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In this paper, we study a kind of the delayed SEIQR infectious disease model with the quarantine and latent, and get the threshold value which determines the global dynamics and the outcome of the disease. The model has a disease-free equilibrium which is unstable when the basic reproduction number is greater than unity. At the same time, it has a unique endemic equilibrium when the basic reproduction number is greater than unity. According to the mathematical dynamics analysis, we show that disease-free equilibrium and endemic equilibrium are locally asymptotically stable by using Hurwitz criterion and they are globally asymptotically stable by using suitable Lyapunov functions for any

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 [

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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.

Theorem 3.1.1. If, the disease-free equilibrium of system (2) is locally asymptotically stable for any in. If, it is unstable for any in.

Proof. The characteristic matrix at the disease-free equilibrium

When the characteristic equation at the disease-free equilibrium of system (2) takes the form

Clearly, system (2) always has two negative real roots

All other roots are given by the roots of equation

Assume

That is,

Because which is contradictory. So Therefore the disease-free equilibrium of system (2) is locally asymptotically stable.

If let

so there is a positive real root at least. The disease-free equilibrium of system (2) is unstable.

When it is easy for us to prove the disease-free equilibrium of system (2) is locally asymptotically stable.

Theorem 3.1.2. If, the disease-free equilibrium of system (2) is globally asymptotically stable for any in.

Proof. For define a differentiable Lyapunov function

Obviously,

Calculating the derivative of along positive solutions of system (2), it follows that

According to the feasible region,

So

That is,

And when

While if and only if,

For all t, it is easy to show that is the largest invariant subset of the set Because of LaSalle’s invariance principle, disease-free equilibrium of system (2) is globally asymptotically stable. This completes the proof.

Theorem 3.2.1. For any, if the endemic equilibrium of system (2) is locally asymptotically stable in

Proof. The characteristic matrix at the endemic equilibrium

Order

The characteristic equation at the endemic equilibrium is

Clearly, system (2) always has a negative real root

When all other roots are given by the roots of equation

so

So according to Hurwitz criterion, the endemic equilibrium of system (2) is locally asymptotically stable.

When all other roots are given by the roots of equation

Simplify, we can get

Let

Then

Let is the root of Equation (3), on substituting to Equation (3), we derive that

Separating real and imaginary parts, it follows that

Then we can get

Order

Letting then Equation (4) becomes

Here

Application of the conclusions of [

Theorem 3.2.2. If when the endemic equilibrium of system (2) is globally asymptotically stable in

Proof. Define a differentiable Lyapunov function

both of them are real numbers. The function is positive definite. Calculating the derivative of along positive solutions of system (2), it follows that

On substituting we have

Let

So In addition, when if and only if

It is easy to show that is the largest invariant subset of the set Because of LaSalle’s invariance principle, the endemic equilibrium of system (2) is globally asymptotically stable when. This completes the proof.

Theorem 3.2.3. If when the endemic equilibrium of system (2) is globally asymptotically stable in

Proof. For define a differentiable Lyapunov function

Order

both of them are real numbers. Let

Then the derivative of along the solution of system (2) satisfies

Then

Simplify, we can get

Order Besides, when if and only if

It is easy to show that is the largest invariant subset of the set Because of LaSalle’s invariance principle, the endemic equilibrium of system (2) is globally asymptotically stable when. This completes the proof.

Zhao et al. studied delay SEIR epidemic model with the nonlinear incidence rate like in the case of pulse. In this paper, the model without pulse is discussed.

It is easy to show disease-free equilibrium is globally asymptotically stable, endemic equilibrium is locally asymptotically stable. The ways we use are similar to that in system (1), here they are omitted.

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

Here see

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

Here see

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

Here see

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

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 [

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