# Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources

Authors: Caijuan Yan; Jianwen Jia
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratioℛ0<1, we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. If ℛ0>1, we obtain sufficient conditions under which the endemic equilibrium E* of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.


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## Body

## 1. Introduction

From an epidemiological viewpoint, it is important to investigate the global dynamics of the disease transmission. In the literature, many authors have researched various epidemic models [1, 2], in which the stability analyses have been carried out extensively. In the recent years, based on SIR epidemic model, in order to investigate the spread of an infectious disease transmitted by a vector, Wang et al. [3] have considered the asymptotic behavior of the following delayed SIR epidemic model:
(1)dS(t)dt=rS(1-Sk)-βSI(t-τ),dI(t)dt=βSI(t-τ)-(μ1+γ)I,dR(t)dt=γI-μ2R.Since nonlinearity in the incidence rates has been observed in disease transmission dynamics, it has been suggested that the standard bilinear incidence rate will be modified into a nonlinear incidence rate by many authors [4–6]. In [7], incidence rate βSI(t-τ) in (1) was replaced by a nonlinear incidence rate of the form βSG(I(t-τ)) with the following system:
(2)dS(t)dt=rS(1-Sk)-βSG(I(t-τ)),dI(t)dt=βSG(I(t-τ))-(μ1+γ)I,dR(t)dt=γI-μ2R.In order to control the spread of epidemic, we consider the new variableZ:
(3)Z(t)=∫-∞t‍S1Texp⁡(-1T(t-τ))dτ,
called information variable which summarizes information about the current state of the disease, that is, depending on current values of state variables, and also summarizes information about past values of state variables. Many authors have used this variable in their models (see, e.g., [8–10]).In this paper, we consider the information variableZ(t), nonlinear incidence rate of the form βSG(I(t-τ)), and limited medical resources h(I)=bI/(ω+I). The model can be described by the following system of equations:
(4)dS(t)dt=rS(1-Sk)-βSG(I(t-τ)),dI(t)dt=βZG(I(t-τ))-(μ1+γ+ɛ)I-bIω+I,Z(t)=∫-∞t‍S1Texp⁡(-1T(t-τ))dτ,dR(t)dt=γI-μ2R+bIω+I,
where S(t), I(t), R(t)>0 and S(t), I(t), R(t) denote the numbers of susceptible, infective, and recovered individuals at time t, respectively. r is the intrinsic growth rate of susceptibles, k is the carrying capacity of susceptibles, α is the saturation factor that measures the inhibitory effect, β is the transmission or contact rate, μ1, μ2 are the natural death rate of the infective and recovered individuals, γ is the natural recovery rate, ɛ is the disease-related mortality, b≥0 is the maximal medical resources supplied per unit time, and ω>0 is half-saturation constant. r, b, μ1, μ2, γ, α, β, k, ω are all positive.We further assume that the functionG is continuous on [0,+∞) and continuously differentiable on (0,+∞) satisfying the following hypotheses:(1)
G(I) is strictly monotone increasing on [0,+∞) with G(0)=0;(2)
I/G(I) is monotone increasing on (0,+∞) with lim⁡I→0+(I/G(I))=1.The organization of this paper is as follows. In Section2, we explore the existence of disease-free equilibria point and the unique existence of the endemic equilibrium point. In Section 3, we analyze the stability of the disease-free equilibria. In Section 4, we obtain sufficient conditions under which the endemic equilibrium E* of system is locally asymptotically stable. In Section 5, we also have discussed the stability and direction of Hopf bifurcations. A numerical analysis and a simple discussion are given to conclude this paper in Section 6.


