease-free periodic solution means that the number of sick people is zero, that is

Let

According to (

By using the second and fifth equations of (

In this subsection, the global stability of the disease-free periodic solution is discussed and the sufficient condition is given accordingly.

If

By the first and fifth equations of (

Subsequently, it follows from (

Consider the comparison system of (

Without loss of generality, assuming that there exists

Similarly, for

In this subsection, the definition of the uniform persistence is first given. Then the sufficient condition is proposed to ensure the uniform persistence of the addressed model.

If there exists a compact set

According to the above definition, we aim to present the analysis result about the uniform persistence for model (

If

It follows from

First, we show the existence of a lower bound for

By

Now we prove that

Two cases to prove are given as follows.

Above all, any positive periodic solution of (

It follows from the first and the fifth equations of (

Because of the second equation of (

Subsequently, it follows from the fourth equation of (

Up till now, we investigate the delay epidemic model with vertical transmission, constant input and nonlinear incidence. The sufficient conditions are given to guarantee the existence of the disease-free periodic solutions, the global stability of disease-free periodic solution, and the uniform persistence of the considered model. It is worth mentioning that, according to the statistics of the suspected patients and patients and using the data identification approach, the parameters in the model can be determined when the disease outbreaks. Subsequently, the illness trend of the epidemic can be predicted. Hence, we can make the reasonable control measures. One of the future research directions would be to apply the developed results to make the control strategy by properly considering the real information on the epidemic data.

For comparisons, we consider the following five cases.

Let the parameters be specified as certain fixed values in (

The global attractability of disease-free periodic solution.

Take the following parameters:

The uniform persistence of disease when

The parameter of (

The uniform persistence of disease when

If the parameter of (

The uniform persistence of disease when

If the parameter of (

It can be seen that

The uniform persistence of disease when

By Figures

In this paper, we have discussed the SEIR model with the pulse vaccination, the constant input item of population, and the vertical transmission. By employing the impulsive differential inequality and the stroboscopic map, the existence conditions of the disease-free periodic solution of the model have been given. Also, the sufficient conditions of globally attractive and uniform persistence have been proposed. Finally, a numerical example has been given to illustrate the validity of the proposed results. One of our future research interests is to extend the main results of the analysis and synthesis of gene regulatory networks or complex dynamical systems as discussed in [

This work was supported in part by the National Natural Science Foundation of Heilongjiang Province under Grant no. A200502 and the Foundation of Educational Commission of Heilongjiang Province under Grant no. 12521099.