Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay

Abstract

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.

1. Introduction

In March 2013, new avian-origin influenza 𝐴(𝐻7𝑁9) virus (𝐴−𝑂𝐼𝑉) broke out in Shanghai and the surrounding provinces of China [1]. 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 [2]. The virus fast took people’s life without timely treatment. Therefore, strong measures should be taken to control the spread of H7N9 viruses.

𝐻7𝑁9 is an infectious disease caused by influenza A virus. Moreover, it is essential to study and to dominate the spread of 𝐻7𝑁9. Mathematical models become important instruments in the analysis and control of infectious diseases. The present study evaluates the possible application of SIR model for 𝐻7𝑁9 spreading.

Let 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) be the population densities of susceptible, infective, and recovered, respectively. Recruitment of new individuals is into the susceptible class at a constant rate 𝐵 [3]. Parameters 𝜇₁, 𝜇₂, and 𝜇₃ are positive constants which represent the death rate of the classes, respectively. 𝜏 is the length of the infectious period; 1/𝛾 is the average time spent in class 𝐼 before recovery [3].

In 1979, Cooke [4] used mass action incidence 𝛽𝑆(𝑡)𝐼(𝑡−𝜏). In 2009, Xu and Ma [5] developed the model with the force of infection given by 𝛽𝑆(𝑡)(𝐼(𝑡−𝜏)/(1+𝛼𝐼(𝑡−𝜏))), where 𝛼 determines the level at which the force of infection saturates and 𝛽 is a contract [5]. Then, the avian influenza virus propagation model based on SIR model has the following form: 𝑆̇(𝑡)=𝐵−𝜇₁𝑆(𝑡)−𝛽𝑆(𝑡)𝐼(𝑡−𝜏)(),𝐼̇(𝑡)=(𝑡)𝐼()1+𝛼𝐼𝑡−𝜏𝛽𝑆𝑡−𝜏()−𝜇1+𝛼𝐼𝑡−𝜏²𝐼(𝑡),𝑅̇(𝑡)(𝑡)+𝛾=𝛾𝐼−𝜇₃𝑅(𝑡). (1)

Since 𝑅 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 𝑆̇(𝑡)=𝐵−𝜇₁𝑆(𝑡)−𝛽𝑆(𝑡)𝐼(𝑡−𝜏)(),𝐼̇(𝑡)=(𝑡)𝐼()1+𝛼𝐼𝑡−𝜏𝛽𝑆𝑡−𝜏()−𝜇1+𝛼𝐼𝑡−𝜏²+𝛾𝐼(𝑡),𝑅̇(𝑡)=𝛾𝐼(𝑡)−𝜇₃𝑅(𝑡), (2)(3) with the following initial condition: 𝑆(0)∈𝑅⁺(𝜃)(𝜃)[],[],𝐼=𝜙for𝜃∈−𝜏,0where𝜙∈𝐶−𝜏,0,𝑅⁺, (4) which was presented and studied in [3].

The steady state of the model and the stability of epidemic models have been studied in many papers. Zhang and Li [6] studied the global stability of an SIR epidemic model with constant infectious periods. Xu and Ma [5] showed the global stability of the endemic equilibrium for the case of the reproduction number 𝑅₀>1. McCluskey [3] 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 𝑅₀=𝐵𝛽/𝜇₁(𝜇₂+𝛾). System (2) always has a disease-free equilibrium 𝐸₁=(𝐵/𝜇₁,0). If 𝐵𝛽>𝜇₁(𝜇₂+𝛾), system (2) has a unique endemic equilibrium 𝐸^∗=(𝑆^∗,𝐼^∗)=((𝐵𝛼+𝜇₂+𝛾)/(𝛽+𝛼𝜇₁),(𝐵𝛽−𝜇₁(𝜇₂+𝛾))/(𝜇₂+𝛾)(𝛽+𝛼𝜇₁)) [3]. The characteristic equation of system (2) at the endemic equilibrium 𝐸^∗ is 𝜆²+𝑝₁𝜆+𝑝₀+𝑞₁𝜆+𝑞₀𝑒^−𝜆𝜏=0, (5) where 𝑝₀=(𝜇₂+𝛾)(𝜇₁+𝛽𝐼^∗/(1+𝛼𝐼^∗)), 𝑝₁=𝜇₁+𝜇₂+𝛾+𝛽𝐼^∗/(1+𝛼𝐼^∗), 𝑞₀=−𝛽𝜇₁𝑆^∗/(1+𝛼𝐼^∗)², and 𝑞₁=−𝛽𝑆^∗/(1+𝛼𝐼^∗)². If 𝑅₀>1 𝑃₁ hold, when 𝜏=0, the endemic equilibrium 𝐸^∗ of system (2) is locally stable [5].

