As we all know, infectious diseases are enemies of human health. For a long time, people have been fighting various infectious disease; and many methods have been used to study the spread of infectious diseases, so as to control and eliminate infectious diseases, see  . Amongst them, mathematical model has become an important tool to analyze the epidemiological characteristics of infectious diseases since the pioneer work of Kermack and McKendrick   , which provides us useful control measures in  . In standard epidemiological models, the incidence rate (the rate of new infections) is bilinear in the infective and susceptible individuals, see    . It has been suggested that the diseases transmission process may have a nonlinear incidence rate, see    .
In real life, epidemics tend to have an incubation period, as susceptible to infection after contacting with infected people. First of all, carrying virus, the virus is not immediately, but after a period of time, to onset and into the herd of infected people. In the paper, on the basis of the work of Yuan and Li in  an SEIR (Susceptible-Exposed-Infected-Removed) epidemic model is considered with a ratio-dependent nonlinear incident rate .
The transmission function plays a key role in determining disease dynamics, see   . Traditionally, the density-dependent transmission (or the bilinear incidence rate, , the proportionality constant) and the frequency-dependent transmission (or the standard incidence rate,
) are two extreme forms of disease transmission, which have
been frequently used in well-know epidemic models   . For example, Capasso and Serio  introduced a saturated transmission rate , where the infectious force is a function of infectious individuals which has been used in many classic disease model. Especially, Yuan and Li  studied a rate-dependent nonlinear incident rate with the following form
where is a parameter which measures the psychological or inhibitory effect. It should be noted that if and , (1) becomes the well-known
frequency-dependent transmission rate .
In the case of , we can obtain the rate-dependent transmission rate as the following form:
which indicates that the transmission rate of disease is approximately governed
by if is small (e.g., at the beginning of disease’s spreading) or it is approximately governed by if is large (e.g., in the endemic when
almost everyone is infected). Therefore, the ratio-dependent transmission rate (2) indeed takes accounts of the crowding effects and behavior changes during epidemics. In this paper, we mainly focus on a SEIR epidemic model with the ratio-dependent incidence rate (2).
We consider the global properties of this SEIR model and show that if the basic reproduction number , the disease-free equilibrium point is globally asymptotically stable, while if , the disease-free equilibrium point is unstable and the unique endemic equilibrium point is globally asymptotically stable.
The organization of this paper is as follows: in the next section, we present the model and derive the disease-free equilibrium point and the endemic equilibrium point. In Section 3 we analyze the global stability of the equilibrium point. A brief discussion and summarize are given in Section 4.
2. Model Formulation
The whole population is divided into four subclasses based on disease status: the susceptible population, the exposed population, the infected population and removed population, denoted by , respectively, and . We assume that infectious disease can cause additional mortality, then the SEIR model can be modeled by the following set of nonlinear differential Equations (3) deterministically:
where is the recruitment rate of the population, is the natural death rate, is the constant rate such that the exposed individuals become infective, is the constant rate for recovery, is the disease inducing death. Since does not appear in the first three equations of system (3), thus (3) reduces to the following three-dimensional system (4)
Because , , , the
non-negativity of the initial value of system (4) in is guaranteed, where
It follows from system (4) that:
Since , the feasible region for system (4) is thus a
bounded set :
The region is positively invariant with respect to systems (4). So, the only solution with the associated initial conditions will be considered inside the region , where the uniqueness of solutions, usual existence, and continuation results are satisfied. Hence, system (4) is considered mathematically and epidemiologically well posed in . Notice that model (4) has a disease-free equilibrium
point for all parameter values. Let , system (4) can
be written as (see  )
The jacobian matrices of and at the disease-free equilibrium point are, respectively,
So the regeneration matrix of system (4) is
the spectral radius of is
Hence the basic reproduction number (see  ) is
Without difficulty we can get unique endemic equilibrium point state of model (4) with
and the endemic equilibrium point is written in the following form:
Through the above analysis, system (4) has no endemic equilibrium point for , and from (19) we know that system (4) has a unique endemic equilibrium point if .
3. Global Stability of the Equilibrium Point
In this section, we first consider the global stability of model (4) at the disease-free equilibrium point .
Theorem 1: If , the disease-free equilibrium point is globally asymptotically stable; if , is unstable.
Proof. The characteristic equation of system (4) at is give by
It is clear that is one root of (20). The other roots of (20) are determined by the following equation
If , , thus (21) has two roots with negative real parts, therefore, the disease-free equilibrium point is locally asymptotically stable.
To complete the proof, we construct the following Lyapunov function
The time derivative of along the solution of (4) is
When , we have . By the LaSalle’s invariance principle, see
 , we conclude that is globally asymptotically stable if .
When , the Jacobian matrix of model (4) evaluated at is
which has an eigenvalue . Denoted by
we find that ( represents the trace of matrix B, which is the sum of the elements of the main diagonal of the matrix B). When , and ( is the determinant of matrix B), the matrix B must have a positive eigenvalue, thus the disease-free equilibrium point is unstable whenever . This completes the proof.
For the stability of endemic equilibrium point of model (4), we have the following theorem:
Theorem 2: If , the unique endemic equilibrium point of model (4) is globally asymptotically stable.
Proof. Suppose that is any positive solution to system (4). Define a Lyapunov function
Obviously and when , . Remember that is the solution to the system (4), the upper right derivative of can be estimated:
and we all know that when , ; , ; , .
In (27), there are 8 kinds of situation for the size of and , and , and , it is enough to analyze the situation of , , , while for the other situations, the discussion is similar.
Firstly there is
Integrate from to on both sides of (28), we have
Since the front set has a boundary, and must have boundaries, and their derivatives are bounded. It means that is uniformly continuous.
By Barbalat Lemma in  , there is
and the unique endemic equilibrium point of model (4) is globally asymptotically stable. This completes the proof.
4. Brief Summary
In this paper, we consider the SEIR epidemic model which is different from the classical nonlinear incident rate. We assume that the infectious force is a function of a number ratio of the infective to that of the susceptible which takes the
form , and the regeneration matrix is used to obtain the basic reproductive number ; the existence of equilibrium is obtained by direct calculation.
By constructing the proper Lyapunov functions, we prove that if , there exists only the disease-free equilibrium point which is globally asymptotically stable, and if , there is a unique endemic equilibrium point and this endemic equilibrium point is globally asymptotically stable.
In the future work, we can further consider adding pulse condition to the model studied in this paper.
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 Korobeinikov, A. and Maini, P.K. (2004) A Lyapunov Function and Global Properties for SIR and SEIR Epidemiological Models with Nonlinear Incidence. Mathematical Bioscience and Engineering, 1, 57-60.
 Driessche, P.V.D. and Wathmough, J. (2002) Reproduction Number and Suc-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180, 29-48.