ABSTRACT

In this paper, a class of discrete vertical and horizontal transmitted disease model under constant vaccination is researched. 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, the hyperbolicity of equilibrium points is determined. By getting the equations of flows on center manifold, the direction and stability of the transcritical bifurcation and flip bifurcation are discussed.

Keywords:

Vertical and Horizontal Transmission, Vaccination, Center Manifold, Transcritical Bifurcation, Flip Bifurcation

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).

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

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).

Cite this paper

Li, M.S., Liu, X.M. and Zhou, X.L. (2016) The Dynamic Behavior of a Discrete Vertical and Horizontal Transmitted Disease Model under Constant Vaccination. International Journal of Mo- dern Nonlinear Theory and Application, 5, 171-184. http://dx.doi.org/10.4236/ijmnta.2016.54017

References