## 2. The Existence of Equilibria

The nonlinear integrodifferential system (4) can be transformed into the following set of nonlinear ordinary differential questions:
(5)dS(t)dt=rS(1-Sk)-βSG(I(t-τ)),dI(t)dt=βZG(I(t-τ))-(μ1+γ+ɛ)I-bIω+I,dZ(t)dt=1T(S-Z),dR(t)dt=γI-μ2R+bIω+I.Since the dynamical behavior of the last equation of the system (5), that is, the dynamics of R, depends only the dynamics of I, we do not consider that equation in our discussion. Here we will study the following nonlinear ordination differential equations:
(6)dS(t)dt=rS(1-Sk)-βSG(I(t-τ)),dI(t)dt=βZG(I(t-τ))-(μ1+γ+ɛ)I-bIω+I,dZ(t)dt=1T(S-Z).
For simplicity, we nondimensionalize system (6) by defining
(7)S~(t~)=S(t)k,I~(t~)=I(t)k,Z~(t~)=Z(t)k,t~=βkt,T~=βkT,ω~=ωk,r~=rβk,G~(I~(t~))=G(I(t))k,μ~1=μ1βk,μ~2=μ2βk,γ~=γβk,ɛ~=ɛβk,b~=bβk2.We note thatG~ also satisfies the hypotheses (1) and (2). Dropping the ~ for convenience of readers, system (6) can be written in the following form:
(8)dS(t)dt=rS(1-S)-SG(I(t-τ)),dI(t)dt=ZG(I(t-τ))-(μ1+γ+ɛ)I-bIω+I,dZ(t)dt=1T(S-Z).
The basic reproduction is ℛ0=1/(μ1+γ+ɛ+b/ω).Theorem 1.
(1) The system (8) has a trivial equilibrium E0=(0,0,0) and the disease-free equilibrium E1=(1,0,1).
(2) Ifℛ0>1, the system (8) has one endemic equilibrium E*=(S*,I*,Z*) except the disease-free equilibria E0 and E1.Proof.
(1) LetI=0; we have S=Z=0, or S=Z=1; it is not easy to find that the system has a trivial equilibrium and the disease-free equilibria E0=(0,0,0) and E1=(1,0,1).
(2) Ifℛ0>1, from the third question of (8), we have Z*=S*; from the second question of (8), we have
(9)S*=Z*=(μ1+γ+ɛ+b/(ω+I*))I*G(I*).
Then substituting them into the first question of (8) yields
(10)r[1-(μ1+γ+ɛ+b/(ω+I*))I*G(I*)]-G(I*)=0.
Let F(I)=r[1-(μ1+γ+ɛ+b/(ω+I*))I/G(I)]-G(I). By hypothesis (2), we obtain
(11)lim⁡I→+0F(I)=r[1-(μ1+γ+ɛ+bω+I*)]>r[1-(μ1+γ+ɛ+bω)]=r(1-1R0)>0.
SinceF(I) is strictly monotone decreasing function on (0,+∞), it suffices to show that F(I)<0 holds for I sufficiently large. From (1), G(I) is either unbounded above or bounded above on [0,+∞).
First, we suppose thatG(I) is unbounded above. Then there exists an I1>0 such that G(I1)=r, from which we have F(I)<0 for all I≥I1. Second we suppose that G(I) is bounded above. Then, from (2), I/G(I) is unbounded above on [0,+∞); that is, there exists an I2>0 such that μ1+γ+ɛ+b/(ω+I2)=G(I2)/I2. This yields F(I)<0 for all I>I2. Therefore, for the both cases, there exists a unique endemic I*>0 such that F(I*)=0. By the second and third equations of (8), there exists a unique endemic equilibrium E* of system (8) if ℛ0>1.
Second, we assumeℛ0≤1; then it is obvious that system (8) has no equilibria. Hence the proof is complete.


