# An SIRS Epidemic Model Incorporating Media Coverage with Time Delay

Authors: Huitao Zhao; Yiping Lin; Yunxian Dai
Journal: Computational and Mathematical Methods in Medicine
 (2014)

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## Abstract

An SIRS epidemic model incorporating media coverage with time delay is proposed.
The positivity and boundedness are studied firstly. The locally asymptotical stability of the disease-free equilibrium and endemic equilibrium is studied in succession. And then, the conditions on which periodic orbits bifurcate are given. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the delay. The obtained results show that the time delay in media coverage can not affect the stability of the disease-free equilibrium when the basic reproduction numberR0<1. However, when R0>1, the stability of the endemic equilibrium will be affected by the time delay; there will be a family of periodic orbits bifurcating from the endemic equilibrium when the time delay increases through a critical value. Finally, some examples for numerical simulations are also included.


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## Body

## 1. Introduction

Since Kermack and Mckendrick proposed the classical SIR epidemic model in 1927, mathematical modeling has become important tools in analyzing the spread and control of infectious diseases. Attempts have been made to develop realistic mathematical models for the transmission dynamics of infectious diseases. In recent years, epidemic models described by ordinary differential equations have been studied by many authors (see, e.g., [1–9] and the references cited therein).One of the most fundamental compartment models based on differential equations is the SIRS model described by (1) below [10–15]. Let S(t) be the number of susceptible individuals, I(t) the number of infective individuals, and R(t) the number of removed individuals at time t, respectively. A general SIRS epidemic model can be formulated as
(1)dSdt=b-dS-g(I)S+γR,dIdt=g(I)S-(d+μ+δ)I,dRdt=μI-(d+γ)R,
where b>0 is the recruitment rate of the population, d>0 is the natural death rate of the population, μ>0 is the natural recovery rate of the infective individuals, γ>0 is the rate at which recovered individuals lose immunity and return to the susceptible class, and δ>0 is the disease-induced death rate. The transmission of the infection is governed by the incidence rate g(I)S, and g(I) is called the infection force.In modelling of communicable diseases, the incidence rateg(I)S may be affected by some factors, such as media coverage, density of population, and life style [16–22]. It is worthy to note that media coverage plays an important role in helping both the government authority make interventions to contain the disease and people respond to the disease [16, 19]. And a number of mathematical models have been formulated to describe the impact of media coverage on the transmission dynamics of infectious diseases. In particular, Cui et al. [16], Tchuenche et al. [18], and Sun et al. [20] incorporated a nonlinear function of the number of infective individuals (2) in their transmission term to investigate the effects of media coverage on the transmission dynamics:
(2)g(I)=β1-β2Im+I,
where β1>0 is the maximal effective contact rate between the susceptible and infective individuals and β2>0 is the maximal reduced effective contact rate due to mass media alert in the presence of infective individuals; the terms β2I/(m+I) measure the effect of reduction of the contact rate when infectious individuals are reported in the media. Because the coverage report cannot prevent disease from spreading completely we have β1⩾β2>0. The half-saturation constant m>0 reflects the impact of media coverage on the contact transmission. The function I/(m+I) is a continuous bounded function which takes into account disease saturation or psychological effects [20–22]. Then model (1) becomes
(3)dSdt=b-dS-(β1-β2Im+I)SI+γR,dIdt=(β1-β2Im+I)SI-(d+μ+δ)I,dRdt=μI-(d+γ)R.On the other hand, delays are ubiquitous in life, so it is in media coverage. Media coverage of an infectious outbreak can be seen as following two major routes [20, 23]. The first route is when the media report directly to the public on facts that they (the media) observe; the second has public health authorities using mass media or the Internet to communicate about the outbreak. For the second route, the number of infections and the number of suspected infections reported by media today are often the statistical result of yesterday or the day before. So the effects of media coverage on the transmission dynamics can be modified as follows:
(4)g(I(t-τ))=β1-β2I(t-τ)m+I(t-τ),
where τ>0 is a time delay representing the latent period of media coverage. Then model (3) can be modified as
(5)dS(t)dt=b-dS(t)-(β1-β2I(t-τ)m+I(t-τ))S(t)I(t)+γR(t),dI(t)dt=(β1-β2I(t-τ)m+I(t-τ))S(t)I(t)-(d+μ+δ)I(t),dR(t)dt=μI(t)-(d+γ)R(t).In the following, we will investigate the effect of time delay on the dynamics of system (5). We suppose that the initial condition for system (5) takes the form
(6)S(θ)=ϕ1(θ),I(θ)=ϕ2(θ),R(θ)=ϕ3(θ),ϕi(θ)≥0,θ∈[-τ,0],ϕi(0)>0,i=1,2,3,
where (ϕ1(θ),ϕ2(θ),ϕ3(θ))∈𝒞([-τ,0],R+03), which is the Banach space of continuous functions mapping the interval [-τ,0] into R+03, where R+03={(x,y,z)∣x≥0,y≥0,z≥0}.By the fundamental theory of functional differential equations [24], system (5) has a unique solution (S(t),I(t),R(t)) satisfying the initial condition (6).The rest of the paper is organized as follows. In Section2, we show the positivity and the boundedness of solutions of system (5) with initial condition (6). In Section 3, we study the local stability of the equilibria and the existence of the Hopf bifurcation at the positive equilibrium. In Section 4, we consider the global existence of bifurcating periodic solutions. In Section 5, we will give some numerical simulations to support the theoretical prediction. In Section 6, a brief discussion is given.