If 𝑖𝜔 (𝜔>0) is a solution of system (2), separating real and imaginary parts, we obtain the following: 𝑝₁𝜔=𝑞₀sin𝜔𝜏−𝑞₁𝜔𝜔cos𝜔𝜏,₂−𝑝₀=𝑞₀cos𝜔𝜏+𝑞₁𝜔sin𝜔𝜏. (6) Then, we get 𝑞cos𝜔𝜏=⁰−𝑝₁𝑞₁𝜔²−𝑝₀𝑞₀𝑞²₀+𝑞²₁𝜔²,𝑝sin𝜔𝜏=¹𝑞₀𝜔+𝜔²−𝑝₀𝑞₁𝜔𝑞²₀+𝑞²₁𝜔². (7) It follows that 𝜔⁴+𝑝²₁−2𝑝₀−𝑞²₁𝜔²+𝑝²₀−𝑞²₀=0. (8) Letting 𝑧=𝜔², we get 𝑧²+𝑝²₁−2𝑝₀−𝑞²₁𝑧+𝑝²₀−𝑞²₀=0. (9) It is easy to show that 𝑝²₁−2𝑝₀−𝑞²₁=𝜇₁+𝛽𝐼^∗1+𝛼𝐼^∗2+𝜇₂+𝛾²−𝜇₂+𝛾²(1+𝛼𝐼^∗)²𝑝>0,₂₀−𝑞²₀=𝜇₂+𝛾𝜇₂+𝛾𝜇₁+𝛽𝐼^∗1+𝛼𝐼^∗+𝛽𝜇₁𝑆^∗(1+𝛼𝐼^∗)²×𝜇₁−𝜇₁1+𝛼𝐼^∗+𝛽𝐼^∗1+𝛼𝐼^∗. (10) The case of 𝛽≥𝜇₁𝛼 𝐻₁ has been discussed in [5]. We obtain global asymptotic stability of the endemic equilibrium when 𝑅₀>1. If 𝛽<𝜇₁𝛼 𝐻₂ hold, that is, (𝛽−𝜇₁𝛼)𝐼^∗/(1+𝛼𝐼^∗)<0, we have 𝑝²₀−𝑞²₀<0. Following the theorem given by Ruan [7], there exists critical value 𝜏_𝑘^(𝑗)=1𝜔_𝑘𝜔arccos^2𝑘−𝑝₀𝑞₀−𝑝₁𝑞₁𝜔²_𝑘𝑞²₀+𝑞²₁𝜔²_𝑘+2𝑗𝜋𝜔_𝑘, (11) with 𝜔_𝑗=2𝑝₀+𝑝²₁−𝑞²₁+2𝑝₀+𝑝²₁−𝑞²₁²−4𝑝²₀−𝑞²₀2^1/2, (12) where 𝑘=1,2,…, 𝑗=0,1,2,…. If 𝑃₁ and 𝐻₂ are satisfied, (6) has a pair of purely imaginary roots ±𝜔₀𝑖 when 𝜏=𝜏₀. Additionally, all roots of (6) have negative real parts when 𝜏∈[0,𝜏₀] and when 𝜏>𝜏₀ (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 Re(𝑑𝜆/𝑑𝜏)⁻¹>0 holds. Denote 𝜆=𝛼(𝜏)+𝑖𝜔(𝜏) as the root of (5) with 𝛼(𝜏)=0,𝜔(𝜏)=𝜔₀. Differentiating (5) with respect to 𝜏 yields 2𝜆+𝑝₁+𝑞₁−𝜏𝑞₁𝜆+𝑞₀𝑒^−𝜆𝜏𝑑𝜆𝑞𝑑𝜏=𝜆₁𝜆+𝑞₀𝑒^−𝜆𝜏. (13) For the sake of simplicity denoting 𝜔₀ and 𝜏₀ by 𝜔,𝜏, respectively, 𝑑𝜆=𝑞𝑑𝜏¹𝜆+𝑞₀𝜆𝑒^−𝜆𝜏2𝜆+𝑝₁+𝑞₁𝑒^−𝜆𝜏−𝑞₁𝜆+𝑞₀𝜏𝑒^−𝜆𝜏 (14) in the following: Re𝑑𝜆𝑑𝜏⁻¹=2𝜆+𝑝₁+𝑞₁𝑒^−𝜆𝜏𝑞₁𝜆+𝑞₀𝜆𝑒^−𝜆𝜏=𝑝₁cos𝜔𝜏−2𝜔sin𝜔𝜏+𝑞₁+𝑖2𝜔cos𝜔𝜏+𝑝₁×sin𝜔𝜏−𝑞₁𝜔²+𝑖𝑞₀𝜔⁻¹=−𝑝₁𝑞₁𝜔²cos𝜔𝜏+2𝑞₁𝜔³sin𝜔𝜏−𝑞²₁𝜔²+2𝑞₀𝜔²cos𝜔𝜏+𝑝₁𝑞₀×𝑞𝜔sin𝜔𝜏₀𝜔²+𝜔⁴⁻¹=−𝑝₁𝑞₁𝜔²+2𝑞₀𝜔²𝑞₀−𝑝₁𝑞₁𝜔²−𝑝₀𝑞₀+2𝑞₁𝜔³+𝑝₁𝑞₀𝜔×𝑝₁𝑞₀𝜔+𝜔²−𝑝₀𝑞₁𝜔−𝑞²₀𝑞²₁𝜔²−𝑞⁴₁𝜔²×𝑞²₀+𝑞²₁𝜔²𝑞₀𝜔²+𝜔⁴⁻¹=2𝑞²₁𝜔⁶+2𝑞²₀+𝑝²₁−2𝑝₀−𝑞²₁𝑞²₁𝜔⁴+𝑝²₁+2𝑝₀−𝑞²₁𝑞²₀𝜔²×𝑞²₀+𝑞²₁𝜔²𝑞₀𝜔²+𝜔⁴⁻¹. (15) From (10), we know 𝑝²₁−2𝑝₀−𝑞²₁>0; then, Re(𝑑𝜆/𝑑𝜏)⁻¹>0 hold. Under this condition, we have the following theorem.

