Bifurcations for a deterministic SIR epidemic model in discrete time
© Zhou et al.; licensee Springer 2014
Received: 14 January 2014
Accepted: 4 June 2014
Published: 23 June 2014
In this paper, we are concerned with the theoretical analysis of the bifurcations for a deterministic SIR epidemic model in discrete time. By deriving equations describing flows on the center manifolds, we discuss the transcritical bifurcation at the disease-free equilibrium point and the direction and stability of the flip bifurcation at the positive endemic equilibrium point. We give explicit conditions to check the stability of equilibrium points and the critical parameter for the emergence of a flip bifurcation. For illustrating the theoretical analysis, we also give some numerical simulation examples.
MSC:37N25, 39A11, 92B05.
Since Kermack and McKendrick  proposed the Susceptible-Infective-Recovered model (or SIR for short) in 1927, a lot of glorious studies on the dynamics of epidemic models have been presented (see [2–5]). The basic and important research subjects for these systems are local and global stability of the disease-free equilibrium point and the endemic equilibrium point, existence of periodic solutions, persistence and extinction of the disease, etc. According to the dependence on the variable (i.e., time), these systems were classified into two types: continuous-time systems and discrete-time systems.
As the value of r increases above 2, we have period doubling and eventually chaos.
What kinds of periodic behavior may occur?
What restriction is enough to guarantee this periodic behavior?
Is the period-doubling behavior possible?
As far as we know, there was no literature of theoretical analysis to answer the above questions up till now. In this paper, we pay attention to the theoretical analysis of structural stabilities of the disease-free equilibrium point and the endemic equilibrium point under certain restrictive conditions of α, β, and γ. By applying center manifold theory, we find and prove the existence of a transcritical bifurcation at a disease-free equilibrium point and flip bifurcation (or period-doubling bifurcation) at a positive endemic equilibrium point. The transcritical bifurcation behavior (see Theorem 3.1) shows that when we have the restrictive condition the SIR system has only one equilibrium point (disease-free equilibrium point), when s is slightly away from zero, another equilibrium point (endemic equilibrium point) occurs, and, moreover, their stabilities exchange at . The flip bifurcation behavior (see Theorem 4.1) demonstrates that when the restrictive parameter γ crosses over the critical value slightly with a given direction, two endemic equilibrium points appear and form a period-two orbit (cycle). From these results we properly answer the above question. For illustrating our theoretical conclusions, we also give some numerical simulation examples.
where , , and represent susceptible, infective, and removed (or isolated) subgroups, respectively, n represents , , Δt is a fixed time interval (e.g., 1 hour or 1 day). It is assumed that , , , and and the parameters are positive, , , . To guarantee the solutions of system (1) to be non-negative for all initial conditions, we further assume (see ). It is easy to see that for all time, i.e. the total population size remains constant. is the value of the force of infection (number of contacts that result in infection per susceptible individual in the time interval Δt), is the number of births or deaths per individual during the time interval Δt (number of births = number of deaths) and is the removal number (number of individuals that recover in the time interval Δt). In addition, it is assumed that there are no deaths due to the disease, no recruitment, and no vertical transmission of the disease (all new-born members are susceptible) and that the individual’s recovery leads to immunity.
In order to discuss the model (1) easily, some preliminary transformations will be made hereafter.
The organization of this paper is as follows. In the next section, we identify all cases of non-hyperbolic and hyperbolic equilibrium points, which is the fundament for all succeeding studies. In Section 3, we discuss the transcritical bifurcation at the disease-free equilibrium point of (1). Section 4 is devoted to the investigation of the direction and stability of the flip bifurcation at the positive endemic equilibrium point by computing a center manifold. In Section 5, some simulations are made to demonstrate our results.
2 Non-hyperbolic and hyperbolic cases
It is non-hyperbolic if and only if lies on the line .
(a) If , it is a saddle node; (b) if , it is a stable node.
From the assumption , we see that . Then non-hyperbolicity happens in the case . In view of and , we know is impossible. From , we get , i.e. , implying that lies on .
(a) When (referred to the case ), the equilibrium point P is a saddle node since . (b) When (referred to the case ), the eigenvalue , then the equilibrium point P is a stable node. The proof is complete. □
It is non-hyperbolic if and only if lies on the curve .
(a) If , it is a stable node; (b) if , it is a saddle node; (c) if , it is a stable focus.
- (1)It is well known that is hyperbolic if and only if none of the eigenvalues , lies on the unit circle . Denote . In the case of , and are both real. Then the non-hyperbolicity happens when or is 1. For whether or , we get
When and , the equilibrium point is hyperbolic.
If , the matrix has a double real eigenvalue . It is obvious that . Considering the line and the curve , we can get two intersection points and where and . Then as . This implies . Therefore, the equilibrium point is a stable node in the cases of and .
we have . Therefore, the equilibrium point is a stable node as .