## 2. Positivity and Boundedness

In this section, we study the positivity and boundedness of solutions of system (5) with initial condition (6).Theorem 1.
Solutions of system (5) with initial condition (6) are positive for all t⩾0.Proof.
Assume(S(t),I(t),R(t)) is a solution of system (5) with initial condition (6). Let us consider I(t) for t⩾0. It follows from the second equation of system (5) that
(7)I(t)=I(0)e∫0t‍((β1-((β2I(s-τ))/(m+I(s-τ))))S(s)-(d+μ+δ))ds.
From the initial condition (6), we have I(t)>0, for t⩾0. Then, from the third equation of system (5), we have
(8)dR(t)dt=μI(t)-(d+γ)R(t)>-(d+γ)R(t).
A comparison argument shows that
(9)R(t)⩾R(0)e∫0t‍(-(d+γ))ds.
From the initial condition (6), we have R(t)>0, for t⩾0.
Next, we prove thatS(t) is positive. Assume the contrary; then, let t1 be the first time such that S(t1)=0. By the first equation of (5) we have
(10)dS(t)dt|t=t1=b+γR(t1)>0.
This means S(t)<0 for t∈(t1-ɛ,t1), where ɛ is an arbitrarily small positive constant. This leads to a contradiction. It follows that S(t) is always positive for t⩾0. This ends the proof.Theorem 2.
Solutions of system (5) with initial condition (6) are ultimately bounded.Proof.
From Theorem1, solutions of system (5) with initial condition (6) are positive for all t⩾0. Let N(t)=S(t)+I(t)+R(t). From (5), we have
(11)dN(t)dt=b-dN(t)-δI(t)<b-dN(t).
Therefore, N(t)<(b/d)+ɛ for all large t, where ɛ is an arbitrarily small positive constant. Thus, S(t), I(t), and R(t) are ultimately bounded.


