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In this paper, a class of discrete deterministic SIR epidemic model with vertical and horizontal transmission is studied. Based on the population assumed to be a constant size, we transform the discrete SIR epidemic model into a planar map. Then we find out its equilibrium points and eigenvalues. From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability. Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point. Finally, we give some numerical simulation examples for illustrating the theoretical analysis and the biological explanation of our theorem.

Since Kermack and McKendrick [

For the epidemic models, there have been a lot of researches focusing on the case of continuous-time (see [

In 1989, Hethcote [

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 1/b. The susceptible becomes infectious at a bilinear rate βI, where I is the proportion of infectious individuals and β is the contact rate. The infectious recover (i.e., acquire lifelong immunity) at a rate r, so that 1/r is the mean infectious period. The constant p, q,

In this paper, we pay attention to the discrete situation of (1). From discussing the influence of the coefficient parameters effected on the eigenvalues, we give the hyperbolicity of equilibrium points and determine which point is saddle, node or focus as well as their stability. Further, by deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the non-hyperbolic equilibrium point and research how does small perturbation of coefficient parameters affect the number and stability of equilibrium points. Moreover, we give some numerical simulation examples for illustrating the theoretical analysis and explain the biological meaning of our theorem.

In this section, we consider the discrete SIR epidemic model with vertical and horizontal transmission:

where

system (2) can be changed into

Rewrite (4) as a planar map F:

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

The organization of this paper is as follows. In next section, we identify all cases of non- and hyperbolic equilibria, which is a fundament for all succeeding studies. In Section 4, we discuss the transcritical bifurcation at the disease-free equilibrium of (1), the direction and stability of the transcritical bifurcation is investigated by computing a center manifold. In Section 5, some simulations are made to demonstrate our results and the biologic explanation of the theorem is also given.

In this section, we will discuss the hyperbolic and non-hyperbolic cases in a two parameters space parameter.

Theorem 1. The equilibrium point

and

Otherwise, the equilibrium point

Remark 1. By Theorem 3.1 the domain

Proof. The Jacobian matrix of (5) at

and its eigenvalues are

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

Saddle | |||

Stable node | |||

Saddle |

From the assumption

Theorem 2. There does not exist non-hyperbolic case for equilibrium point

(I) When

Where

and

(II) When

Where

Remark 2. By Theorem 3.2, when

When

tricts

Proof. Performing a coordinate shift as follows:

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

Stable node | |||

Stable node | |||

Saddle | |||

Stable node | |||

Stable focus | |||

Stable node |

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

Stable node | |||

Saddle | |||

Stable node | |||

Stable focus |

and letting

where

It is known that

Case (I). When discriminant_{1} and λ_{2} are both real. Because non- hyperbolicity happens if and only if _{1} = 1 or λ_{2} = 1, we can get_{1} = 1 nor λ_{2} = 1 is possible. Next, let’s examine

In the case of

If

If

Since

we have

and

we have

For the case

and

Therefore, the equilibrium Q is a stable node as

Finally, we study the case of

Then, we have

This means that the equilibrium Q is a stable node for

Case (II). When discriminant

When

If

If

We know

Therefore, the equilibrium Q is a saddle as

Finally, we study the case of

The proof is complete. □

In this section we consider the case that

Lemma 1. ([

satisfies that A is

where f and g are

for

Lemma 2. ([

having a nonhyperbolic fixed point, i.e.,

undergoes a transcritical bifurcation at

Theorem 3. A transcritical bifurcation occurs at the equilibrium P when w = 1. More concretely, for w < 1 slightly there are two equilibriums: a stable point P and an unstable negative equilibrium which coalesce at w = 1 and for w > 1 slightly there are also two equilibriums: an unstable equilibrium P and a stable positive equilibrium Q. Thus an exchange of stability has occurred at w = 1.

Proof. For

and it has eigenvectors

corresponding to

First, we put the matrix

with inverse

which transform system (5) into

Rewrite system (13) in the suspended form with assumption

where

Thus, from Lemma 4.1, 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 (15) into (16) and comparing coefficients of

from which we solve

Therefore the expression of (15) is approximately determined. Substituting (15) into (14), we obtain a one dimensional map reduced to the center manifold

It is easy to check that

The condition (18) implies that in the study of the orbit structure near the bifurcation point terms of

Map (19) can be viewed as truncated normal form for the transcritical bifurcation (see Lemma 4.2). The stability of the two branches of equilibriums lying on both sides of

In this section, we will give a simulation to illustrate the result obtained in the above section.

Example 1. Let

If let

If let

The conclusion in Theorem 4.1 reveals a fact that the topological structure changes at disease-free equilibrium point will take place when system (2.1) encounters small perturbation for coefficient parameters. Concretely, when parameter

If let

If let

Therefore, in reality we may control the factors of contact rate, birth rate, recovery rate, etc., to achieve the aim of prevention and treatment of disease.

We thank the Editor and the referee for their comments. This work has been supported by the Science Innovation Project (Grant 2013KJCX0125) and the Innovation and Developing School Project (Grant 2014KZDXM065) of Department of Education of Guangdong province, the NSF of Guangdong province (Grant S2013010013385) and the NSFP of Lingnan Normal University (Grant ZL1303).

XiaodanLiao,HongboWang,XiaohuaHuang,WenboZeng,XiaoliangZhou, (2015) The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time. Applied Mathematics,06,1665-1675. doi: 10.4236/am.2015.610148