- (b)We discuss the case that , i.e. . In this case we have
- (c)In the case of , and are a pair of conjugate complex. Since
and lie inside of and the equilibrium point Q is a stable focus for the case . □
3 Transcritical bifurcation
In this section we consider the case that , where the transcritical bifurcation at equilibrium point will happen.
Theorem 3.1 A transcritical bifurcation occurs at the equilibrium point P when . More concretely, for a parameter s being slightly less than zero there are two equilibrium points: a stable point P and an unstable negative equilibrium point which coalesce at ; for parameter s being slightly greater than zero there are also two equilibrium points: an unstable equilibrium point P and a stable positive equilibrium point Q. Thus an exchange of stability has occurred at .
corresponding to and , respectively, where T means the transpose of the matrices. Our goal is to determine the nature of the stability of for s near zero. First, we must put the matrix into a diagonal form.
for sufficiently small u and s.
The map (15) can be viewed as a truncated normal form for the transcritical bifurcation (see [, p.365]). The stability of the two branches of equilibrium points lying on both sides of are easily verified. □
Remark 3.1 (The biological explanation of Theorem 3.1)
Because the epidemic model (1) cannot have a negative equilibrium point in real life, when (i.e. ), (1) has only a disease-free equilibrium point which is stable. In this case, for any given initial value with , the state will finally tend to , namely, the final situation of epidemic is free from disease. However, when , a positive equilibrium point will occur. It is an endemic equilibrium point and stable, meanwhile, the disease-free equilibrium point changes to unstable. For any given initial value with , the state will finally tend to .
4 Flip bifurcation
Then we have the following theorem.
Theorem 4.1 If , then a flip bifurcation occurs at the equilibrium point when , i.e. and . More concretely, for , an attractive 2-periodic orbit of map F emerges near the equilibrium point when , but the 2-periodic orbit does not exist when , for , a repellent 2-periodic orbit of map F emerges near the equilibrium point when , but the 2-periodic orbit does not exist when .
as assumed in our theorem. Thus, the conditions () and () of Theorem 3.5.1 in  are checked by (22) and (23), respectively. Therefore a flip bifurcation occurs at and a 2-periodic orbit arises as stated in the theorem. □
Remark 4.1 (The biological explanation of Theorem 4.1)
When , the epidemic model (1) has only one positive equilibrium point, i.e. the endemic equilibrium point . If the parameters cross the curve slightly with a given direction, two new positive equilibrium points (assumed to be , ) of model (1) will emerge and form a 2-periodic orbit, i.e. and . Their stabilities are determined by the negative and positive values of , concretely, when they are attractive, when they are repellent.
In this section, we will give three simulation examples to illustrate the results obtained in the above sections.
From Figures 3 and 4, we see that the conclusion of Theorem 3.1 is well verified by numerical simulation. Namely, for given various initial values for , if slightly, there are a stable point and an unstable negative point which coalesce as , if slightly, point is unstable and positive point is stable. Thus a transcritical bifurcation occurs at the equilibrium point when .
Additionally, we see from Figure 8 that the flip bifurcation giving a 4-periodic orbit occurs at parameter . The next period doubling takes place at , and so on. But from Figure 5 we do not see this phenomenon because of different parameters being between in Examples 5.2 and 5.3. Indeed, a 4-periodic orbit, an 8-periodic orbit etc. may occur in the region in Example 5.2. However, exceeds the restriction of the parameter γ in our model. Therefore, in Example 5.2 we can only see the emergence of a stable equilibrium point and an attractive 2-periodic orbit.
Discrete-time epidemic models are useful for modeling situations of epidemic. They always exhibit richer and more complicated dynamical behaviors than continuous-time models, though some of them may be considered as approximations to the continuous-time models. Allen  gave a systematical comparison between the discrete-time models and the corresponding continuous-time models and showed the periodic behavior (which does not occur in the corresponding continuous cases) for the case of discrete-time model SIR with births and deaths by numerical simulations. To reveal the reason for the resulting periodic behavior of the discrete-time models SIR with births and deaths, we give a sufficient theoretical investigation of this model. Our theoretical analysis focuses on the transcritical bifurcation at the disease-free equilibrium point and the period-doubling bifurcation at endemic equilibrium point. Our analytic conclusions well answer the questions presented in the Introduction section. Using our results, one can check the stability of the above-mentioned equilibrium points and calculate the critical parameter γ for the emergence of a flip bifurcation. Finally, we also present some numerical simulation examples for illustrating our theoretical analysis.
This work has been supported by the NNSF of China (Grant 11161018), the NSF of Guangdong province (Grant s2013010013385), the Science Innovation Project of Department of Education of Guangdong province (Grant 2013KJCX0125) and NSFP of Zhanjiang Normal University (Grant ZL1303). The authors thank the anonymous reviewers for their detailed and insightful comments and suggestions for improvement of the manuscript.
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