Theorem 1. (i) If 𝑃₁ and 𝐻₁ holds, the equilibrium (𝑆^∗,𝐼^∗) of system (2) is asymptotically stable for any 𝜏>0.
(ii) If 𝑃₁ and 𝐻₂ holds, (𝑆^∗,𝐼^∗) is asymptotically stable for 𝜏∈[0,𝜏₀) and unstable for 𝜏∈(𝜏₀,+∞). System (2) exhibits the Hopf bifurcation at the equilibrium (𝑆^∗,𝐼^∗) for 𝜏=𝜏_𝑗, 𝑗=0,1,2,….

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 𝜏. In this section, we investigate the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution at critical values 𝜏₀, using techniques of the normal form theory and center manifold theorem.

Let 𝑢₁=𝑆(𝑡)−𝑆^∗ and let 𝑢₂=𝐼(𝑡)−𝐼^∗. The Taylor expansion of system (2) at 𝐸^∗ is 𝑢̇₁(𝑡)=𝑎₁𝑢₁(𝑡)−𝑎₂𝑢₂(𝑡−𝜏)+𝑎₆𝑢₂(𝑡)²(𝑡−𝜏)+𝑎₇𝑢₁(𝑡)𝑢₂(),𝑢̇𝑡−𝜏₂(𝑡)=𝑎₃𝑢₁(𝑡)+𝑎₄𝑢₂(𝑡)+𝑎₅𝑢₂(𝑡−𝜏)−𝑎₇𝑢₁(𝑡)𝑢₂(𝑡−𝜏)−𝑎₆𝑢²₂(𝑡−𝜏), (16) where 𝑎₁=−𝜇₁+𝛽𝐼^∗/(1+𝛼𝐼^∗), 𝑎₂=−𝛽𝑆²/(1+𝛼𝐼^∗), 𝑎₃=𝛽𝐼^∗/(1+𝛼𝐼^∗), 𝑎₄=−𝜇₂−𝛾, 𝑎₅−𝛽𝑆²/(1+𝛼𝐼^∗), 𝑎₆=−𝛼𝛽𝑆^∗/(1+𝛼𝐼^∗)³, 𝑎₇=𝛽/(1+𝛼𝐼^∗)², 𝜏=𝜏₀+𝜇, and 𝑢_𝑡=𝑢(𝑡+𝜃)∈𝐶₁ for 𝜃∈[−1,0]. System (2) is transformed into FDE as 𝑢̇(𝑡)=𝐿_𝜇+𝐹𝑢_𝑡,𝜇, (17) with 𝐿_𝜇𝜙=𝜏₀+𝜇𝐵₁𝜙(0)+𝐵₂𝜙(,𝐹=𝜏−1)𝜙,𝜇₀𝑎+𝜇⁶𝜙²₂(−1)+𝑎₇𝜙₁(0)𝜙₂(−1)−𝑎₆𝜙₁(0)𝜙₂(−1)−𝑎₇𝜙²₂(),−1 (18) where 𝐵₁=𝑎₁0𝑎₃𝑎₄,𝐵₂=0𝑎₂0𝑎₅. (19) By Riesz representation theorem, there exists a function 𝜂(𝜃,𝜇) of bounded variation, for 𝜃∈[−1,0], such that 𝐿_𝜇𝜙=⁰₋₁𝑑𝜂𝜃,𝜇𝜙(𝜃)for𝜙∈𝐶. (20) In fact, we can choose 𝜂𝜃,𝜇=𝜏₀+𝜇𝐵₁𝛿(𝜃)+𝐵₂𝛿(𝜃+1), (21) where 𝛿(𝜃) is a delta function.