## 3. The Stability Analysis of Disease-Free Equilibrium Point

In this section, we will examine the local stability of the equilibria by analyzing the eigenvalues of the Jacobian matrices of (8) at the equilibria and using Routh-Hurwitz criterion.LetE¯=(S¯,I¯,R¯) be the arbitrarily equilibrium point of system (8); then the Jacobian matrix of (8) at E¯ is(12)J(S¯,I¯,Z¯)=(r(1-2S¯)-G(I¯)-S¯G′(I¯)e-λτ00Z¯G′(I¯)e-λτ-(μ1+γ+ɛ+bω(ω+I¯)2)G(I¯)1T0-1T).Then the characteristic equation of the system (8) at equilibrium E¯ is
(13)|λE-J(E¯)|=(λ+1T)[λ+G(I¯)-r(1-2S¯)]×[λ+μ1+γ+ɛ+bω(ω+I¯)2]-[(λ+1T)(λ+G(I¯)-r(1-2S¯))-1TG(I¯)]×Z¯G′(I¯)e-λτ=0.Theorem 2.
The trivial equilibriumE0 of system (8) is always unstable.Proof.
The characteristic equation (13) at E0=(0,0,0) becomes as follows:
(14)(λ+1T)(λ-r)(λ+μ1+γ+ɛ+bω)=0.
Since (14) has a positive root λ=r>0, E0 is unstable.Theorem 3.
The disease-free equilibriumE1 of system (8) is locally asymptotically stable if ℛ0<1 and it is unstable if ℛ0>1.Proof.
ForE1=(1,0,1), the characteristic equation (13) at E1 becomes as follows:
(15)(λ+1T)(λ+r)[λ+(μ1+γ+ɛ+bω)-e-λτ]=0.
It is clear that both λ=-1/T and λ=-r are all the negative root of (15). Then the other root of (15) is determined as the following equation:
(16)λ+(μ1+γ+ɛ+bω)-e-λτ=0.
For the case ℛ0<1, we suppose on the contrary that E1 is not locally asymptotically stable; that is, Reλ~>0. Then, there exists a root λ=λ~, such that Reλ~≥0. However, from (16), we obtain
(17)Reλ~=(μ1+γ+ɛ+bω)(ℛ0e-Reλ~τcos⁡(Im⁡λ~τ)-1)≤(μ1+γ+ɛ+bω)(ℛ0-1)<0,
which is a contradiction. Hence, ifℛ0<1, the disease-free equilibrium E1 of system (8) is locally asymptotically stable.
Now, we put(18)P(λ)=λ+(μ1+γ+ɛ+bω)-e-λτ=0.
For the caseℛ0>1, we have P(0)=μ1+γ+ɛ+b/ω-1<0 and lim⁡λ→+∞P(λ)=+∞; then P(λ)=0 has at least one positive root. Hence, E1 is unstable if and only if ℛ0>1. The proof is complete.