## 3. Local Stability and Hopf Bifurcation Analysis

### 3.1. Previous Results

We now state some key results from [17], which provide the context for the main results of this paper. The basic reproduction number [17, 21] for the model is
(12)R0=bβ1d(d+μ+δ).
From [21], when τ=0, system (5) has a disease-free equilibrium E0=(b/d,0,0), which exists for all parameter values. When R0>1, system (5) has a unique endemic equilibrium E*=(S*,I*,R*), where
(13)S*=(d+μ+γ)(m+I*)β1(m+I*)-β2I*,R*=μI*d+γ,H1I*2+H2I*+H3=0,H1=-1d+γ(β1-β2)[γ(d+δ)+d(d+μ+δ)],H2=-dβ1mμd+γ-β1m(d+δ)-bβ2+bβ1(1-1R0),H3=dm(d+μ+δ)(R0-1).DenotingΓ={(S,I,R)∈ℛ+3∣0<S+I+R⩽b/d}, the following results in [17, 21] are here just recalled.Lemma 3.
Forτ=0, we have the following.(i)
The disease-free equilibriumE0 is globally asymptotically stable if R0<1 and unstable if R0>1 in the set Γ.(ii)
The endemic equilibriumE* is globally asymptotically stable if R0>1 in the set Γ.


### 3.2. Local Stability atE0

The characteristic equation of system (5) at E0 is
(14)det⁡[λ+dβ1bd-γ0λ+(d+μ+δ-β1bd)00-μλ+(d+γ)]=0,
which is equivalent to
(15)(λ+d)(λ+d+γ)(λ+d+μ+δ-β1bd)=0.
It is easy to see that, when R0<1, (15) has three negative roots and that, when R0>1, (15) has one positive root and two negative roots. Thus, we have the following.Theorem 4.
For any time delayτ⩾0, we have the following:(i)
the disease-free equilibriumE0 is locally asymptotically stable if R0<1.(ii)
the disease-free equilibriumE0 is unstable if R0>1.