For 𝜙∈𝐶^′[−1,0], the operators 𝐴 and 𝑅 are defined as follows: 𝐴𝜇𝜙(𝜃)=𝑑𝜙(𝜃)[),𝑑𝜃,𝜃∈−1,0₀₋₁𝑑𝜂𝑡,𝜇𝜙(𝑡),𝜃=0,𝑅𝜇𝜙(𝜃)=[),𝑓0,𝜃∈−1,0𝜇,𝜃,𝜃=0. (22)(23) The adjoint operator 𝐴^∗(0) corresponding to 𝐴(0) is defined as follows: 𝐴^∗𝜓(𝑠)=−𝑑𝜓(𝑠)(],𝑑𝑠,𝑠∈0,1₀₋₁𝑑𝜂^𝑇(𝑡,0)𝜓(−𝑡),𝑠=0 (24) and an adjoint bilinear is as follows: 𝜓,𝜙=𝜓(0)𝜙(0)−⁰₋₁^𝜃₀𝜓(𝜉−𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉, (25) where 𝜂(𝜃)=𝜂(𝜃,0).

From the preceding discussion, we know that 𝑞(𝜃) and 𝑞^∗(𝜃) be the eigenvectors of 𝐴 and 𝐴^∗ corresponding to 𝑖𝜏₀𝜔₀ and −𝑖𝜏₀𝜔₀, respectively. Next, we calculate 𝑞(𝜃) and 𝑞^∗(𝑠) to determine the normal form of operator 𝐴.

Proposition 2. Let 𝑞(𝜃) and 𝑞^∗(𝑠) be eigenvectors of 𝐴 and 𝐴^∗ corresponding to 𝑖𝜏₀𝜔₀ and −𝑖𝜏₀𝜔₀, respectively, satisfying ⟨𝑞^∗,𝑞⟩=1 and ⟨𝑞^∗,𝑞⟩=0.
Then, 𝑞(𝜃)=(1,𝛼)^𝑇𝑒^{𝑖𝜔0𝜏0𝜃}=𝜔1,⁰𝑖−𝑎₁𝑎₂𝑒^{−𝑖𝜔0𝜏0𝑇}𝑒^{𝑖𝜔0𝜏0𝜃},𝑞^∗(𝑠)=𝐷1,𝛽^∗𝑒^{−𝑖𝜔0𝜏0𝑠}=1,−𝜔₀𝑖−𝑎₁𝑎₃^𝑇𝑒^{−𝑖𝜔0𝜏0𝑠}, (26) where 1𝐷=1+𝛼𝛽^∗−𝜏₀𝛼𝑎₂+𝛽^∗𝑎₅𝑒^{−𝑖𝜔0𝜏0}. (27)

Proof . Without loss of generality, we just consider the eigenvector 𝑞(𝜃). By the definition of 𝐴 and 𝑞(𝜃) with 𝜃∈[−1,0), we get 𝑞(𝜃)=(1,𝛼)^𝑇𝑒^{𝑖𝜔0𝜏0} (here, 𝛼 is a parameter). In what follows, notice that 𝑞(0)=(1,𝛼)^𝑇 and 𝐴𝑞(0)=∫⁰₋₁𝑑(𝜂(𝑡,𝜇)𝜙(𝑡))=𝑖𝜔₀𝜏₀𝑞(0); we have 𝛼=(𝜔₀𝑖−𝑎₁)/𝑎₂𝑒^{−𝑖𝜔0𝜏0}. Using a proof procedure similar to that in [8], by direct computation, we get 𝑞(𝜃) and 𝑞^∗(𝑠). Bring 𝑞(𝜃) and 𝑞^∗(𝑠) into ⟨𝑞^∗,𝑞⟩=1; it is not hard to obtain the parameter 𝐷. The detailed procedure of proof refers to [9]. The proof is completed.

