Epidemic models described by ordinary differential equations have become important tools in analyzing the spread and control of infectious diseases. In recent years, more and more delayed models have been investigated during the study of epidemic models [1-8].
When the diseases spread quickly, the population remains constant. Furthermore the population remains constant if the birth is nearly equal to the natural death when the disease we consider it over many years. In fact, in many diseases the birth of the population can not be balanced by the natural death, and then we need to assume that the population is a function of time.
Varying total population has become one of the most important areas in the mathematical theory of epidemicology [3-8]. Anderson and May [1,2] have done a lot of work about varying total population, and Michael Y. Li et al. considered a SEIR model with varying total population in [
During the study of the dynamical behaviors of the disease, the standard incidence rate 
and the bilinear incidence rate 
are frequently used [3,5,7-11]. In recent years, more and more researchers are interested in the nonlinear incidence rate; especially the saturate incidence rate has been investigated by many authors in [12-22], in which the recruitment rate of the population is considered as a constant.
A class of delayed SIR models has been investigated with nonlinear incidence rate. Capasso and Serio [
into epidemic models, where 
tends to a saturation level when 
reach the maximum number of effective contacts between infective individuals and susceptible individuals may saturate at high infective levels due to crowding of infective individuals or due to the protection measures by the susceptible individuals[
In [
on the dynamics of a SIR epidemic model as follows:
Here parameters 
are positive constants representing the death rates of susceptibles, infectives, and recovered, respectively. The parameters 
and 
are positive constants representing the birth rate of the population and the recovery rate of infectives, respectively. By analyzing the corresponding characteristic equations, they discussed the local stability of an endemic equilibrium and a disease free equilibrium. By comparison arguments, they analyzed the globally asymptotically stable of the disease free equilibrium, and by means of an iteration technique and Lyapunov functional technique, respectively, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium.
In [
as follows:
The characteristic of this model is: the saturated incidence rate
, which includes the three forms: 
(if
), 
(if
) and 
(if
) saturated with the susceptible and the infective individuals. The inclusion of time delay into susceptible and infective individuals in incidence rate, only on the first equation, because susceptible individuals infected at time 
is able to spread the disease at time
.
In the SIR model (2), they consider the period in the evolution of susceptible class, and not in the evolution of infectious class. They discuss the local stability and the existence of Hopf bifurcation. At last some numerical simulations are given to illustrate the theoretical analysis.
In this paper we consider a delayed SIR model with the saturation incidence rate 
and exponential birth rate. We also analyze the stability and the existence conditions of Hopf bifurcation. The organization of this paper is as follows: In Section 2, we consider a delayed SIR model with saturation incidence rates and exponential birth rate. Then we also consider an exceptional case. In this case the saturation incidence rate becomes a bilinear incidence rate. Numerical simulations with different values of the delay are given in Section 3.
. Then we consider the parameter measure
, the saturation incidence rate will become a bilinear incidence rate
, we consider it in Section 2.2.
and a time delay describing a latent period. Let 
is the number of susceptible individuals, 
is the number of infective individuals, and 
is the number of recovered individuals, then we have the following model:
is the rate of natural birth, 
is the rate of natural death
, 
is the rate of disease-related death, 
is the rate of recovery, 
is the incubation period. 
is the parameter that measure in infections with the inhibitory effect. Define the basic reproduction number by
, the solution of system (3) is 
with
. If
, system (3) has a unique endemic equilibrium 
, where:
and
. If
, from the third equation in (3), we get
, then from the first equation in (3), it follows that
, we have
. Then we can get that the solution of system (3)
.
, from the third equation in (3), we get:
, we must have
. Thus we get if
, system (3) has a unique endemic equilibrium.
with
. We analyze the stability by the characteristic equation.
is of the form
, suppose 
and when
, the endemic equilibrium
is stable, and when
, it is unstable.
, the characterristic Equation (4) reads as:
, endemic equilibrium 
of system (3) is stability.
, we suppose system (3) has a purely imaginary root
, then separating real and imaginary parts, we have
, let
, then
. Then, we have 
and
. Thus Equation (8) has at least one positive root
, Equation (7) has at least one positive root, denoted by
.
as bifurcation parameter. We view the solutions of Equation (4) as functions of the bifurcation parameter
. Let 
be the eigenvalue of Equation (4) such that for some initial value of the bifurcation parameter
, we have
, and 
(we assume
). From Equations (5) and (6) we have
. By continuitythe real part of 
becomes positive when 
and the steady state becomes unstable. A Hopf bifurcation occurs when 
passes through the critical value 
[
, we have the standard incidence rate
. Then, we can get the following model:
, the solution of system (9) is 
with
. When
, the system has a unique endemic equilibrium
, where:
, consider two case: 
and
.
, we can get 
from the third Equation in (9), then we have
, we have
. Then, we can get that system (9) always has
with
.
, from the third equation of system (9), we get
.
We must have
. This implies that there exists the endemic equilibrium with
.
with
.
and when
, the endemic equilibrium 
is stable, and when
, 
is unstable.
, the characteristic equation of system (9) read as:
, 
and
. According to Hurwitz criterion, we can know the system is stable with
.
, use the same method as system (3). Suppose it has a purely imaginary root
, then we can get:
. Then, we have 
and
. Thus Eq.(10) has at least one positive root 
. Also, we can have
. A Hopf bifurcation occurs when 
passes through the critical value
.
of supporting the theoretical analysis in Section 2.
, we know system (3) has 
with
; and when 
system (9) has
. In Figures 1(a) and (b), we give appropriate parameter values of system (3) and system (9) with
.
(and 
). The system (3) (or system (9)) exist unique endemic equilibrium with 
(or
).
, we give the same parameters for endemic equilibrium in system (3) and (9) as follows:
in system (3) is higher, and the proportions of
, 
are lower.
, we give the parameters with different 
for endemic equilibrium in system (3) as follows:
, we find endemic equilibrium in system (3) is stable. When
, from 
, the incidence rate will become to the standard incidence rate, then we have system (9). We give parameters with different 
for endemic equilibrium in system (9) as follows:
, we find it is stable; from 
; and from 
reduce the infective proportion. From Figures 3 and 4, when
, the endemic equilibrium in system (3) and (9) is locally asymptotically stable; when
, the endemic equilibrium in system (3) and (9) exists periodic solutions. It showed that endemic equilib-
is suitably small. Furthermore, there exist periodic solutions with appropriate 
for two models. When
, the endemic equilibriums is locally asymptotically stable; and unstable when
, and there exists a Hopf bifurcation.