AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/242410 242410 Research Article Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay http://orcid.org/0000-0003-3704-9919 Zhai Yanhui http://orcid.org/0000-0003-0873-9371 Xiong Ying Ma Xiaona Bai Haiyun Yang Zhichun School of Science, Tianjin Polytechnic University, Tianjin 300387 China tjpu.edu.cn 2014 1472014 2014 01 01 2014 14 06 2014 14 7 2014 2014 Copyright © 2014 Yanhui Zhai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 . 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.

H 7 N 9 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 . 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 .

In 1979, Cooke  used mass action incidence βS(t)I(t-τ). In 2009, Xu and Ma  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 . Then, the avian influenza virus propagation model based on SIR model has the following form: (1)S˙(t)=B-μ1S(t)-βS(t)I(t-τ)1+αI(t-τ),I˙(t)=βS(t)I(t-τ)1+αI(t-τ)-(μ2+γ)I(t),R˙(t)=γI(t)-μ3R(t).

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 (2)S˙(t)=B-μ1S(t)-βS(t)I(t-τ)1+αI(t-τ),I˙(t)=βS(t)I(t-τ)1+αI(t-τ)-(μ2+γ)I(t),(3)R˙(t)=γI(t)-μ3R(t), with the following initial condition: (4)S(0)R+,I(θ)=ϕ(θ)for  θ[-τ,0],whereϕC([-τ,0],R+), 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 R0>1. 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 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)) . The characteristic equation of system (2) at the endemic equilibrium E* is (5)λ2+p1λ+p0+(q1λ+q0)e-λτ=0, 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 (P1)R0>1 hold, when τ=0, the endemic equilibrium E* of system (2) is locally stable .

If iω (ω>0) is a solution of system (2), separating real and imaginary parts, we obtain the following: (6)p1ω=q0sinωτ-q1ωcosωτ,ω2-p0=q0cosωτ+q1ωsinωτ. Then, we get (7)cosωτ=(q0-p1q1)ω2-p0q0q02+q12ω2,sinωτ=p1q0ω+(ω2-p0)q1ωq02+q12ω2. It follows that (8)ω4+(p12-2p0-q12)ω2+p02-q02=0. Letting z=ω2, we get (9)z2+(p12-2p0-q12)z+p02-q02=0. It is easy to show that (10)p12-2p0-q12=(μ1+βI*1+αI*)21111111111111+(μ2+γ)2-(μ2+γ)2(1+αI*)2>0,p02-q02=(μ2+γ)[(μ2+γ)(μ1+βI*1+αI*)+βμ1S*(1+αI*)2]×(μ1-μ11+αI*+βI*1+αI*). The case of (H1)βμ1α has been discussed in . We obtain global asymptotic stability of the endemic equilibrium when R0>1. If (H2)β<μ1α hold, that is, (β-μ1α)I*/(1+αI*)<0, we have p02-q02<0. Following the theorem given by Ruan , there exists critical value (11)τk(j)=1ωkarccos(ωk2-p0)q0-p1q1ωk2q02+q12ωk2+2jπωk, with (12)ωj=[2p0+p12-q12+(2p0+p12-q12)2-4(p02-q02)2]1/2, 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 (13)[2λ+p1+(q1-τ(q1λ+q0)e-λτ)]dλdτ=λ(q1λ+q0)e-λτ. For the sake of simplicity denoting ω0 and τ0 by ω,τ, respectively, (14)dλdτ=(q1λ+q0)λe-λτ2λ+p1+q1e-λτ-((q1λ+q0)τe-λτ) in the following: (15)Re(dλdτ)-1=2λ+p1+q1e-λτ(q1λ+q0)λe-λτ=((p1cosωτ-2ωsinωτ+q1)1111111+i(2ωcosωτ+p1sinωτ))×(-q1ω2+iq0ω)-1=(-p1q1ω2cosωτ+2q1ω3sinωτ1111111-q12ω2+2q0ω2cosωτ+p1q0ωsinωτ)1111111×(q0ω2+ω4)-1=((-p1q1ω2+2q0ω2)(q0-p1q1)ω2-p0q01111+(2q1ω3+p1q0ω)1111×[p1q0ω+(ω2-p0)q1ω]-q02q12ω2-q14ω2)×((q02+q12ω2)(q0ω2+ω4))-1=(2q12ω6+[2q02+(p12-2p0-q12)q12]ω4+(p12+2p0-q12)q02ω2)×((q02+q12ω2)(q0ω2+ω4))-1. From (10), we know p12-2p0-q12>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=τj, 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 (16)u˙1(t)=a1u1(t)-a2u2(t-τ)+a6u2(t)2(t-τ)+a7u1(t)u2(t-τ),u˙2(t)=a3u1(t)+a4u2(t)+a5u2(t-τ)-a7u1(t)u2(t-τ)-a6u22(t-τ), 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 ut=u(t+θ)C1 for θ[-1,0]. System (2) is transformed into FDE as (17)u˙(t)=Lμ+F(ut,μ), with (18)Lμ(ϕ)=(τ0+μ)[B1ϕ(0)+B2ϕ(-1)],F(ϕ,μ)=(τ0+μ)(a6ϕ22(-1)+a7ϕ1(0)ϕ2(-1)-a6ϕ1(0)ϕ2(-1)-a7ϕ22(-1)), where (19)B1=(a10a3a4),B2=(0a20a5). By Riesz representation theorem, there exists a function η(θ,μ) of bounded variation, for θ[-1,0], such that (20)Lμϕ=-10dη(θ,μ)ϕ(θ)forϕC. In fact, we can choose (21)η(θ,μ)=(τ0+μ)[B1δ(θ)+B2δ(θ+1)], where δ(θ) is a delta function.

