Analysis of an SEIS Epidemic Model with a Changing Delitescence

and Applied Analysis 3 2. Existence of Equilibria Now, we study equilibria of model 1.3 , note that m d ε, n d γ δ . Steady states of model 1.3 satisfy the following equations: μ βI 1 αI N − E − I − d ε E 0, ( 1 − μ βSI 1 αI N − E − I εE − d γ δI 0, A − dN − δI 0. 2.1 If I 0, then E 0 and N A/d, so 2.1 has the disease-free equilibrium P0 0, 0, A/d . If I / 0, from the third equation of 2.1 , we obtain


Introduction
Infectious diseases have tremendous influence on human life.Every year millions of human beings suffer from or die of various infectious diseases.It has been an increasingly complex issue to control infectious diseases.In order to predict the spreading of infectious diseases among regions, many epidemic models have been proposed and analyzed in recent years see 1-14 .Bilinear and standard incidence rates have been frequently used in classical epidemiological models see 5 .However, it is more effective by using nonlinear incidence.Mathematical models describing the population dynamics of infectious diseases have been playing an important role in disease control for a long time.Many scholars have studied mathematical models which describe the dynamical behavior of the transmission of infectious diseases see also 1-14 and the references therein .A variety of nonlinear incidence rates have been used in epidemiological models see 6-14 .However, the models with a changing delitescence have seldom been studied.
In recent years, the spread of infectious diseases is diversiform, such as H1N1 disease.The diversity of the delitescence period in each infected body who is infected with H1N1 virus is mainly due to the variation of the virus and the distinct constitution of different people.The study of the models with a changing delitescence plays an important role in controlling infectious diseases.In this paper, we consider an SEIS epidemic model with a changing delitescence and a nonlinear incidence rate, and we study the existence and stability of the equilibriums of the SEIS epidemic model.
By a standard nonlinear incidence rate βSI/ 1 αI and a changing delitescence μ, we consider an SEIS epidemic model which consists of the susceptible individuals S , exposed individuals but not yet infected E , infectious individuals I , and the total population N .
The model is given as follows: Form 1.2 , in the absence of the disease, that is, I 0, N → A/d.Since the spread of the disease in the population will lead to the decrease of N, it follows that N ∈ 0, A/d .Note that D is a positively invariant region for the original model: and model 1.3 is obviously well-pased in D.

Existence of Equilibria
Now, we study equilibria of model 1.3 , note that m d ε, n d γ δ .Steady states of model 1.3 satisfy the following equations:

2.1
If I 0, then E 0 and N A/d, so 2.1 has the disease-free equilibrium P 0 0, 0, A/d .
If I / 0, from the third equation of 2.1 , we obtain From the first and second equations of 2.1 , we obtain By substituting 2.2 and 2.3 into the first equation of 2.1 , we obtain the following equation for I:

2.4
Let R 0 βA m 1 − μ μ /mnd.It is easy to see that 2.4 has a positive root if and only if R 0 > 1.So 2.1 has a unique endemic equilibrium P * E * , I * , N * with

2.5
Then, we have the following theorem.
3 has a unique endemic equilibrium P * E * , I * , N * except the disease-free equilibrium P 0 0, 0, A/d .

Stability of Equilibria
Theorem 3.1.If R 0 < 1, then the disease-free equilibrium P 0 0, 0, A/d is locally asymptotically stable Proof.The linearization of model 1.3 about the equilibrium P 0 0, 0, A/d gives Thus, we have

3.2
Assume that λ 1 , λ 2 , and λ 3 are the roots of the above equation.Then, we know that λ 1 −d < 0, and λ 2 , λ 3 are the roots of the following equation: According to Viete theorem, we have

3.5
It is easy to see that all the roots of 3.3 have negative real parts if and only if R 0 < 1.If R 0 1, it is obvious that one of eigenvalue of 3.3 is 0. If R 0 > 1, one of the roots of 3.3 has positive real part.This completes the proof.
Proof.Consider a Liapunov function as V εE mI, then we have

3.6
If R 0 < 1, V ≤ 0, then V 0 if and only if E I 0. Hence, according to Theorem 3.1, if R 0 < 1, the disease-free equilibrium P 0 0, 0, A/d is globally asymptotically stable.This completes the proof.Now, we study the local stability of the endemic equilibrium P * E * , I * , N * .Substituting E * μn/ m 1 − μ με I * into the first equation of 2.1 , we have Then, the Jacobi matrix of 1.3 about P * E * , I * , N * is
From the above discussion, we get the following conclusion.
3 has a unique endemic equilibrium P * E * , I * , N * , which is locally asymptotically stable.
From the third equation of 1.3 , we have

3.11
It is easy to see that system 3.11 has a disease-free equilibrium P 0 0, 0 .If R 0 > 1, system 3.11 has a unique endemic equilibrium P * E * , I * with

3.12
Consider the Dulac function B 1/I.Note that < 0, there is no limit cycle in the first quadrant of the I-E plane.System 3.11 has a unique endemic equilibrium P * E * , I * , then we prove that P * E * , I * is globally asymptotically stable.
From the above discussion, we get the following conclusion.

Numerical Simulation
For the initial values 20,30,500 , 30,20,700 , 20,20,500 , 30,40,800 , 20,50,900 , 50,40,900 , 20,40,900 , Figures 3, 4 From Figures 1 and 2, it is easy to see that the value of μ decides the development trend of the infectious disease, that is, whether the infectious disease dies out or does not exist forever when the other conditions are the same.From Figures 3-5, with a unique endemic equilibrium of system 1.3 , the value of μ will affect the numbers of exposed individuals who are not yet infectious, infectious individuals, and the total population.With β 0.5, d 0.01, m 0.3, n 0.5, μ 0.5, α 0.4, ε 0.1, Figure 6 shows that system 3.11 has no limit cycle.

Conclusion
Because of distinct constitutions of individuals, some infected individuals who are not yet infectious become exposed individuals, while other infected individuals immediately become infectious.So, we establish an epidemic model with a changing delitescence between SEIS and SIS model by using the proportional number μ.

Figures 1 -
Figures 1-5 are drawn by MATLAB.With different parameters, they describe the changes of the number of exposed individuals who are not yet infectious, infectious individuals, and the total population.