### 3.3. Local Stability and Hopf Bifurcation atE*

In this subsection, we suppose thatR0>1. In what follows, using time delay as the bifurcation parameter, we investigate the Hopf bifurcation for system (5) and the stability of E* by using the method in [25, 26].The characteristic equation of system (5) at E* is
(16)det⁡[λ-a1-a2-a6e-λτ-γ-a3λ-a4+a6e-λτ00-μλ-a5]=0,
where a1=-d-β1I*+(β2I*2/(m+I*)), a2=-β1S*+(β2S*I*/(m+I*)), a3=β1I*-(β2I*2/(m+I*)), a4=-(d+μ+β)+β1S*-(β2S*I*/(m+I*)), a5=-(d+γ), and a6=(mβ2S*I*/(m+I*)2). Equation (16) is equivalent to
(17)λ3+b1λ2+b2λ+b3+(b4λ2+b5λ+b6)e-λτ=0,
where b1=-(a1+a4+a5), b2=a1a4+a1a5+a4a5-a2a3, b3=-a1a4a5+a2a3a5-a3γμ, b4=a6, b5=-a6(a1+a3+a5), and b6=a5a6(a1+a3).Obviously,iω is a root of (17) if and only if ω satisfies
(18)-ω3i-b1ω2+b2ωi+b3+(-b4ω2+b5ωi+b6)(cos⁡ωτ-isinωτ)=0.
Separating the real and imaginary parts, we have
(19)b1ω2-b3=(b6-b4ω2)cos⁡ωτ+b5ωsinωτ,-ω3+b2ω=(b6-b4ω2)sinωτ-b5ωcos⁡ωτ,
which is equivalent to
(20)ω6+(b12-b42-2b2)ω4+(b22-2b1b3-b52+2b4b6)ω2+b32-b62=0.
Let z=ω2 and denote p=b12-b42-2b2, q=b22-2b1b3-b52+2b4b6, and r=b32-b62. Then (20) becomes
(21)z3+pz2+qz+r=0.
Next, we need to seek the conditions under which (21) has at least one positive root. Denote
(22)h(z)=z3+pz2+qz+r.
Since lim⁡z→+∞⁡h(z)=+∞, we conclude that if r<0, then (21) has at least one positive root.From (22), we have
(23)h′(z)=3z2+2pz+q.
Clearly, if Δ=p2-3q⩽0, then the function h(z) is monotone increasing in z∈[0,+∞). Thus, when r⩾0 and Δ⩽0, (21) has no positive real root. On the other hand, when r⩾0 and Δ>0, the following equation
(24)3z2+2pz+q=0
has two real roots
(25)z1*=-p+Δ3,z2*=-p-Δ3.
It is easy to see that h′′(z1*)=2Δ>0 and h′′(z2*)=-2Δ<0. It follows that z1* and z2* are the local minimum and the local maximum of h(z), respectively. Hence, we have the following simple property.Lemma 5.
Suppose thatr⩾0 and Δ>0. Then (21) has positive root if and only if z1*>0 and h(z1*)⩽0.From Lemma5 and the discussion above, we have the following.Lemma 6.
For the polynomial equation (21), we have the following results. (i)
Ifr<0, then (21) has at least one positive root.(ii)
Ifr⩾0 and Δ=p2-3q⩽0, then (21) has no positive root.(iii)
Ifr⩾0 and Δ=p2-3q>0, then (21) has positive roots if and only if z1*=((-p+Δ)/3)>0 and h(z1*)⩽0.Suppose that (21) has positive root. Without loss of generality, we assume that it has three positive roots, defined by z1, z2, and z3, respectively. Then (20) has three positive roots
(26)ω1=z1,ω2=z2,ω3=z3.
From (19), we have
(27)cos⁡ωτ=b5ω2(ω2-b2)-(b1ω2-b3)(b4ω2-b6)(b4ω2-b6)2+b12ω2.
Thus, if we denote
(28)τk(j)=1ωk{cos⁡-1(b5ωk2(ωk2-b2)-(b1ωk2-b3)(b4ωk2-b6)(b4ωk2-b6)2+b12ωk2)fffff+2jπ(b5ωk2(ωk2-b2)-(b1ωk2-b3)(b4ωk2-b6)(b4ωk2-b6)2+b12ωk2)},
where k=1,2,3 and j=0,1,2,…, then ±iωk is a pair of purely imaginary roots of (17) with τ=τk(j). Define
(29)τ0=τk0(0)=min⁡k∈{1,2,3}{τk(0)},ω0=ωk0.
Note that, from Lemma 3, when τ=0, the endemic equilibrium E* is stable if R0>1. Till now, we can employ a result from Ruan and Wei [25] to analyze (17), which is stated as follows.Lemma 7.
Consider the exponential polynomial(30)P(λ,e-λτ1,…,e-λτm)=λn+p1(0)λn-1+⋯+pn-1(0)λ+pn(0)+(p1(1)λn-1+⋯+pn-1(1)λ+pn(1))e-λτ1+⋯+(p1(m)λn-1+⋯+pn-1(m)λ+pn(m))e-λτm,
where τi⩾0(i=1,2,…,m) and pj(i)(i=0,1,…,m;j=1,2,…,n) are constants. As (τ1,τ2,…,τm) vary, the sum of the order of the zeros of P(λ,e-λτ1,…,e-λτm) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.Applying Lemmas6 and 7 and the discussion above, we obtain the following lemma.Lemma 8.
For the third degree transcendental equation (17), we have the following:(i)
ifr⩾0 and Δ=p2-3q⩽0, then all roots of (17) have negative real parts for all τ⩾0;(ii)
if eitherr<0 or r⩾0, Δ=p2-3q>0, z1*=((-p+Δ)/3)>0, and h(z1*)⩽0, then all roots of (17) have negative real parts for τ∈[0,τ0).Let(31)λ(τ)=α(τ)+iω(τ)
be the root of (17) near τ=τk(j) satisfying α(τk(j))=0 and ω(τk(j))=ωk. Then, from Lemma  8 in [26], we have the following transversality condition.Lemma 9.
Suppose thatzk=ωk2 and h′(zk)≠0, where h(z) is defined by (22). Then
(32)d(Reλ(τk(j)))dτ≠0,
and d(Reλ(τk(j)))/dτ has the same sign with h′(zk).The proof of Lemma9 is similar to that in the proof of Lemma  8 in [26], and here we omit it.Then, from the above discussion and Lemmas8 and 9, we have the following theorem.Theorem 10.
SupposeR0>1 holds, and τk(j), ω0, and τ0 are defined by (28) and (29), respectively. Then (i)
ifr⩾0 and Δ=p2-3q⩽0, the endemic equilibrium E* of system (5) is locally asymptotically stable for all τ⩾0;(ii)
if eitherr<0 or r⩾0, Δ=p2-3q>0, z1*=((-p+Δ)/3)>0, and h(z1*)⩽0, the endemic equilibrium E* of system (5) is locally asymptotically stable for τ∈[0,τ0);(iii)
if the conditions of (ii) are satisfied andh′(zk)≠0, then system (5) exhibits Hopf bifurcation at the endemic equilibrium E* when τ pass through τ=τk(j).