Then, we construct the coordinates of the center manifold 𝐶₀ at 𝜇=0. Let 𝑧(𝑡)=𝑞^∗,𝑢_𝑡,𝑊(𝑡,𝜃)=𝑢_𝑡(𝜃)−2Re𝑧(𝑡)𝑞(𝜃). (28) On the center manifold 𝐶₀, we have 𝑊(𝑡,𝜃)=𝑊𝑧(𝑡),𝑧(𝑡),𝜃, (29) where 𝑊(𝑧,𝑧,𝜃)=𝑊₂₀(𝜃)𝑧²2+𝑊₁₁(𝜃)𝑧𝑧+𝑊₀₂𝑧²2+𝑊₃₀𝑧³6,…; (30) and 𝑧 and 𝑧 are local coordinates for the center manifold 𝐶₀ in the direction of 𝑞 and 𝑞^∗, respectively. Since 𝜇=0, we have 𝑧^′(𝑡)=𝑖𝜏₀𝜔₀𝑧(𝑡)+𝑞^∗(𝜃),𝑓𝑊+2Re𝑧(𝑡)𝑞(𝜃)=𝑖𝜏₀𝜔₀𝑧(𝑡)+𝑞^∗(0)𝑓𝑊(𝑧,𝑧,0)+2Re𝑧(𝑡)𝑞(0)≜𝑖𝜏₀𝜔₀𝑧(𝑡)+𝑞^∗(0)𝑓₀(𝑧,𝑧), (31) where 𝑓₀(𝑧,𝑧)=𝑓_𝑧2𝑧²2+𝑓_𝑧²𝑧²2+𝑓_𝑧𝑧𝑧𝑧+⋯. (32) We rewrite this as 𝑧^′(𝑡)=𝑖𝜏₀𝜔₀𝑧+𝑔(𝑧,𝑧), (33) with 𝑔(𝑧,𝑧)=𝑞^∗(0)𝑓₀(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2𝑔(+⋯,𝑧,𝑧)=𝐷𝜏₀1,𝛽^∗𝑎₆𝜙²₂(−1)+𝑎₇𝜙₁(0)𝜙₂(−1)−𝑎₆𝜙₁(0)𝜙₂(−1)−𝑎₇𝜙²₂(),−1 (34) where 𝜙₁(0)=𝑧+𝑧+𝑊⁽¹⁾₂₀(0)𝑧²2+𝑊⁽¹⁾₁₁(0)𝑧𝑧+𝑊⁽¹⁾₀₂(0)𝑧2,𝜙₂(−1)=𝑧𝛼𝑒^{−𝑖𝜔0𝜏0}+𝑧𝛼𝑒^{𝑖𝜔0𝜏0}+𝑊⁽²⁾₂₀()𝑧−1²2+𝑊⁽²⁾₁₁(−1)𝑧𝑧+𝑊⁽²⁾₀₂(−1)𝑧2. (35)

