The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination ()
1. Introduction
The SIR infections disease model is an important model and has been studied by many authors [1] - [8] . The basic and important research subjects for these systems are local and global stability of the disease-free equilibrium and the endemic equilibrium, existence of periodic solutions, persistence and extinction of the disease, etc. In recent years, the study of vaccination, treatment, and associated behavioral changes related to disease transmission has been the subject of intense theoretical analysis [4] [9] [10] [11] [12] . In 2008, Meng and Chen [13] considered a class of continuous vertical and horizontal transmitted epidemic model under constant vaccination
(1)
where S represents the proportion of individuals susceptible to the disease, who are born (with b) and die (with d) at the same rate b (b = d) and have mean life expectancy
. The susceptible become infectious at a bilinear rate
, where I is the proportion of infectious individuals and
is the contact rate. The infectious recover (i.e. acquire lifelong immunity) at a rate r, so that
is the mean infectious period. The constant p, q, 0 < p < 1, 0 < q < 1, and p + q = 1, where p is the proportion of the offspring of infective parents that are susceptible individuals, and q is the proportion of the offspring of infective parents that are infective individuals. In their work, the basic reproductive rate determining the stability of disease-free equilibrium point and endemic equilibrium point was found out and the local and global stability of the equilibrium points have been researched by using Lyapunov function and Dulac function.
Due to a lot of discrete-time models are not trivial analogues of their continuous ones and simple discrete-time models can even exhibit complex behavior (see [14] ), in this paper, we pay attention to the discrete situation of Equation (1) as follows
(2)
where
,
and
represent susceptible, infective and recovered subgroups, n represent a fixed time. Under the hypothesis of population being constant size, the model is transformed into a planar map and its equilibrium points and the corresponding eigenvalues are solved out. By discussing the influence of coefficient parameters on the eigenvalues, we determine the hyperbolicity of equilibrium points. Further, we get the equations of flows on center manifold and discuss the direction and stability of the transcritical bifurcation and flip bifurcation.
2. Hyperbolic and Non-Hyperbolic Cases
In this section, we will discuss the hyperbolic and non-hyperbolic cases in a two parameters space parameter. In view of assumption that population is a constant size, i.e.,
(3)
system Equation (2) can be changed into
(4)
Rewrite Equation (4) as a planar map F:
(5)
It is obvious that this map has a disease-free equilibrium point
and an endemic equilibrium point
where
,
,
.
Theorem 1. The equilibrium point
is non-hyperbolic if and only if
lies on the lines:
![]()
And
.
Otherwise, the equilibrium point
is an one of the following types: (See Table 1).
Proof. The Jacobian matrix of map (5) at
is:
![]()
And its eigenvalues are
,
.
From the assumption
, we see that
. Then non-hyperbolic will be happened in the case
. From
and
, we get that
and
lies on
. Also, from
, we know
which means
lies on
. When
(referred to the case
), the eigenvalue
satisfies
, then the equilibrium point P is a saddle. When
(referred to the case
), the eigenvalue
satisfie
,
so the equilibrium point P is a stable node and meanwhile when
(referred to the case
), the equilibrium point P is a saddle since
. The proof is complete.
Theorem 2. We select s, r as parameters. There does not exist non-hyperbolic case for the equilibrium
. But the hyperbolicity can be divided into the following cases (I), (II).
(I) When
, there exist six types for hyperbolic equilibrium point Q: (See Table 2).
![]()
Table 1. Types of hyperbolic equilibrium point
.
![]()
Table 2. Types of hyperbolic equilibrium point
.
Where
satisfy
![]()
![]()
respectively.
(II) When
, there exist four types for hyperbolic equilibrium point Q: (See Table 3).
Where
satisfies
.
Proof. Performing a coordinate shift as follows:
, ![]()
and letting
denote the transformed F, we translate equilibrium
into
and discuss equilibrium point
of the map
. The matrix of linearization of
at
is
![]()
where
,
. Its eigenvalues are
![]()
![]()
Table 3. Types of hyperbolic equilibrium
.