## 3.1. Previous Results

We now state some key results from [17], which provide the context for the main results of this paper. The basic reproduction number [17, 21] for the model is
(12)R0=bβ1d(d+μ+δ).
From [21], when τ=0, system (5) has a disease-free equilibrium E0=(b/d,0,0), which exists for all parameter values. When R0>1, system (5) has a unique endemic equilibrium E*=(S*,I*,R*), where
(13)S*=(d+μ+γ)(m+I*)β1(m+I*)-β2I*,R*=μI*d+γ,H1I*2+H2I*+H3=0,H1=-1d+γ(β1-β2)[γ(d+δ)+d(d+μ+δ)],H2=-dβ1mμd+γ-β1m(d+δ)-bβ2+bβ1(1-1R0),H3=dm(d+μ+δ)(R0-1).DenotingΓ={(S,I,R)∈ℛ+3∣0<S+I+R⩽b/d}, the following results in [17, 21] are here just recalled.Lemma 3.
Forτ=0, we have the following.(i)
The disease-free equilibriumE0 is globally asymptotically stable if R0<1 and unstable if R0>1 in the set Γ.(ii)
The endemic equilibriumE* is globally asymptotically stable if R0>1 in the set Γ.


## 3.2. Local Stability atE0

The characteristic equation of system (5) at E0 is
(14)det⁡[λ+dβ1bd-γ0λ+(d+μ+δ-β1bd)00-μλ+(d+γ)]=0,
which is equivalent to
(15)(λ+d)(λ+d+γ)(λ+d+μ+δ-β1bd)=0.
It is easy to see that, when R0<1, (15) has three negative roots and that, when R0>1, (15) has one positive root and two negative roots. Thus, we have the following.Theorem 4.
For any time delayτ⩾0, we have the following:(i)
the disease-free equilibriumE0 is locally asymptotically stable if R0<1.(ii)
the disease-free equilibriumE0 is unstable if R0>1.


