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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.

The SIR infections disease model is an important model and has been studied by many authors [

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

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 [

where

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.,

system Equation (2) can be changed into

Rewrite Equation (4) as a planar map F:

It is obvious that this map has a disease-free equilibrium point

Theorem 1. The equilibrium point

And

Otherwise, the equilibrium point

Proof. The Jacobian matrix of map (5) at

And its eigenvalues are

From the assumption

so the equilibrium point P is a stable node and meanwhile when

Theorem 2. We select s, r as parameters. There does not exist non-hyperbolic case for the equilibrium

(I) When

Cases | Conditions | Eigenvalues | Properties | |
---|---|---|---|---|

saddle | ||||

stable nod | ||||

saddle | ||||

Cases | Conditions | Eigenvalues | Properties |
---|---|---|---|

stable node | |||

stable node | |||

saddle | |||

stable node | |||

stable focus | |||

_{ } | stable node |

Where

respectively.

(II) When

Where

Proof. Performing a coordinate shift as follows:

and letting

where

Cases | Conditions | Eigenvalues | Properties | ||
---|---|---|---|---|---|

stable node | |||||

saddle | |||||

stable node | |||||

stable focus | |||||

It is known that

When discriminant

whether

diction with

When

Therefore,

When

The matrix has a double real eigenvalue

If

and

We have

and

We have

For the case

and

We assume

tradiction with

i.e.,

Finally, we study the case of

Then, we have

and

We have

When discriminant

When

Therefore,

When

double real eigenvalue

it is obvious that

We have

Therefore, the equilibrium Q is a saddle as

Finally, we study the case of

The following lemmas were be derived from reference [

Lemma 1. ( [

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

For

Lemma 2. ( [

maps

Having a non-hyperbolic fixed point, i.e.,

Undergoes a transcritical bifurcation at

Theorem 3. A transcritical bifurcation occurs at the equilibrium

Proof. For

and it has eigenvectors

, (9)

Corresponding to

with inverse

which transform system Equation (5) into

where

Rewrite system (12) in the suspended form with assumption

where

Thus, from Lemma 1, the stability of equilibrium

for sufficiently small v and

We now want to compute the center manifold and derive the mapping on the center manifold. We assume

near the origin, where

Substituting (16)into (15) and comparing coefficients of

from which we solve

Therefore, the expression of (15) is approximately determined:

Substituting (17) into (14), we obtain a one dimensional map reduced to the center manifold

It is easy to check that

The condition (19) implies that in the study of the orbit structure near the bifurcation point terms of

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

This section is devoted to the analysis for the case

have

bifurcation happens at the equilibrium point

Theorem 4. For map (5) when

Proof. Performing a coordinate shift as follows

We translate equilibrium

Therefore, we discuss equilibrium point

For

The matrix have eigenvectors

to

where

Therefore, we obtain the inverse of transformation (23)

Therefore

where

Rewrite system (25) in the suspended form

where

Equivalently, the suspended system (26) has a two-dimensional center manifold of the form

Near the origin, where

Then

Comparing coefficients of

from which we solve

Thus, the expression of (27)is determined, i.e.,

Substituting (30) into the first equation in (26), we obtain a one-dimensional map

From (31), we can check that

Thus, the conditions

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 [

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).

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