The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time

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.


Introduction
Since Kermack and McKendrick [1] proposed the Susceptible-Infective-Recovered model (or SIR for short) in 1927, a lot of glorious studies on the dynamics of the epidemic models have been presented (see [2]- [9]).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.According to the dependence on variable (i.e., time), these systems were classified into two types: continuoustime system and discrete-time system.
For the epidemic models, there have been a lot of researches focusing on the case of continuous-time (see [2]- [6] and that cited therein).However, discrete-time models (or called difference equations) are also useful for modeling situations of epidemic.They can not only have the basic features of the corresponding continuoustime models but also provide a substantial reduction of computer time (see [10]).What is more, a lot of discretetime models are not trivial analogues of their continuous ones and simple models can even exhibit complex behavior (see [5] [10]).
In 1989, Hethcote [7] considered a class of continuous epidemic model with vertical and horizontal transmission.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, 0 1 p < < , 0 1 q < < , and 1 p q + = , 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.Because of biological meanings, a natural constraint is 0 b pd − > .A similarly detailed description of the model and its dynamics may be found in [7].In recent, Meng and Chen [8] have also studied the epidemic system (1).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.
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.

Discrete SIR Epidemic Model with Vertical and Horizontal Transmission
In this section, we consider the discrete SIR epidemic model with vertical and horizontal transmission: where n S , n I and n R represent susceptible, infective and removed (or isolated) subgroups respectively, n represents a fixed time, 0,1, 2, n =  .It is assumed that 0 0 S > , 0 0 I ≥ , 0 0 R ≥ and 0 0 0 1 S I R + + = .In view of assumption that population is a constant size in [3], i.e., 0 0 0 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.

Hyperbolic and Non-Hyperbolic Cases
In this section, we will discuss the hyperbolic and non-hyperbolic cases in a two parameters space parameter.
Theorem 1.The equilibrium point ( )  and 3  for equilibrium point P (see Figure 1).
Proof.The Jacobian matrix of ( 5) at ( ) and its eigenvalues are Types of hyperbolic equilibrium point ( ) From the assumption 0 1 b < < , we see that 1 0 1 λ < < .Then non-hyperbolicity will be happened in the case λ < − , then the equilibrium point P is a stable node.When ( )  ), the equilibrium point P is a node since (I) When b β < , there exist six types for hyperbolic equilibrium point Q (see Table 2).Where 1 s and 2 s satisfy ( ) , , respectively.
(II) When 1 b b β < ≤ , there exist four types for hyperbolic equilibrium point Q (see Table 3).
Where 3 ,  and 7  for equilibrium point Q (see Figure 2).
In the case of 0 ∆ < , 1 λ and 2 λ are a pair of conjugate complex.Since  λ > .Therefore, the equilibrium Q is a stable node as ( ) 4 , s r ∈  .For the case 1 0 s s < < , i.e., ( ) 5 , s r ∈  , we have Finally, we study the case of 2 s s Then, we have  λ are a pair of conjugate complex.Since ( ) 1, a r a s , s r ∈  by same methods as in case (1).This means that the equilibrium Q is a stable node for ( ) 9 , s r ∈  .The proof is complete.□

Transcritical Bifurcation
In this section we consider the case that ( ) 2 , w b ∈  , where the transcritical bifurcation at equilibrium point ( ) 1, 0 P will happen.The following lemmas were be derived from reference [11].Lemma 1. ([11], Theorem 2.1.4)The map x Ax f x y x y y Bx g x y satisfies that A is c c × matrix with eigenvalues of modulus one, and B is s s × matrix with eigenvalues of modulus less than one, and where f and g are ( ) in some neighborhood of the origin.Then there exists a r C center manifold for (7) which can be locally represented as a graph as follows for δ sufficiently small.Moreover, the dynamics of (7) restricted to the center manifold is, for u sufficiently small, given by the c-dimensional map 11], in page 365) A one-parameter family of ( ) having a nonhyperbolic fixed point, i.e., ( ) ( ) 2 0, 0 0, 0, 0 0, 0, 0 0.
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 ( ) 1 , w b ∈  , we have 1 1 λ = and 2 0 1 λ < < .Consider w as the bifurcation parameter and write F as w F to emphasize the dependence on w.One can easily see that the matrix ( ) and it has eigenvectors corresponding to 1 λ and 2 λ respectively, where T means the transpose of matrices.
First, we put the matrix ( ) 0, 0 w DF into a diagonal form.Using the eigenvectors (10), we obtain the transformation which transform system (5) into Rewrite system (13) in the suspended form with assumption ( ) ( ) We now want to compute the center manifold and derive the mapping on the center manifold.We assume ( ) ( ) It is easy to check that ( ) ( ) ( ) 2 0, 0 0, 0, 0 0, 0, 0 0. The condition (18) 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, (18) shows that the orbit structure of (17) near ( ) ( ) , 0, 0 v w = is qualitatively the same as the orbit structure near ( ) ( ) , 0, 0 v w = of the map

Simulations
In this section, we will give a simulation to illustrate the result obtained in the above section.( ) 0.2,0.63,0.17Q is stable for case of 5 1 w = > .
as a planar map F:

Figure 1 .
Figure 1.Districts for equilibrium point P.

Theorem 2 .
and meanwhile when 1 w < , the equilibrium P is a saddle since 1 1 λ > .The proof is complete.□ There does not exist non-hyperbolic case for equilibrium point ( ) * * , Q S I .But the hyperbolicity can be divided into the following cases:

Figure 2 .
Figure 2. Districts for equilibrium point Q when b < β.

Figure 3 .
Figure 3. Districts for equilibrium point Q when b 1 < β ≤ b. and letting F  denote the transformed F, we translate equilibrium ( ) * * , Q S I hyperbolic if and only if none of eigenvalues 1 λ , 2 λ lies on the unit circle 1 S .In the following we discuss the eigenvalues in two cases, i.e., λ 1 and λ 2 are both real.Because non- hyperbolicity happens if and only if 1

λ lie inside of 1 S 4 , 1 λ< < for ( ) 4 ,
and the equilibrium point Q is a stable focus referred to the case ( ) , i.e. ( ) 4 , s r ∈  .In this case we have s r ∈  .On the other hand, there also exists 1 0 s r ∈  .In fact, since

λ lie inside of 1 S 8 ,
and the equilibrium point Q is a stable focus referred to the case ( ) , equilibrium Q is stable node in the cases of 3  .If 0 ∆ > , We first discuss the case that 0 b s − < < , i.e. ( ) Therefore, the equilibrium Q is a saddle as ( ) 8 , s r ∈  .Finally, we study the case of 3 0 s s < < , i.e. ( ) 9 , s r ∈  .We easily prove by studying a one-parameter family of map on a center manifold which can be represented as follows, v and w .
Therefore the expression of (15) is approximately determined.Substituting (15) into (14), we obtain a one dimensional map reduced to the center manifold w other negative equilibrium point (no meaning in biology) is unstable.The simulation result is presented in Figure4.and and two equilibrium points of system (2) occurs, where disease-free equilibrium point ( ) 1, 0, 0 P is unstable and other positive equilibrium point ( ) 0.2, 0.63, 0.17 Q is unstable.The simulation result is presented in Figure 5.

Table 2 .
Types of hyperbolic equilibrium 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 0 s = are easily verified.□