## 3.3. Local Stability and Hopf Bifurcation atE*

In this subsection, we suppose thatR0>1. In what follows, using time delay as the bifurcation parameter, we investigate the Hopf bifurcation for system (5) and the stability of E* by using the method in [25, 26].The characteristic equation of system (5) at E* is
(16)det⁡[λ-a1-a2-a6e-λτ-γ-a3λ-a4+a6e-λτ00-μλ-a5]=0,
where a1=-d-β1I*+(β2I*2/(m+I*)), a2=-β1S*+(β2S*I*/(m+I*)), a3=β1I*-(β2I*2/(m+I*)), a4=-(d+μ+β)+β1S*-(β2S*I*/(m+I*)), a5=-(d+γ), and a6=(mβ2S*I*/(m+I*)2). Equation (16) is equivalent to
(17)λ3+b1λ2+b2λ+b3+(b4λ2+b5λ+b6)e-λτ=0,
where b1=-(a1+a4+a5), b2=a1a4+a1a5+a4a5-a2a3, b3=-a1a4a5+a2a3a5-a3γμ, b4=a6, b5=-a6(a1+a3+a5), and b6=a5a6(a1+a3).Obviously,iω is a root of (17) if and only if ω satisfies
(18)-ω3i-b1ω2+b2ωi+b3+(-b4ω2+b5ωi+b6)(cos⁡ωτ-isinωτ)=0.
Separating the real and imaginary parts, we have
(19)b1ω2-b3=(b6-b4ω2)cos⁡ωτ+b5ωsinωτ,-ω3+b2ω=(b6-b4ω2)sinωτ-b5ωcos⁡ωτ,
which is equivalent to
(20)ω6+(b12-b42-2b2)ω4+(b22-2b1b3-b52+2b4b6)ω2+b32-b62=0.
Let z=ω2 and denote p=b12-b42-2b2, q=b22-2b1b3-b52+2b4b6, and r=b32-b62. Then (20) becomes
(21)z3+pz2+qz+r=0.
Next, we need to seek the conditions under which (21) has at least one positive root. Denote
(22)h(z)=z3+pz2+qz+r.
Since lim⁡z→+∞⁡h(z)=+∞, we conclude that if r<0, then (21) has at least one positive root.From (22), we have
(23)h′(z)=3z2+2pz+q.
Clearly, if Δ=p2-3q⩽0, then the function h(z) is monotone increasing in z∈[0,+∞). Thus, when r⩾0 and Δ⩽0, (21) has no positive real root. On the other hand, when r⩾0 and Δ>0, the following equation
(24)3z2+2pz+q=0
has two real roots
(25)z1*=-p+Δ3,z2*=-p-Δ3.
It is easy to see that h′′(z1*)=2Δ>0 and h′′(z2*)=-2Δ<0. It follows that z1* and z2* are the local minimum and the local maximum of h(z), respectively. Hence, we have the following simple property.Lemma 5.
Suppose thatr⩾0 and Δ>0. Then (21) has positive root if and only if z1*>0 and h(z1*)⩽0.From Lemma5 and the discussion above, we have the following.Lemma 6.
For the polynomial equation (21), we have the following results. (i)
Ifr<0, then (21) has at least one positive root.(ii)
Ifr⩾0 and Δ=p2-3q⩽0, then (21) has no positive root.(iii)
Ifr⩾0 and Δ=p2-3q>0, then (21) has positive roots if and only if z1*=((-p+Δ)/3)>0 and h(z1*)⩽0.Suppose that (21) has positive root. Without loss of generality, we assume that it has three positive roots, defined by z1, z2, and z3, respectively. Then (20) has three positive roots