## 4. The Stability Analysis of the Endemic Equilibrium Point

Theorem 4.
Ifτ=0, ℛ0>1, and r-G′(I*)>rTG(I*), then the positive equilibrium E* of system (8) is locally asymptotically stable.Proof.
The characteristic equation of (13) at E* becomes as follows:
(19)(λ+1T)(λ+rS*)[λ+μ1+γ+ɛ+bω(ω+I*)2]-[(λ+1T)(λ+rS*)-1TG(I*)]Z*G′(I*)e-λτ=0.
The above equation can be rewritten as
(20)P(λ)+Q(λ)e-λτ=0,
where P(λ)=λ3+b1λ2+b2λ+b3, Q(λ)=b4λ2+b5λ+b6, and
(21)b1=1T+rS*+μ1+γ+ɛ+bω(ω+I*)2,b2=1TrS*+1T(μ1+γ+ɛ+bω(ω+I*)2)+rS*(μ1+γ+ɛ+bω(ω+I*)2),b3=1TrS*(μ1+γ+ɛ+bω(ω+I*)2),b4=-S*G′(I*),b5=-S*G′(I*)[1T+rS*],b6=-1TS*G′(I*)(rS*-G(I*)).
LetC=μ1+γ+ɛ+bω/(ω+I*)2-S*G′(I*).
Then ifτ=0, (20) becomes P(λ)+Q(λ)=0; that is,
(22)λ3+a1λ2+a2λ+a3=0,
where
(23)a1=b1+b4=1T+rS*+C,a2=b2+b5=1T(rS*+C)+rS*C,a3=b3+b6=1T[rS*C+S*G(I*)G′(I*)].
Letr-G′(I*)>rTG(I*); we have a1>0, a3>0, and
(24)a1a2-a3=(1T+rS*+C)[1T(rS*+C)+rS*C]-1T[rS*C+S*G(I*)G′(I*)]=1T(rS*+C)(1T+rS*+C)+(rS*+C)rS*C-1TS*G(I*)G′(I*)>1T(rS*+C)(1T+rS*+C)+(rS*+C)rS*C-1TrS*G(I*)=1T2(rS*+C)+1T(rS*+C)(rS*+C+rTS*C)-1TrS*G(I*)=1T2(μ1+γ+ɛ+bω(ω+I*)2)+1T(rS*+C)(rS*+C+rTS*C)+1T2S*(r-G′(I*))-1TrS*G(I*)>0.
By using the Routh-Hurwitz theorem,λ has negative real part for τ=0. So the positive equilibrium E* is locally asymptotically stable.In the following, we investigate the existence of purely imaginary rootsλ=iω(ω>0) to (19). Equation (19) takes the form of a third-degree exponential polynomial in λ, with all the coefficients of P and Q depending on τ. Beretta and Kuang [11] established a geometrical criterion which gives the existence of purely imaginary root of a characteristic equation with delay dependent coefficients.Now we letλ=iω(ω>0) be a root of (20) from which we have that
(25)P(λ)+Q(λ)e-λτ=-iω3-b1ω2+b2ωi+b3+(-b4ω2+b5ωi+b6)(cos⁡(ωτ)-isin(ωτ))=-b1ω2+b3-[b4ω2cos⁡(ωτ)-b5ωsin(ωτ)-b6cos⁡(ωτ)]+i[-ω3+b2ω+b4ω2sin(ωτ)+b5ωcos⁡(ωτ)-b6sin(ωτ)-ω3+b2ω+b4ω2sin(ωτ)]=0.Hence, we have that(26)-b1ω2+b3=b4ω2cos⁡(ωτ)-b5ωsin(ωτ)-b6cos⁡(ωτ)=(b4ω2-b6)cos⁡(ωτ)-b5ωsin(ωτ),ω3-b2ω=b4ω2sin(ωτ)+b5ωcos⁡(ωτ)-b6sin(ωτ)=(b4ω2-b6)sinωτ+b5ωcos⁡(ωτ).From (26), it follows that(27a)cos⁡(ωτ)=b5ω(ω3-b2ω)+(b3-b1ω2)(b4ω2-b6)(b4ω2-b6)2+(b5ω)2,(27b)sin(ωτ)=(ω3-b2ω)(b4ω2-b6)-b5ω(b3-b1ω2)(b4ω2-b6)2+(b5ω)2.By the definitions of P(λ), Q(λ) is as in (20), and applying the property (1), (27a) and (27b) can be written as
(28)sin(ωτ)=Im⁡P(iω)Q(iω),cos⁡(ωτ)=-ReP(iω)Q(iω),which yields|P(iω)|2=|Q(iω)|2.Assume thatD∈R0+ is the set where ωτ is a positive root of
(29)F(ω)=|P(iω)|2-|Q(iω)|2.
From
(30)|P(iω)|2=|-iω3-b1ω2+b2ωi+b3|2=|(b3-b1ω2)+(b2ω-ω3)|2=ω6+(b12-2b2)ω4+(b22-2b1b3)ω2+b32,|Q(iω)|2=|(-b4ω2+b5ωi+b6)(cos⁡(ωτ)-sin(ωτ))|2=(b6-b4ω2)2+(b5ω)2=b42ω4+(b52-2b4b6)ω2+b62,we haveF(ω)=ω6+a1ω4+a2ω2+a3, where a1=b12-2b2-b42, a2=b22-2b1b3-b52+2b4b6, a3=b32-b62, and, for τ∈D, ωτ is not defined. Then, for all τ in D, ωτ satisfied F(ω)=0.Letω2=h; then we have that
(31)F(h)=h3+a1h2+a2h+a3=0.Assume thatF(ω) has only one positive real root; we denote by h+ this positive real root. Thus, (29) has only one positive real root ω=h+. And the critical values of τ and ω(τ) are impossible to solve explicitly, so we will use the procedure described in Beretta and Kuang [11] and Song et al. [12]. According to this procedure, we define θ(τ)∈[0,2π) such that sinθ(τ) and cos⁡θ(τ) are given by the right-hand sides of (27a) and (27b), respectively, with θ(τ) given by (19).And the relation between the argumentθ and ω(τ) in (28) for τ>0 must be
(32)ω(τ)=θ+2nπ,n=0,1,2,….
Hence we can define the maps τn:D→R+0 given by
(33)τn=θ(τ)+2nπω(τ),τn>0,n=1,2,…,
where a positive root ω(τ) of (31) exists in D. Let us introduce the functions
(34)Sn(τ):D⟶R,Sn(τ)=τ-θ(τ)+2nπω(τ),n=0,1,2,…,
which are continuous and differentiable in τ. Thus, we give the following theorem which is due to Beretta and Kuang [11].Theorem 5.
Assume thatω(τ) is a positive root of (19) defined for τ∈D, D⊆R+0, and, at some τ*∈D, Sn(τ*)=0 for some n∈N0. Then a pair of simple conjugate pure imaginary roots λ=±iω exists at τ=τ* which crosses the imaginary axis from left to right if
(35)δ(τ*)=sign⁡{Fω′(ωτ*,τ*)}sign⁡{dSn(τ)dτ|τ=τ*}.Applying Theorems2 and 3 and the Hopf bifurcation theorem for functional differential equation [13], we can conjecture the existence of a Hopf bifurcation as stated in Theorem 6.Theorem 6 (a conjecture).
For system (2), there exists τ*∈D, such that the equilibrium E* is asymptotically stable for 0≤τ<τ*, and it becomes unstable for τ staying in some tight neighborhood of τ*, with a Hopf bifurcation occurring when τ=τ*.