For ϕC[-1,0], the operators A and R are defined as follows: (22)A(μ)ϕ(θ)={dϕ(θ)dθ,θ[-1,0),-10d(η(t,μ)ϕ(t)),θ=0,(23)R(μ)ϕ(θ)={0,θ[-1,0),f(μ,θ),θ=0. The adjoint operator A*(0) corresponding to A(0) is defined as follows: (24)A*ψ(s)={-dψ(s)ds,s(0,1],-10d(ηT(t,0)ψ(-t)),s=0 and an adjoint bilinear is as follows: (25)ψ,ϕ=ψ¯(0)ϕ(0)--100θψ¯(ξ-θ)dη(θ)ϕ(ξ)dξ, 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, (26)q(θ)=(1,α)Teiω0τ0θ=(1,ω0i-a1a2e-iω0τ0)Teiω0τ0θ,q*(s)=D(1,β*)e-iω0τ0s=(1,-ω0i-a1a3)Te-iω0τ0s, where (27)D=11+αβ¯*-τ0α(a2+β¯*a5)e-iω0τ0.

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)Teiω0τ0 (here, α is a parameter). In what follows, notice that q(0)=(1,α)T and Aq(0)=-10d(η(t,μ)ϕ(t))=iω0τ0q(0); we have α=(ω0i-a1)/a2e-iω0τ0. Using a proof procedure similar to that in , 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 . The proof is completed.

Then, we construct the coordinates of the center manifold C0 at μ=0. Let (28)z(t)=q*,ut,W(t,θ)=ut(θ)-2Re{z(t)q(θ)}. On the center manifold C0, we have (29)W(t,θ)=W(z(t),z(t)¯,θ), where (30)W(z,z¯,θ)=W20(θ)z22+W11(θ)zz¯+W02z¯22+W30z36,; 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 (31)z(t)=iτ0ω0z(t)+q*(θ),f(W+2Re{z(t)q(θ)})=iτ0ω0z(t)+q*(0)¯f(W(z,z¯,0)+2Re{z(t)q(0)})iτ0ω0z(t)+q*(0)¯f0(z,z¯), where (32)f0(z,z¯)=fz2z22+fz¯2z¯22+fzz¯zz¯+. We rewrite this as (33)z(t)=iτ0ω0z+g(z,z¯), with (34)g(z,z¯)=q*¯(0)f0(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+,g(z,z¯)=Dτ0(1,β*¯)(a6ϕ22(-1)+a7ϕ1(0)ϕ2(-1)-a6ϕ1(0)ϕ2(-1)-a7ϕ22(-1)), where (35)ϕ1(0)=z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯2,ϕ2(-1)=zαe-iω0τ0+z-α-eiω0τ0+W20(2)(-1)z22+W11(2)(-1)zz¯+W02(2)(-1)z¯2.

Comparing the coefficients of the above equation with (22), we obtain (36)g20=2Dτ0[(a6-β*¯a7)α2e-2iω0τ0+(a7-β*¯a6)αe-iω0τ0],g11=Dτ0[2(a6-β*¯a7)αα¯+(a7-β*¯a6)×(α¯eiω0τ0+αe-iω0τ0)(a6-β*¯a7)],g02=2Dτ0[(a6-β*¯a7)α2¯e2iω0τ0200111000+(a7-β*¯a6)α¯e-iω0τ0],g21=2Dτ0[(a6-β*¯a7)11111111111×[W11(2)α¯eiω0τ0W20(2)(-1)+2αe-iω0τ0W11(2)(-1)]+(a7-β*¯a6)×[12α¯eiω0τ0W20(1)(0)+αe-iω0τ0W11(1)(0)111111111+12W20(2)(-1)+W11(2)(-1)12],(37)W˙=u˙t-z˙q-z-˙q-={AW-2Req*¯(0)f0q(θ),θ[-1,0],AW-2Req*¯(0)f0q(θ)+f0,θ=0.AW+H(z,z¯,θ), where (38)H(z,z¯,θ)=H20(θ)z22+H11(θ)zz¯+H02(θ)z¯22+. Expanding the above series and comparing the coefficients, we get (39)(A-2iω0τ0I)W20(θ)=-H20(θ),AW11(θ)=-H11(θ),(A+2iω0τ0I)W02(θ)=-H02(θ). Comparing the coefficients with (38), we obtain (40)H20(θ)=-g20q(θ)-g¯02q¯(θ),H11(θ)=-g11q(θ)-g¯11q¯(θ).

