Global Stability Analysis of a Delayed SEIQR Epidemic Model with Quarantine and Latent

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 .  Besides, the SEIQR model with nonlinear incidence rate is studied, and the 0  that the basic reproduction number is a unity can be found out. Finally, numerical simulations are performed to illustrate and verify the conclusions that will be useful for us to control the spread of infectious diseases. Meanwhile, the 1, k 3 k will effect changing trends of , S , E , I , Q R in system (1), which is obvious in simulations. Here, we take 3 k as an example to explain that.


Introduction
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][2][3][4][5][6].So many epidemics were controlled.Generally speaking, when epidemics spread, there are many kinds of delays, which include immunity period delay [7][8][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 La-Salle'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.

Establishment of the Model
We establish the following SEIQR epidemic model, Here The initial conditions for system (1) are : 0 .
And the feasible region of the model with the initial conditions above is Here, we presume that   k is the recovery rate of E 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 , 0 , 0 0, , 0 , , And the feasible region of the model with the initial condition above is min , , .

    
It is easy to show that 2  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) , .e e

The Stability of 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.

Stability of Disease-Free Equilibrium
Theorem 3.1.1.If 0 1 R  , the disease-free equilibrium 0 E of system (2) is locally asymptotically stable for any  in 2 Proof.The characteristic matrix at the disease-free equilibrium 0 e e 0 0 the characteristic equation at the disease-free equilibrium 0 E 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 21 e e 0, A a e e , 0 0, 0, so there is a positive real root at least.The disease-free equilibrium 0 it is easy for us to prove the disease-free equilibrium 0 E of system ( 2) is locally asymptotically stable.
Proof.For 0, t  define a differentiable Lyapunov function Calculating the derivative of   V t along positive solutions of system (2), it follows that The characteristic equation at the endemic equilibrium 1 e e e e 0.
Clearly, system (2) always has a negative real root all other roots are given by the roots of equation So according to Hurwitz criterion, the endemic equilibrium * E of system (2) is locally asymptotically stable.

 
all other roots are given by the roots of equation 1 e e e e 0. b c a bc Separating real and imaginary parts, it follows that .
Application of the conclusions of [17], we can know that positive z doesn't exist.Hence  also doesn't exist.There are not pure imaginary roots in system (2).Therefore all the roots have negative real component.So endemic equilibrium * E of system ( 2) is locally asymptotically stable.
both of them are real numbers.The function is positive definite.Calculating the derivative of V along positive solutions of system (2), it follows that .

S V S S I S SI S S I S S I SI S I S SI S I
Then the derivative of V along the solution of system (2) satisfies .

S I S I S S I S I S
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.

The Numerical Simulations
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 0 R increases.For example, Here 0 0.7502, R  see Figure 1.Here 0 0.8656, R  see Figure 2. When take different 3 k , we can get different simulation images.In other words, the R increases when 3 k increases, which is obvious in Figures 3 and 4.And then it is easy for us to find that how 3 k effects changing trends of S , E , I , Q , R .
Here 0 1.9 , 581 R  see Here 0 1.9 , 581 R  see Figure 4.At last, if the basic reproduction number is much larger and we will get new diagrams.For example, let

Discussions
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 E is quarantined and can recover in this model, which will effect changing trends of S , E , I , Q , R .Here, we take 3 k as an example to explain that.Meanwhile, the simulation image which 0 R changes as  can be obtained and we can find out 0  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.

see Figure 5 .
At the same time, the changing trends of I and E are shown in Figures 6 and 7.And the time which I comes peak will become large as  increases, the * I will decrease.For example, 3   , see Figure 8.In addition, when  changes, 0 R will change.And we can find out that when

. The Stability of Endemic Equilibrium Theorem 3.2.1.
For any  , if 01,