## 5. Stability and Direction of Hopf Bifurcations

In this section, we will study the direction of the Hopf bifurcation and stability of bifurcating periodic solutions by using the normal theory and center manifold theorem due to Hassard et al. [14]. Letting u1=S-S*, u2=I-I*, u3=Z-Z*, u~i(t)=ui(τt)(i=1,2,3), τ=ν+τ*, and dropping the bars for simplification of notations, system (8) becomes functional differential equations in C=C([-1,0],ℝ3) as
(36)u1′(t)=(τ*+ν)×[S*G′(I*)u2(t-1)-G′(I*)u1(t)u2(t-1)r(1-2S*)u1(t)-G(I*)u1(t)-S*G′(I*)u2(t-1)-G′(I*)u1(t)u2(t-1)],u2′(t)=(τ*+ν)×[bω(ω+I*)3G′(I*)u3(t)u2(t-1)+G(I*)u3(t)+Z*G′(I*)u2(t-1)-(μ1+γ+ɛ)u2(t)-bω(ω+I*)2u2(t)+bω(ω+I*)3u22(t)-⋯],u3′(t)=1T(τ*+ν)(u1(t)-u3(t)).
Then system (36) is equivalent to
(37)u′(t)=Lνu(t)+f(ν,u(t)),
where u(t)=(u1(t),u2(t),u3(t))T∈ℝ3 and Lν:C→ℝ3, f:ℝ×C→ℝ3 are given, respectively, by
(38)Lν(ϕ)=(τ*+ν)×(r(1-2S*)-G(I*)000-(μ1+γ+ɛ)-bω(ω+I*)2G(I*)1T0-1T)×(ϕ1(0)ϕ2(0)ϕ3(0))+(τ*+ν)(0-S*G′(I*)00S*G′(I*)0000)(ϕ1(-1)ϕ2(-1)ϕ3(-1)),(39)f(ν,ϕ)=(τ*+ν)×{(-G′(I*)ϕ1(0)ϕ2(-1)G′(I*)ϕ3(0)ϕ2(-1)0)+(0bω(ω+I*)3ϕ22(0)0)+⋯}.
By the Riesz representation theorem, there exists a function η(θ,ν) of bounded variation for θ∈[-1,0], such that
(40)Lν(ϕ)=∫-10‍dη(θ,ν)ϕ(θ),forθ∈C.
In fact, we can choose
(41)η(θ,ν)=(τ*+ν)×(r(1-2S*)-G(I*)000-(μ1+γ+ɛ)-bω(ω+I*)2G(I*)1T0-1T)×δ(θ)-(τ*+ν)(0-S*G′(I*)00S*G′(I*)0000)δ(θ+1),
where δ denote the Dirac delta function. For ϕ∈C([-1,0],ℝ3), define
(42)A(ν)ϕ={dϕ(θ)dθ,θ∈[-1,0),∫-10‍dη(θ,ν)ϕ(θ),θ=0,R(ν)(ϕ)={0,θ∈[-1,0),f(ν,ϕ),θ=0.Then system (37) is equivalent to
(43)u˙(t)=A(ν)ut+R(ν)ut,
where ut=u(t+θ) for θ∈[-1,0].Forψ∈C([0,1],(ℝ3)*), define
(44)A*ψ(s)={-dψ(s)ds,s∈(0,1],∫-10‍ψ(-t)dη(t,0),s=0,
and a bilinear inner product
(45)〈ψ(s),ϕ(θ)〉=ψ¯(0)ϕ(0)-∫-10‍∫σ=0θ‍ψ¯(σ-θ)dη(θ)ϕ(σ)dσ,
where η(θ)=η(θ,0). Then A(0) and A* are adjoint operators. By the discussion in (20), we know that ±iω*τ* are eigenvalues of A(0). Hence, they are also eigenvalues of A*. We first need to compute the eigenvectors of A(0) and A* corresponding to iω*τ* and -iω*τ*, respectively.Suppose thatq(θ)=(1,q1,q2)Teiω*τ*θ is the eigenvectors of A(0) corresponding to iω*τ*; then A(0)q(θ)=iω*τ*q(θ). Then, from the definition of A(0) and (38), (40), (41), and q(-1)=q(0)e-iω*τ*, we have
(46)(r(1-2S*)-G(I*)-S*G′(I*)00S*G′(I*)-(μ1+γ+ɛ)-bω(ω+I*)2G(I*)1T0-1T)×(1q1(0)q2(0))=iω*(1q1(0)q2(0)).