Comparing the coefficients of the above equation with (22), we obtain 𝑔₂₀=2𝐷𝜏₀𝑎₆−𝛽^∗𝑎₇𝛼²𝑒^{−2𝑖𝜔0𝜏0}+𝑎₇−𝛽^∗𝑎₆𝛼𝑒^{−𝑖𝜔0𝜏0},𝑔₁₁=𝐷𝜏₀2𝑎₆−𝛽^∗𝑎₇𝛼𝛼+𝑎₇−𝛽^∗𝑎₆×𝛼𝑒^{𝑖𝜔0𝜏0}+𝛼𝑒^{−𝑖𝜔0𝜏0},𝑔₀₂=2𝐷𝜏₀𝑎₆−𝛽^∗𝑎₇𝛼²𝑒^{2𝑖𝜔0𝜏0}+𝑎₇−𝛽^∗𝑎₆𝛼𝑒^{−𝑖𝜔0𝜏0},𝑔₂₁=2𝐷𝜏₀𝑎₆−𝛽^∗𝑎₇×𝛼𝑒^{𝑖𝜔0𝜏0}𝑊⁽²⁾₂₀(−1)+2𝛼𝑒^{−𝑖𝜔0𝜏0}𝑊⁽²⁾₁₁(+𝑎−1)₇−𝛽^∗𝑎₆×12𝛼𝑒^{𝑖𝜔0𝜏0}𝑊⁽¹⁾₂₀(0)+𝛼𝑒^{−𝑖𝜔0𝜏0}𝑊⁽¹⁾₁₁(0)+12𝑊⁽²⁾₂₀(−1)+𝑊⁽²⁾₁₁(−1),𝑊̇=𝑢̇_𝑡−𝑧𝑞̇−𝑧̇𝑞=𝐴𝑊−2Re𝑞^∗(0)𝑓₀𝑞(𝜃),𝜃∈[−1,0],𝐴𝑊−2Re𝑞^∗(0)𝑓₀𝑞(𝜃)+𝑓₀(,𝜃=0.≜𝐴𝑊+𝐻𝑧,𝑧,𝜃), (36)(37) where 𝐻(𝑧,𝑧,𝜃)=𝐻₂₀(𝜃)𝑧²2+𝐻₁₁(𝜃)𝑧𝑧+𝐻₀₂(𝜃)𝑧²2+⋯. (38) Expanding the above series and comparing the coefficients, we get 𝐴−2𝑖𝜔₀𝜏₀𝐼𝑊₂₀(𝜃)=−𝐻₂₀(𝜃),𝐴𝑊₁₁(𝜃)=−𝐻₁₁(𝜃),𝐴+2𝑖𝜔₀𝜏₀𝐼𝑊₀₂(𝜃)=−𝐻₀₂(𝜃). (39) Comparing the coefficients with (38), we obtain 𝐻₂₀(𝜃)=−𝑔₂₀𝑞(𝜃)−𝑔₀₂𝑞(𝜃),𝐻₁₁(𝜃)=−𝑔₁₁𝑞(𝜃)−𝑔₁₁𝑞(𝜃). (40)

It follows from (39), (40), and the definition of 𝐴 that we have 𝑊̇₂₀(𝜃)=2𝑖𝜏₀𝜔₀𝑊₂₀(𝜃)+𝑔₂₀𝑞(𝜃)+𝑔₂₀𝑞(𝜃),𝑊̇₁₁(𝜃)=𝑔₁₁𝑞(𝜃)+𝑔₁₁𝑞(𝜃). (41)

So, 𝑊₂₀(𝜃)=𝑖𝑔₂₀𝜏₀𝜔₀𝑞(0)𝑒^{𝑖𝜏0𝜔0𝜃}−𝑔₀₂3𝑖𝜏₀𝜔₀𝑞(0)𝑒^{−𝑖𝜏0𝜔0𝜃}+𝐸₁𝑒^{2𝑖𝜏0𝜔0𝜃},𝑊₁₁(𝜃)=−𝑖𝑔₁₁𝜏₀𝜔₀𝑞(0)𝑒^{𝑖𝜏0𝜔0𝜃}+𝑖𝑔₁₁𝜏₀𝜔₀𝑞(0)𝑒^{−𝑖𝜏0𝜔0𝜃}+𝐸₂, (42) where 𝐸₁=22𝑖𝜔₀−𝑎₁−𝑎₂𝑒^{−2𝑖𝜔0𝜏0}−𝑎₃2𝑖𝜔₀−𝑎₄−𝑎₅𝑒^{−2𝑖𝜔0𝜏0−1}×𝑎₆𝛼²𝑒^{−2𝑖𝜔0𝜏0}+𝑎₇𝛼𝑒^{−𝑖𝜔0𝜏0}−𝑎₇𝛼²𝑒^{−2𝑖𝜔0𝜏0}−𝑎₆𝛼𝑒^{−𝑖𝜔0𝜏0},𝐸₂=−𝑎₁−𝑎₂−𝑎₃−𝑎₄−𝑎₅⁻¹×2𝑎₆𝛼𝛼+𝑎₇𝛼𝑒^{𝑖𝜔0𝜏0}+𝛼𝑒^{−𝑖𝜔0𝜏0}−2𝑎₇𝛼𝛼−𝑎₆𝛼𝑒^{𝑖𝜔0𝜏0}+𝛼𝑒^{−𝑖𝜔0𝜏0}. (43) According to the discussion above, we can compute the following parameters: 𝐶₁(0)=𝑖2𝜏₀𝜔₀𝑔₂₀𝑔₁₁−2𝑔₁₁−13𝑔₀₂²+𝑔₂₁2,𝜇₂𝐶=−Re₁(0)Re𝜆^′𝜏₀,𝛽₂=2Re𝐶₁(0,𝑇₂)𝐶=−Im₁(0)+𝜇₂Im𝜆^′𝜏₀𝜔₀, (44) where 𝜇₂ determines the directions of the Hopf bifurcations, 𝛽₂ determines the stability of the bifurcation periodic solutions, and 𝑇₂ determines the period of the bifurcating periodic solutions [9]. By lemma (5), we know that Re{𝜆^′(𝜏₀)}>0; we have the following theorem.

