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Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay
The paper investigated an avian influenza virus propagation model with nonlinear incidence rate and delay based on SIR epidemic model. We regard delay as bifurcating parameter to study the dynamical behaviors. At first, local asymptotical stability and existence of Hopf bifurcation are studied; Hopf bifurcation occurs when time delay passes through a sequence of critical values. An explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions is derived by applying the normal form theory and center manifold theorem. What is more, the global existence of periodic solutions is established by using a global Hopf bifurcation result.
In March 2013, new avian-origin influenza virus broke out in Shanghai and the surrounding provinces of China . During the first week of April, this virus had been detected in six provinces and municipal cities; this virus has caused global concern as a potential pandemic threat . The virus fast took people’s life without timely treatment. Therefore, strong measures should be taken to control the spread of H7N9 viruses.
is an infectious disease caused by influenza A virus. Moreover, it is essential to study and to dominate the spread of . Mathematical models become important instruments in the analysis and control of infectious diseases. The present study evaluates the possible application of SIR model for spreading.
Let , , and be the population densities of susceptible, infective, and recovered, respectively. Recruitment of new individuals is into the susceptible class at a constant rate . Parameters , , and are positive constants which represent the death rate of the classes, respectively. is the length of the infectious period; is the average time spent in class before recovery .
In 1979, Cooke  used mass action incidence . In 2009, Xu and Ma  developed the model with the force of infection given by , where determines the level at which the force of infection saturates and is a contract . Then, the avian influenza virus propagation model based on SIR model has the following form:
Since does not appear in the first two equations, and avoid excessive use of parentheses in some of the latter calculations, the avian influenza virus propagation model is transformed into the following form with the following initial condition: which was presented and studied in .
The steady state of the model and the stability of epidemic models have been studied in many papers. Zhang and Li  studied the global stability of an SIR epidemic model with constant infectious periods. Xu and Ma  showed the global stability of the endemic equilibrium for the case of the reproduction number . McCluskey  shown that the endemic equilibrium is globally asymptotically stable whenever it exists. In this paper, we investigated the Hopf bifurcation and the global existence of periodic solutions of model (2), which have not been reported yet.
The organization of this paper is as follows. In Section 2, we will investigate the local asymptotical stability and existence of Hopf bifurcation by analyzing the associated characteristic equation. In Section 3, an explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions will be derived by applying the normal form theory and center manifold theorem. In Section 4, existence of global periodic solutions will be established by using a global Hopf bifurcation result. In Section 5, a brief discussion is offered to conclude this work.
2. Local Stability and Hopf Bifurcation
Some results can be directly obtained from [3, 5]. The basic reproduction number for the model is . System (2) always has a disease-free equilibrium . If , system (2) has a unique endemic equilibrium . The characteristic equation of system (2) at the endemic equilibrium is where , , , and . If hold, when , the endemic equilibrium of system (2) is locally stable .
If () is a solution of system (2), separating real and imaginary parts, we obtain the following: Then, we get It follows that Letting , we get It is easy to show that The case of has been discussed in . We obtain global asymptotic stability of the endemic equilibrium when . If hold, that is, , we have . Following the theorem given by Ruan , there exists critical value with where , . If and are satisfied, (6) has a pair of purely imaginary roots when . Additionally, all roots of (6) have negative real parts when and when (5) has at least a pair of roots with positive real part. In order to give the main results, it is necessary to prove the transversality condition holds. Denote as the root of (5) with ,. Differentiating (5) with respect to yields For the sake of simplicity denoting and by ,, respectively, in the following: From (10), we know ; then, hold. Under this condition, we have the following theorem.
Theorem 1. (i) If and holds, the equilibrium of system (2) is asymptotically stable for any .
(ii) If and holds, is asymptotically stable for and unstable for . System (2) exhibits the Hopf bifurcation at the equilibrium for , .
3. Direction and Stability of the Bifurcating Periodic Solutions
In Section 2, we obtain the conditions under which a family of periodic solutions bifurcate from the steady state at the critical value of . In this section, we investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution at critical values , using techniques of the normal form theory and center manifold theorem.
Let and let . The Taylor expansion of system (2) at is where , , , , , , , , and for . System (2) is transformed into FDE as with where By Riesz representation theorem, there exists a function of bounded variation, for , such that In fact, we can choose where is a delta function.