We obtain
(47)q1=r-2rS*-G(I*)-iω*S*G′(I*),q2=11+Tiω*.
Similarly, we can obtain the eigenvector q*(s)=D(1,q1*,q2*)Teiω*τ* of A* corresponding to -iω*τ*, where
(48)q1*=S*G′(I*)S*G′(I*)-(μ1+γ+ɛ)-bω/(ω+I*)2+iω*,q2*=T[-iω-r(1-2S*)+G(I*)].
In order to assure that 〈q*(s),q(θ)〉=1, 〈q*(s),q-(θ)〉=0, we need to determine the value of D. By (45), we have
(49)〈q*(s),q(θ)〉=D¯(1,q¯1*,q¯2*)(1,q1,q2)T-∫-10‍∫σ=0θ‍D¯(1,q¯1*,q¯2*)e-iω*τ*(σ-θ)dηaaaaaaaaaa×(θ)(1,q1,q2)Teiω*τ*σdσ=D¯{∫-10‍1+q1q¯1*+q2q¯2*-∫-10‍(1,q¯1*,q¯2*)θeiω*τ*θdη(θ)(1,q1,q2)T}=D¯{1+q1q¯1*+q2q¯2*+τ*q1S*G′(I*)(-1+q¯1*)e-iω*τ*}.
Therefore, we can choose D as
(50)D=11+q¯1q1*+q¯2q2*+τ*q¯1S*G′(I*)(-1+q1*)e-iω*τ*.
We use the way of [14] and similarly way of [3]; we obtain that the coefficients are
(51)g20=2τ*D¯(q¯1*-1)G′(I*)q1+2τ*D¯q¯1*bω(ω+I*)3q12,g11=2τ*D¯(q¯1*-1)G′(I*)Re{q1}+2τ*D¯q¯1*bω(ω+I*)3|q1|2;g02=2τ*D¯(q¯1*-1)G′(I*)q¯1+2τ*D¯q¯1*bω(ω+I*)3q¯12;g21=τ*D¯(q¯1*-1)G′(I*)+[W20(1)(0)q¯1+2q1W11(1)(0)+W20(2)(0)+2W11(2)(0)]+2τ*D¯q¯1*bω(ω+I*)3[q¯1W20(2)(0)+2q1W11(2)(0)],
where
(52)W20(θ)=ig20ω*τ*q(0)eiω*τ*θ+ig¯023ω*τ*q¯(0)e-iω*τ*θ+[W20(0)+g20iω*τ*q(0)+g¯023iω*τ*q¯(0)]e2iω*τ*θ≜ig20ω*τ*q(0)eiω*τ*θ+ig¯023ω*τ*q¯(0)e-iω*τ*θ+E1e2iω*τ*θ,W11(θ)=ig11ω*τ*q¯(0)e-iω*τ*θ+E2.Besides,E1, E2 are satisfied with the following equation:(53)(2iω*-r(1-2S*)+G(I*)S*G′(I*)002iω*-S*G′(I*)+(μ1+γ+ɛ)+bω(ω+I*)2-G(I*)-1T02iω*+1T)E1=2τ*G′(I*)q1(-1,1,0)T+2τ*bω(ω+I*)3q12(0,1,0)T,(-r(1-2S*)+G(I*)S*G′(I*)00-S*G′(I*)+(μ1+γ+ɛ)+bω(ω+I*)2-G(I*)-1T01T)E2=2τ*G′(I*)q1(-1,1,0)T+2τ*bω(ω+I*)3q12(0,1,0)T.Thus, we can determineW20(θ) and W11(θ) from (52). Furthermore, we can compute g21 by (51). Thus we can compute the following values:
(54)c1(0)=i2ω*τ*(g20g11-2|g11|2-|g02|3)+g212,ν2=-Re{c1(0)}Re{dλ(τ*)/dτ},β2=2Re{c1(0)},T2=-Re{c1(0)}+ν2Re{dλ(τ*)/dτ}ω*τ*.
By the result of Hassard et al. [14], we have the following.Theorem 7.
In (54), the following results hold:(i)
the sign ofν2 determines the directions of the Hopf bifurcation: if ν2>0(ν2<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ>τ*(τ<τ*);(ii)
the sign ofβ2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2<0(β2>0);(iii)
the sign ofT2 determines the period of the bifurcating periodic solutions: the period is increasing (decreasing) if β2>0(β2<0).