Theorem 3. If Re{𝐶₁(0)}<0(>0), the direction of the Hopf bifurcation of the system (1) at the equilibrium (0,0) when 𝜏=𝜏₀ 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 𝐸^∗=(𝑆^∗,𝐼^∗) when 𝜏=𝜏_𝑗(𝑗=0,1,2,…). A natural question is that if the bifurcating periodic solutions of system (2) exist when is 𝜏 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 [10], we obtain the global continuation of periodic solutions bifurcating from the points (𝐸^∗,𝜏_𝑗)(𝑗=0,1,2,…). First of all, we define ]),()∶()𝑋=𝐶([−𝜏,0,𝑅Σ=𝐶𝑙𝑥,𝜏,𝑙𝑥,𝜏,𝑙∈𝑋×𝑅₊×𝑅₊,()∶𝑥isa𝑙-periodicsolutionofsystem,𝑁=𝑥,𝜏,𝑙𝑥=0or𝑣. (45)Let 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) denote the connected component of 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) in Σ and Proj_𝜏(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) its projection on 𝜏 component. From theorem (5), we know that 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) is nonempty. 𝜔₀ and 𝜏_𝑗 are defined in (10) and (11).

Lemma 4. All periodic solutions of system (2) are uniformly bounded.

Proof . Let (𝑆(𝑡),𝐼(𝑡)) be a nonconstant periodic solution of system (2), and let 𝑆(𝑡₁), 𝑆(𝑡₂) and 𝐼(𝑡₃), 𝐼(𝑡₄) be the maximum and minimum of 𝑆(𝑡) and 𝐼(𝑡), respectively. Using a proof procedure similar to that in [8], we can obtain (𝑡)<𝐵0<𝑆𝜇₁(𝑡)<𝐵,0<𝐼𝜇₂.+𝛾 (46) 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 𝜏.

Proof . For a contradiction, if system (5) has a 𝜏-periodic solution, say (𝑆(𝑡),𝐼(𝑡)), then it satisfies the ODES as follows: 𝑆̇(𝑡)=𝐵−𝜇₁𝑆(𝑡)−𝛽𝑆(𝑡)𝐼(𝑡)(𝑡)(),𝐼̇(𝑡)=(𝑡)𝐼(𝑡)1+𝛼𝐼=𝑃𝑆,𝐼𝛽𝑆(𝑡)−𝜇1+𝛼𝐼²+𝛾𝐼(𝑡)=𝑄(𝑆,𝐼). (47) We can get 𝜕𝑃+𝜕𝑆𝜕𝑄𝜕𝐼=−𝜇₁−𝛽𝐼(𝑡)(𝑡)−11+𝛼𝐼(1+𝛼𝐼(𝑡))²−𝜇₂+𝛾<0. (48) By Bendixson’s criterion, we know that system (2) has no nonconstant periodic solutions, which prove the lemma.

Theorem 6. Suppose that the condition 𝐻₁ and 𝑃₁ is satisfied. Then, for each 𝜏>𝜏_𝑗,𝑗=0,1,2,…, system (2) has at least 𝑗−1 periodic solutions.