(26)ω1=z1,ω2=z2,ω3=z3.
From (19), we have
(27)cos⁡ωτ=b5ω2(ω2-b2)-(b1ω2-b3)(b4ω2-b6)(b4ω2-b6)2+b12ω2.
Thus, if we denote
(28)τk(j)=1ωk{cos⁡-1(b5ωk2(ωk2-b2)-(b1ωk2-b3)(b4ωk2-b6)(b4ωk2-b6)2+b12ωk2)fffff+2jπ(b5ωk2(ωk2-b2)-(b1ωk2-b3)(b4ωk2-b6)(b4ωk2-b6)2+b12ωk2)},
where k=1,2,3 and j=0,1,2,…, then ±iωk is a pair of purely imaginary roots of (17) with τ=τk(j). Define
(29)τ0=τk0(0)=min⁡k∈{1,2,3}{τk(0)},ω0=ωk0.
Note that, from Lemma 3, when τ=0, the endemic equilibrium E* is stable if R0>1. Till now, we can employ a result from Ruan and Wei [25] to analyze (17), which is stated as follows.Lemma 7.
Consider the exponential polynomial(30)P(λ,e-λτ1,…,e-λτm)=λn+p1(0)λn-1+⋯+pn-1(0)λ+pn(0)+(p1(1)λn-1+⋯+pn-1(1)λ+pn(1))e-λτ1+⋯+(p1(m)λn-1+⋯+pn-1(m)λ+pn(m))e-λτm,
where τi⩾0(i=1,2,…,m) and pj(i)(i=0,1,…,m;j=1,2,…,n) are constants. As (τ1,τ2,…,τm) vary, the sum of the order of the zeros of P(λ,e-λτ1,…,e-λτm) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.Applying Lemmas6 and 7 and the discussion above, we obtain the following lemma.Lemma 8.
For the third degree transcendental equation (17), we have the following:(i)
ifr⩾0 and Δ=p2-3q⩽0, then all roots of (17) have negative real parts for all τ⩾0;(ii)
if eitherr<0 or r⩾0, Δ=p2-3q>0, z1*=((-p+Δ)/3)>0, and h(z1*)⩽0, then all roots of (17) have negative real parts for τ∈[0,τ0).Let(31)λ(τ)=α(τ)+iω(τ)
be the root of (17) near τ=τk(j) satisfying α(τk(j))=0 and ω(τk(j))=ωk. Then, from Lemma  8 in [26], we have the following transversality condition.Lemma 9.
Suppose thatzk=ωk2 and h′(zk)≠0, where h(z) is defined by (22). Then
(32)d(Reλ(τk(j)))dτ≠0,
and d(Reλ(τk(j)))/dτ has the same sign with h′(zk).The proof of Lemma9 is similar to that in the proof of Lemma  8 in [26], and here we omit it.Then, from the above discussion and Lemmas8 and 9, we have the following theorem.Theorem 10.
SupposeR0>1 holds, and τk(j), ω0, and τ0 are defined by (28) and (29), respectively. Then (i)
ifr⩾0 and Δ=p2-3q⩽0, the endemic equilibrium E* of system (5) is locally asymptotically stable for all τ⩾0;(ii)
if eitherr<0 or r⩾0, Δ=p2-3q>0, z1*=((-p+Δ)/3)>0, and h(z1*)⩽0, the endemic equilibrium E* of system (5) is locally asymptotically stable for τ∈[0,τ0);(iii)
if the conditions of (ii) are satisfied andh′(zk)≠0, then system (5) exhibits Hopf bifurcation at the endemic equilibrium E* when τ pass through τ=τk(j).