![]()
It is known that
is hyperbolic if and only if none of eigenvalues
,
lies on the unit circle
. In the following we discuss the eigenvalues in two case, i.e.,
and
.
(I) ![]()
When discriminant
, then
and
are both real . Because non-hyperbolicity happens if and only if
or
. For whether
or
, we can get
. By condition
and
, we see that
. This is a contradiction with
and
, so
and
are impossible. Next, let’s examine
and
. From
whether
or
, we can get
, By condition
we see that
,
, This is a contra-
diction with
, so
and
are impossible.
When
,
and
are a pair of conjugate complex. Since
![]()
Therefore,
and
lie inside of
and the equilibrium point Q is a stable focus referred to the case
.
When
, the equilibrium point Q Is hyperbolic. If
, i.e.
![]()
The matrix has a double real eigenvalue
. From the constraint condition
, it is obvious that
. Therefore, equilibrium point Q is stable node in the case of
and
.
If
, i.e.,
and
, the eigenvalue
and
are different real numbers. We first discuss the case that
, i.e.,
, In this case we have
![]()
and
![]()
We have
for
, On the other hand, there also exists
for
. In fact, since
![]()
and
![]()
We have
. Therefore, the equilibrium Q is a stable node as
.
For the case
, i.e.,
, we have
and
![]()
and
,
(6)
We assume
, by condition
, we see that
, i.e.,
and by condition
. This is a con-
tradiction with
and
. So
are impossible,
i.e.,
. Therefore, we have
. Therefore, the equilibrium Q is a stable node as
.
Finally, we study the case of
,
. We have
![]()
![]()
Then, we have
for
. Moreover, there also has
for
. In fact that,
![]()
![]()
and
![]()
We have
. This means that the equilibrium Q is a stable node for
.
(II) ![]()
When discriminant
, because non-hyperbolicity happens if and only if
or
. Similar to the proof in case (I), neither
nor
is possible.
When
,
and
are a pair of conjugate complex. Since
![]()
Therefore,
and
lie inside of
and the equilibrium point Q is a stable node referred to the case
.
When
, the equilibrium point Q is hyperbolic. If
, the matrix has a
double real eigenvalue
. From the constraint condition
,
it is obvious that
. Therefore, equilibrium point Q is stable node in the case of
. If
, we first discuss the case that
, i.e.,
, In this case we have
![]()
We have
for
, On the other hand, there also exists
for
. In fact, since
Therefore, we have
.
Therefore, the equilibrium Q is a saddle as
.
Finally, we study the case of
, i.e.
, We easily prove
by same methods as in case (I). This means that the equilibrium Q is a stable node for
. The proof is complete.
3. Transcritical Bifurcation of the Model
The following lemmas were be derived from reference [15] .
Lemma 1. ( [15] , Theorem 2.1.4) The map
(7)
satisfies that A is cxc matrix with eigenvalues of modulus one, and B is sxs matrix with eigenvalues of modulus less than one, and
![]()
where f and g are
(
) in some neighborhood of the origin. Then there exists a
center manifold for equation (7) which can be locally represented as a graph as follows
![]()
For
sufficiently small. Moreover, the dynamics of equation (4.1) restricted to the center manifold is, for
sufficiently small, given by the c-dimensional map
![]()
Lemma 2. ( [15] , in page 365) A one-parameter family of
(
) one-dimensional
maps
(8)
Having a non-hyperbolic fixed point, i.e.,
![]()
Undergoes a transcritical bifurcation at
if
![]()
Theorem 3. A transcritical bifurcation occurs at the equilibrium
when
. More concretely, for
slightly there are two equilibriums: a stable point P and an unstable negative equilibrium which coalesce at
, for
slightly there are also two equilibriums: an unstable equilibrium P and a stable positive equilibrium Q. Thus an exchange of stability has occurred at
.