For , the operators and are defined as follows: The adjoint operator corresponding to is defined as follows: and an adjoint bilinear is as follows: where .
From the preceding discussion, we know that and be the eigenvectors of and corresponding to and , respectively. Next, we calculate and to determine the normal form of operator .
Proposition 2. Let and be eigenvectors of and corresponding to and , respectively, satisfying and .
Proof . Without loss of generality, we just consider the eigenvector . By the definition of and with , we get (here, is a parameter). In what follows, notice that and ; we have . Using a proof procedure similar to that in , by direct computation, we get and . Bring and into ; it is not hard to obtain the parameter . The detailed procedure of proof refers to . The proof is completed.
Then, we construct the coordinates of the center manifold at . Let On the center manifold , we have where and and are local coordinates for the center manifold in the direction of and , respectively. Since , we have where We rewrite this as with where
So, where According to the discussion above, we can compute the following parameters: where determines the directions of the Hopf bifurcations, determines the stability of the bifurcation periodic solutions, and determines the period of the bifurcating periodic solutions . By lemma (5), we know that ; we have the following theorem.
Theorem 3. If , the direction of the Hopf bifurcation of the system (1) at the equilibrium when is supercritical (subcritical) and the bifurcating periodic solutions are orbitally asymptotically stable (unstable).
4. Global Existence of Periodic Solution
From the above discussion, we know that system (2) undergoes a local Hopf bifurcation at when . A natural question is that if the bifurcating periodic solutions of system (2) exist when is far away from critical values? In this section, we will study the global existence of periodic solutions of system (2). Through use of a global Hopf bifurcation theorem given by Wu , we obtain the global continuation of periodic solutions bifurcating from the points . First of all, we define Let denote the connected component of in and its projection on component. From theorem (5), we know that is nonempty. and are defined in (10) and (11).
Lemma 4. All periodic solutions of system (2) are uniformly bounded.
Proof . Let be a nonconstant periodic solution of system (2), and let , and , be the maximum and minimum of and , respectively. Using a proof procedure similar to that in , we can obtain It is shown that all periodic solutions of system (2) are uniformly bounded. This completes the proof.
Lemma 5. System (2) has no nonconstant periodic solution of period .
Proof . For a contradiction, if system (5) has a -periodic solution, say , then it satisfies the ODES as follows: We can get By Bendixson’s criterion, we know that system (2) has no nonconstant periodic solutions, which prove the lemma.
Theorem 6. Suppose that the condition and is satisfied. Then, for each ,, system (2) has at least periodic solutions.
Proof. The characteristic matrix of system (2) at the equilibrium is in the following form:
Using a proof procedure similar to that in , it is easy to obtain that , , is an isolated center.
From the proof procedure of Lemmas 4 and 5, it is easy to know that there exist , , smooth curve such that for all , and
Define , and let . Obviously, if and such that , if and only if , set We obtain the crossing number as follows: We conclude that Since the first crossing number of each center is always , by [10, Theorem ], we conclude that is unbounded. By the definition of given in (10), we know that, for , automatically hold.
Again, the population densities of susceptible and infective are ultimately uniformly bounded, implying that the projection of onto the -space is bounded. Meanwhile, system (2) with has no nonconstant periodic solutions; if there exits such that the projection of onto the -space ins with , then, the projection of onto the -space is bounded. Since and from Lemma 5, we can obtain for with ; that is to say, onto -space is also bounded. Because is unbounded, must be unbounded. Consequently, include for . That is to say, for each , system (2) at least has nonconstant period solutions. The proof is complete.
In this paper we have analytically studied an avian influenza virus propagation model with nonlinear incidence rate and time delay depending on SIR epidemic model. Some previous efforts in epidemic models have been mainly concerned with the global stability and asymptotical stability. However, it is a new idea to study the bifurcation periodic solutions and global existence of periodic solutions. The theoretical analysis for the avian influenza virus propagation models is given. Then, Hopf bifurcation occurs when time delay passes through a sequence of critical values. Furthermore, bifurcations and stability of the bifurcation periodic solutions are derived. Finally, global existence of periodic solutions is established.
Conflict of Interests
The authors declare that they have no financial and personal relationships with other people or organizations that can inappropriately influence their work; there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, in this paper.
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