## 6. Numerical Simulations

To demonstrate the theoretical results obtained from this paper, lettingG(I(t-τ))=I(t-τ)/(1+αI(t-τ)), we will give some numerical simulations. We consider the hypothetical set of parameter values as follows.(1)
Considerμ1=0.1; r=3; b=1; γ=0.05; T=4; ɛ=0.02; α=0.2; ω=0.4. By directly computing, we obtain R0=0.1786<1. According to Theorem 4, we know that the disease-free equilibrium of system (8) is locally asymptotically stable for this case (see Figures 1(a)–1(d)).(2)
Considerμ1=0.01; r=3; b=0.1; γ=0.05; T=10; ɛ=0.02; α=0.2; ω=0.4; τ=1.56. By directly computing, we obtain R0=3.0303>1. According to Theorem 4, we know that the disease-free equilibrium of system (8) is locally asymptotically stable for this case (see Figures 2(a)–2(d)).(3)
Considerμ1=0.01; r=3; b=0.1; γ=0.05; T=10; ɛ=0.02; α=0.2; ω=0.4, τ=2.56. By directly computing, we obtain R0=3.0303>1. According to Theorem 6, we know that the disease-free equilibrium of system (8) is unstable for this case (see Figures 3(a)–3(d)).(a)–(d) showed that the equilibriumE1 of system (8) with initial condition S(0)=3; I(0)=1; Z(0)=1; R0=0.3745<1; and T=4 is locally asymptotically stable.
(a)
Time series ofS(t) of the system
(b)
Time series ofI(t) of the system
(c)
Time series ofZ(t) of the system
(d)(a)–(d) showed that equilibriumE* of system (8) with initial condition S(0)=2; I(0)=2; Z(0)=4.5; R0=3.0303>1; and τ=1.56<τ* is locally asymptotically stable; that is, the trajectory converges to the positive equilibrium at τ=1.56.
(a)
Time series ofS(t) of the system
(b)
Time series ofI(t) of the system
(c)
Time series ofZ(t) of the system
(d)(a)–(d) showed that equilibriumE* of system (8) with initial condition S(0)=2; I(0)=2; Z(0)=4.5; R0=3.0303>1; and τ=2.56>τ* is unstable, that is, a periodic behavior at τ=2.56.
(a)
Time series ofS(t) of the system
(b)
Time series ofI(t) of the system
(c)
Time series ofZ(t) of the system
(d)


## 7. Conclusion

In this paper, we formulate and analyze a new delayed epidemic model with information variable and limited medical resources, the conditions for Hopf bifurcation to occur are derived. By analyzing the model, we have found the existence of disease-free equilibriaE0 and E1 and have a unique positive equilibrium E*. The basic reproduction number ℛ0 changes the stability of the disease-free equilibrium. When ℛ0<1, we discuss the stability of the disease-free equilibrium by analyzing the corresponding characteristic equations and constructing a Lyapunov functional, respectively. The conclusion reveals that the disease dies out and when ℛ0>1, we also get the sufficient criteria of stability switch at the positive equilibrium. Using the time delay (i.e., incubation time) as a bifurcation parameter, the local stability of the endemic equilibrium is investigated, and the conditions for Hopf bifurcation to occur are derived. Using the normal form theory and the center manifold theorem introduced by Hassard et al., we have studied the direction and stability of the bifurcating periodic solutions. Our theoretical results show that the time delay τ must be responsible for the observed regular cycles of disease incidence.Lastly, a numerical simulation provided that whenℛ0 is less than 1, the disease-free equilibrium is stable and while ℛ0 is more than 1, the disease-free equilibrium is unstable; that is, the endemic equilibrium exists (see Figure 1). Further, for τ>0, there will exist τ*∈I, such that the endemic equilibrium is asymptotically stable for 0<τ<τ* (see Figure 2) and becomes unstable for τ staying in some right neighborhood of τ*, with a Hopf bifurcation occurring when τ=τ*. If τ>τ*, the endemic equilibrium is unstable (see Figure 3).



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*Source: 109372-2014-04-09.xml*