Proof. For
, we have
and
. Consider
as the bifurcation parameter and write F as
to emphasize the dependence on
. Performing a coordinate shift as follows
,
. One can easily see that the matrix
is
![]()
and it has eigenvectors
,
(9)
Corresponding to
and
respectively, where T means the transpose of matrices. First, we put the matrix
into a diagonal form. Using the eigenvectors (9), we obtain the transformation
(10)
with inverse
(11)
which transform system Equation (5) into
(12)
where
(13)
Rewrite system (12) in the suspended form with assumption
,
(14)
where
![]()
Thus, from Lemma 1, the stability of equilibrium
near
can be determined by studying an one parameter family of map on a center manifold which can be represented as follows,
![]()
for sufficiently small v and
.
We now want to compute the center manifold and derive the mapping on the center manifold. We assume
(15)
near the origin, where
means terms of order
. By Lemma 1, those coefficients
can be determined by the equation
(16)
Substituting (16)into (15) and comparing coefficients of
and
in (15), we get
![]()
from which we solve
![]()
Therefore, the expression of (15) is approximately determined:
(17)
Substituting (17) into (14), we obtain a one dimensional map reduced to the center manifold
(18)
It is easy to check that
(19)
The condition (19) implies that in the study of the orbit structure near the bifurcation point terms of
do not qualitatively affect the nature of the bifurcation, namely they do not affect the geometry of the curves of equilibriums passing through the bifurcation point. Thus, the orbit structure of (18) near
is qualitatively the same as the orbit structure near
of the map
(20)
Map (20) can be viewed as truncated normal form for the transcritical bifurcation (see Lemma 2). The stability of the two branches of equilibriums lying on both sides of
are easily verified.
4. Degenerate Flip Bifurcation of the Model
This section is devoted to the analysis for the case
. From section 2, we
have
for
. For this case, degenerate flip
bifurcation happens at the equilibrium point
.
Theorem 4. For map (5) when
, degenerate flip bifurcation happens at the equilibrium point
.
Proof. Performing a coordinate shift as follows
,
,
We translate equilibrium
into
, and letting
denote the transformed ![]()
(21)
Therefore, we discuss equilibrium point
of the map
. The matrix of linearization of
at
is
![]()
For
, considering
as the bifurcation parameter and write
as
to emphasize the dependence on w. Therefore, we have
(22)
The matrix have eigenvectors
and
corresponding
to
and
. Therefore, by transformation
(23)
where
.
Therefore, we obtain the inverse of transformation (23)
(24)
Therefore
can be changed into the maps: ![]()
(25)
where
,
.
Rewrite system (25) in the suspended form
(26)
where
,
,
,
.
Equivalently, the suspended system (26) has a two-dimensional center manifold of the form
(27)
Near the origin, where
means terms of order
. By Lemma 1, those coefficients
can be determined by the equation
(28)
Then
(29)
Comparing coefficients of
,
and
in (27), we get
![]()
from which we solve
![]()
Thus, the expression of (27)is determined, i.e.,
(30)
Substituting (30) into the first equation in (26), we obtain a one-dimensional map
, where
(31)
From (31), we can check that
(32)
(33)
Thus, the conditions
and
of Theorem 3.5.1 in [16] are not satisfied. Therefore, this is a degenerate flip bifurcation.
5. Conclusion
Due to a lot of discrete-time models are not trivial analogues of their continuous ones and simple discrete-time models can even exhibit complex behavior (see [14] ), motivated mainly by Meng and Chen [13] considering a class of continuous vertical and horizontal transmitted epidemic model (1) under constant vaccination, we study a class of discrete vertical and horizontal transmitted disease model (2) under constant vaccination. By detailed studies, we found discrete model (2) has a flip bifurcation which did not occurred for continuous model. However, the result of flip bifurcation in current paper is a degenerate situation, for which the more in-depth research needs to be continued.
Acknowledgements
This work has been supported by the Innovation and Developing School Project of Department of Education of Guangdong province (Grant No. 2014KZDXM065) and the Key project of Science and Technology Innovation of Guangdong College Students (Grant No. pdjh2016a0301).