It follows from (39), (40), and the definition of A that we have (41)W˙20(θ)=2iτ0ω0W20(θ)+g20q(θ)+g¯20q¯(θ),W˙11(θ)=g11q(θ)+g¯11q¯(θ).

So, (42)W20(θ)=ig20τ0ω0q(0)eiτ0ω0θ-g¯023iτ0ω0q¯(0)e-iτ0ω0θ+E1e2iτ0ω0θ,W11(θ)=-ig11τ0ω0q(0)eiτ0ω0θ+ig¯11τ0ω0q¯(0)e-iτ0ω0θ+E2, where (43)E1=2(2iω0-a1        -a2e-2iω0τ0-a32iω0-a4-a5e-2iω0τ0)-1×(a6α2e-2iω0τ0+a7αe-iω0τ0-a7α2e-2iω0τ0-a6αe-iω0τ0),E2=(-a1        -a2-a3        -a4-a5)-1×(2a6αα¯+a7(α¯eiω0τ0+αe-iω0τ0)-2a7αα¯-a6(α¯eiω0τ0+αe-iω0τ0)). According to the discussion above, we can compute the following parameters: (44)C1(0)=i2τ0ω0(g20g11-2|g11|-13|g02|2)+g212,μ2=-Re{C1(0)}Re{λ(τ0)},β2=2Re{C1(0)},T2=-Im{C1(0)}+μ2(Im{λ(τ0)})ω0, 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 . 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=τj(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 , we obtain the global continuation of periodic solutions bifurcating from the points (E*,τj)(j=0,1,2,). First of all, we define (45)X=C([-τ,0],R),Σ=Cl(x,τ,l):(x,τ,l)X×R+×R+,111111xisal-periodicsolutionofsystem,N=(x^,τ,l):x^=0orv.Let C(E*,τj,2π/ω0) denote the connected component of C(E*,τj,2π/ω0) in Σ and Projτ(E*,τj,2π/ω0) its projection on τ component. From theorem (5), we know that C(E*,τj,2π/ω0) is nonempty. ω0 and τj 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 , we can obtain (46)0<S(t)<Bμ1,0<I(t)<Bμ2+γ. 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: (47)S˙(t)=B-μ1S(t)-βS(t)I(t)1+αI(t)=P(S,I),I˙(t)=βS(t)I(t)1+αI(t)-(μ2+γ)I(t)=Q(S,I). We can get (48)PS+QI=-μ1-βI(t)1+αI(t)-1(1+αI(t))2-(μ2+γ)<0. 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>τj,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: (49)Δ(z¯,τ,l)(λ)=λI-DϕF(z¯,τ¯,l¯)(eλId); that is, (50)Δ(z¯,τ,l)=(λ+μ1+βz¯(2)e-λτ1+αz¯(2)e-λτβz¯(1)(1+αz¯(2)e-λτ)2-βz¯(2)e-λτ1+αz¯(2)e-λτλ+βz¯(1)(1+αz¯(2)e-λτ)2). Using a proof procedure similar to that in , it is easy to obtain that (E*,τj,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 ɛ>0, δ>0, smooth curve λ:(τj-δ,τj+δ)C such that (51)Δ(λ(τ))=Δ(v,τ,T)(λ(τ))=0,|λ(τ)-iω0|<ɛ, for all τ[τj-δ,τj+δ], and (52)λ(τj)=iω0,dRe(λ(τ))dτ|τ=τj>0.

Define lj=2π/ω0, and let Ωɛ={(0,l):0<u<ɛ,|l-lj|<ɛ}. Obviously, if |τ-τj|δ and (u,l)Ωɛ such that Δ(E*,τ,l)(u+2πi/l)=0, if and only if τ=τj,u=0,l=lj, set (53)H±(E*,τj,2πω0)(u,l)=Δ(E*,τj±δ,l)(u+2πil). We obtain the crossing number as follows: (54)γ1(E*,τj,2πω0)=degB(H-(E*,τj,2πω0),Ωɛ)-degB(H+(E*,τj,2πω0),Ωɛ)=-1. We conclude that (55)(E*,τ,l)C(E*,τj,2π/ω0)γ1(E*,τ,l)<0. Since the first crossing number of each center is always -1, by [10, Theorem 3.3], we conclude that C(E*,τj,2π/ω0) is unbounded. By the definition of τj given in (10), we know that, for j1, (τj/(j+1))<2π/ω0<τj automatically hold.

Again, the population densities of susceptible and infective are ultimately uniformly bounded, implying that the projection of C(E*,τj,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*,τj,2π/ω0) onto the τ-space ins (0,τ0) with τ*>τj, then, the projection of C(E*,τj,2π/ω0) onto the τ-space is bounded. Since (2π/ω0)<τj and from Lemma 5, we can obtain 0<l<τ* for (E,τ,T)C(E*,τj,2π/ω0) with l<τ*; that is to say, C(E*,τj,2π/ω0) onto l-space is also bounded. Because C(E*,τj,2π/ω0) is unbounded, Projτ(E,τj,2π/ω0) must be unbounded. Consequently, Projτ(E,τj,2π/ω0) include [τj,) for j1. That is to say, for each τ>τj(j1), 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.

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