Many people have been paying attention to the study of some epidemics, and have accumulated a lot of experience. By establishing reasonable mathematical models, they put forward the measures which controlled the spread of epidemics effectively. And many scholars researched specific diseases and considered the diseases with incubation period, recovery time, quarantine and so on [1-6]. So many epidemics were controlled. Generally speaking, when epidemics spread, there are many kinds of delays, which include immunity period delay [7-9], infectious period delay, incubation period delay. In [10], Enatsu et al. studied stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates, at the same time, they proved disease-free equilibrium was globally asymptotically stable and endemic equilibrium was permanent under certain conditions. At the same time, global stability of an SIR (where S, I, R denote the number of susceptible individuals, infectious individuals, recovery individuals) epidemic model with constant infectious period was studied by Zhang et al.

[11], they showed the endemic equilibrium was globally asymptotically stable with appropriate Lyapunov functions. And in [12], Gao et al. discussed pulse vaccination of an SEIR (E denote the number of exposed individuals) epidemic model with delay and bilinear incidence. Meanwhile, impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate was researched by Zhao et al. [13], and showed the pulse system that was similar to the pulse system with bilinear incidence rate. Besides, on the basis of [13], Xu and Ma introduced the saturated incidence rate. Meanwhile, they showed disease-free equilibrium and endemic equilibrium were globally asymptotically stable under certain condition in [14]. However, in addition to the bilinear incidence rate, nonlinear incidence rate and saturated incidence rate, there were some scholars who studied the non-monotone incidence. For example, an SIRS epidemic model with pulse vaccination and non-monotonic incidence rate was discussed by Zhang et al. [15], and they proved the disease-free equilibrium and endemic equilibrium were asymptotically stable under certain conditions. Besides, some scholars studied a delayed SEIQR (Q denote the number of quarantined individuals) epidemic model with pulse vaccination and the quarantine measure, and they showed that the disease-free equilibrium of the system was globally attractive and endemic equilibrium was permanent under certain conditions. In this paper, we study a delayed SEIQR epidemic model without pulse on the basis of [14,16].

The organization of this paper is as follows: In Section 2, SIQR epidemic model and its basic reproduction number and existence of equilibrium are given. In Section 3, the local stability of endemic equilibrium and disease-free equilibrium is showed by using Hurwitz criterion. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than unity. At the same time, the system with the nonlinear incidence rate is discussed in Section 3. In Section 4, presents the numerical simulations of the system followed by a conclusion in Section 3. At last, a brief discussion is given in Section 5 to conclude this work.

We establish the following SEIQR epidemic model, Here represents the number of individuals who are susceptible to disease, that is, who are not yet infected at time t. is the number of individuals who are infected but hardly infectious. So we think they can’t infect other people, but they need to be quarantined. represents the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals. is the number of infectious individuals who are quarantined at time t. represents the number of recovered individuals at time t.

The initial conditions for system (1) are

And the feasible region of the model with the initial conditions above is

Here, we presume that

It is easy to show that is positively invariant with respect to system (1).

Where all the parameters are positive constants, is the recruitment rate of the susceptible population, , are the natural death rate of the susceptible, exposed, infectious, quarantine and recovered respectively, is the disease transmission coefficient, is the death rate due to disease without quarantine, is the death rate due to disease after quarantine, is the recovery rate after quarantine, is the recovery rate without quarantine, , are quarantine rate of, respectively, is the recovery rate of and is the latent period of the epidemic.

Because the variables R and Q do not appear in the first three equations in system (1), we further simplify system (1) and then obtain the following model

In this paper, we are concerned with system (2).

The initial conditions for system (2) are

And the feasible region of the model with the initial condition above is

Here, we presume that

It is easy to show that is positively invariant with respect to system (2).

According to the practical significance of the epidemic model, system (2) always has a disease-free equilibrium

Denote the basic reproduction number of system (2)

Define If the basic reproductive number system (2) has an unique endemic equilibrium

In this section, we discuss the local stability of endemic equilibrium and disease-free equilibrium of system (2) by analyzing the corresponding characteristic equations respectively. By defining reasonable Lyapunov functions, we resolve the global dynamics of equilibriums without requiring any extra conditions. In addition, system (2) with nonlinear incidence is studied.

In this section, we study system (1) numerically. According to the different datas that can reflect the actual situation, we get the different simulation images to prove our conclusions obviously (Figures 1-9).

Here, according to the different actual situations, while take different parameters, we can get different simulation diagrams of the disease-free equilibrium. At the same time, we find out the disease will die out after much more time when increases. For example,

Here see Figure 1.

Here see Figure 2.

When take different, we can get different simulation images. In other words, the increases when increases, which is obvious in Figures 3 and 4. And then it is easy for us to find that how effects changing trends of, , , ,.

Here see Figure 3.

Here see Figure 4.

At last, if the basic reproduction number is much larger and we will get new diagrams. For example, let

Here see Figure 5.

At the same time, the changing trends of and are shown in Figures 6 and 7. And the time which comes peak will become large as increases, the will decrease. For example, , see Figure 8. In addition, when changes, will change. And we can find out that when That is, disease will be endemic disease while see Figure 9.

In this paper, a kind of a delayed SEIQR epidemic model with the quarantine and latent is studied. Using Hurwitz criterion, the local stability of the disease-free equilibrium and endemic equilibrium of system (2) is proved. For any time delay we prove the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than unity by means of suitable Lyapunov functions and LaSalle’s invariance principle. So the delay is harmless to system (2). From the biological point of view, the delay here has no influence on the transmission of diseases. However, in [16], the disease-free equilibrium is periodic and globally attractive. At the same time, the disease will be endemic after some period of time. Above all, we consider that is quarantined and can recover in this model, which will effect changing trends of, , , ,. Here, we take as an example to explain that. Meanwhile, the simulation image which changes as can be obtained and we can find out which the basic reproduction number is a unity. Those are useful for us to control epidemics. At last, the conclusions above are verified by numerical simulations.

This research was supported by the National Science Foundation of China (10471040) and the National Sciences Foundation of Shanxi Province (2009011005-1).