Proof. The characteristic matrix of system (2) at the equilibrium 𝑧=[𝑧⁽¹⁾,𝑧⁽²⁾]∈𝑅² is in the following form: Δ(𝑧,𝜏,𝑙)(𝜆)=𝜆𝐼−𝐷_𝜙𝐹𝑧,𝜏,𝑙𝑒^𝜆𝐼𝑑; (49) that is, Δ()=𝑧,𝜏,𝑙𝜆+𝜇₁+𝛽𝑧⁽²⁾𝑒^−𝜆𝜏1+𝛼𝑧⁽²⁾𝑒^−𝜆𝜏𝛽𝑧⁽¹⁾1+𝛼𝑧⁽²⁾𝑒^−𝜆𝜏2−𝛽𝑧⁽²⁾𝑒^−𝜆𝜏1+𝛼𝑧⁽²⁾𝑒^−𝜆𝜏𝛽𝜆+𝑧⁽¹⁾1+𝛼𝑧⁽²⁾𝑒^−𝜆𝜏2. (50) Using a proof procedure similar to that in [9], it is easy to obtain that (𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀), 𝑗=0,1,2,…, is an isolated center.
From the proof procedure of Lemmas 4 and 5, it is easy to know that there exist 𝜀>0, 𝛿>0, smooth curve 𝜆∶(𝜏_𝑗−𝛿,𝜏_𝑗+𝛿)→𝐶 such that Δ(𝜆(𝜏))=Δ_{(𝑣,𝜏,𝑇)}(𝜆(𝜏))=0,𝜆(𝜏)−𝑖𝜔₀<𝜀, (51) for all 𝜏∈[𝜏_𝑗−𝛿,𝜏_𝑗+𝛿], and 𝜆𝜏_𝑗=𝑖𝜔₀,𝑑Re(𝜆(𝜏))𝑑𝜏_{𝜏=𝜏𝑗}>0. (52)
Define 𝑙_𝑗=2𝜋/𝜔₀, and let Ω_𝜀={(0,𝑙)∶0<𝑢<𝜀,|𝑙−𝑙_𝑗|<𝜀}. Obviously, if |𝜏−𝜏_𝑗|≤𝛿 and (𝑢,𝑙)∈𝜕Ω_𝜀 such that Δ_(𝐸^∗_,𝜏,𝑙)(𝑢+2𝜋𝑖/𝑙)=0, if and only if 𝜏=𝜏_𝑗,𝑢=0,𝑙=𝑙_𝑗, set 𝐻^±𝐸^∗,𝜏_𝑗,2𝜋𝜔₀(𝑢,𝑙)=Δ_(𝐸^∗_{,𝜏𝑗±𝛿,𝑙)}𝑢+2𝜋𝑖𝑙. (53) We obtain the crossing number as follows: 𝛾₁𝐸^∗,𝜏_𝑗,2𝜋𝜔₀=deg_𝐵𝐻⁻𝐸^∗,𝜏_𝑗,2𝜋𝜔₀,Ω_𝜀−deg_𝐵𝐻⁺𝐸^∗,𝜏_𝑗,2𝜋𝜔₀,Ω_𝜀=−1. (54) We conclude that _{(𝐸∗)𝐸,𝜏,𝑙∈𝐶∗,𝜏𝑗,2𝜋/𝜔0}𝛾₁𝐸^∗,𝜏,𝑙<0. (55) Since the first crossing number of each center is always −1, by [10, Theorem 3.3], we conclude that 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) is unbounded. By the definition of 𝜏_𝑗 given in (10), we know that, for 𝑗≥1, (𝜏_𝑗/(𝑗+1))<2𝜋/𝜔₀<𝜏_𝑗 automatically hold.
Again, the population densities of susceptible and infective are ultimately uniformly bounded, implying that the projection of 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) onto the 𝜏-space is bounded. Meanwhile, system (2) with 𝜏=0 has no nonconstant periodic solutions; if there exits 𝜏^∗>0 such that the projection of 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) onto the 𝜏-space ins (0,𝜏₀) with 𝜏^∗>𝜏_𝑗, then, the projection of 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) onto the 𝜏-space is bounded. Since (2𝜋/𝜔₀)<𝜏_𝑗 and from Lemma 5, we can obtain 0<𝑙<𝜏^∗ for (𝐸,𝜏,𝑇)∈𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) with 𝑙<𝜏^∗; that is to say, 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) onto 𝑙-space is also bounded. Because 𝐶(𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) is unbounded, Proj_𝜏(𝐸,𝜏_𝑗,2𝜋/𝜔₀) must be unbounded. Consequently, Proj_𝜏(𝐸,𝜏_𝑗,2𝜋/𝜔₀) include [𝜏_𝑗,∞) for 𝑗≥1. That is to say, for each 𝜏>𝜏_𝑗(𝑗≥1), system (2) at least has 𝑗−1 nonconstant period solutions. The proof is complete.

5. Conclusion

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.