## 4. Global Continuation of Local Hopf Bifurcations

In this section, we study the global continuation of periodic solutions bifurcating from the positive equilibriumE* of system (5).Throughout this section, we follow closely the notations in [27]. For simplification of notations, setting z(t)=(z1(t),z2(t),z3(t))T=(S(t),I(t),R(t))T, we may rewrite system (5) as the following functional differential equation:
(33)z˙(t)=ℱ(zt,τ,p),
where zt(θ)=(z1t(θ),z2t(θ),z3t(θ))T = (z1(t+θ),z2(t+θ),z3(t+θ))T∈𝒞([-τ,0],R3). It is obvious that if R0>1 holds, then system (5) has a semitrivial equilibrium E0(b/d,0,0) and a positive equilibrium E*(S*,I*,R*). Following the work of [27], we need to define
(34)X=𝒞([-τ,0],R2),Γ=Cl{(z,τ,p)∈X×R×R+;111zisanonconstantperiodicsolutionof(33)},𝒩={(z¯,τ¯,p¯);ℱ(z¯,τ¯,p¯)=0}.
Let ℓ(E*,τj,2π/ω0) denote the connected component passing through (E*,τj,2π/ω0) in Γ, where τj and ω0 are defined by (28) and (29). From Theorem 10, we know that ℓ(E*,τj,2π/ω0) is nonempty.We first state the global Hopf bifurcation theory due to Wu [27] for functional differential equations.Lemma 11.
Assume that(z*,τ,p) is an isolated center satisfying the hypotheses (A1)–(A4) in [27]. Denote by ℓ(z*,τ,p) the connected component of (z*,τ,p) in Γ. Then either(i)
ℓ(z*,τ,p) is unbounded, or(ii)
ℓ(z*,τ,p) is bounded, ℓ(z*,τ,p)∩Γ is finite, and
(35)∑(z,τ,p)∈ℓ(z*,τ,p)∩𝒩‍γm(z*,τ,p)=0,for all m=1,2,…, where γm(z*,τ,p) is the mth crossing number of (z*,τ,p) if m∈J(z*,τ,p), or it is zero if otherwise.Clearly, if (ii) in Lemma11 is not true, then ℓ(z*,τ,p) is unbounded. Thus, if the projections of ℓ(z*,τ,p) onto z-space and onto p-space are bounded, then the projection onto τ-space is unbounded. Further, if we can show that the projection of ℓ(z*,τ,p) onto τ-space is away from zero, then the projection of ℓ(z*,τ,p) onto τ-space must include interval [τ,+∞). Following this ideal, we can prove our results on the global continuation of local Hopf bifurcation.From Theorems1 and 2, it is easy to have the following.Lemma 12.
If the conditionR0>1 holds, then all nonconstant periodic solutions of (5) with initial condition (6) are uniformly bounded.From [21], we know the following lemma.Lemma 13.
If the conditionR0>1 holds, then when τ=0, the positive equilibrium E* is globally stable in R+3.Lemma 14.
IfR0>1, then system (5) has no nonconstant periodic solution with period τ.Proof.
Suppose for a contradiction that system (5) has nonconstant periodic solution with period τ. Then the following system of ordinary differential equations has nonconstant periodic solution:
(36)dS(t)dt=b-dS(t)-(β1-β2I(t)m+I(t))S(t)I(t)+γR(t),dI(t)dt=(β1-β2I(t)m+I(t))S(t)I(t)-(d+μ+δ)I(t),dR(t)dt=μI(t)-(d+γ)R(t).
System (36) has the same equilibria as system (5), that is, E0(Λ/d,0,0) and a positive equilibrium E*(S*,I*,R*). Note that I-axis and R-axis are the invariable manifold of system (36) and the orbits of system (36) do not intersect each other. Thus, there is no solution that crosses the coordinate axis. On the other hand, note the fact that if system (36) has a periodic solution, then there must be the equilibrium in its interior and E0 are located on the coordinate axis. Thus, we conclude that the periodic orbit of system (36) must lie in the first quadrant. From Lemma 13, the positive equilibrium is asymptotically stable and globally stable in R+3; thus, there is no periodic orbit in the first quadrant. This ends the proof.Theorem 15.
Letω0 and τj(j=0,1,…) be defined in (28) and (29). If R0>1, then system (5) has at least j-1 periodic solutions for every τ>τj(j=1,2,…).Proof.
It is sufficient to prove that the projection ofℓ(E*,τj,2π/ω0) onto τ-space is [τ¯,+∞) for each j>0, where τ¯≤τj.
The characteristic matrix of (33) at an equilibrium z¯=(z¯(1),z¯(2),z¯(3))∈R3 takes the following form:
(37)Δ(z¯,τ,p)(λ)=λId-Dℱ(z¯,τ¯,p¯)(eλId),
where (z¯,τ¯,p¯) is called a center if ℱ(z¯,τ¯,p¯)=0 and det⁡(Δ(z¯,τ¯,p¯)((2π/p)i))=0. A center is said to be isolated if it is the only center in some neighborhood of (z¯,τ¯,p¯). It follows from (37) that
(38)det⁡(Δ(E0,τ,p)(λ))=(λ+d)(λ+d+γ)(λ+d+μ+δ-β1bd)=0,det⁡(Δ(E*,τ,p)(λ))=λ3+b1λ2+b2λ+b3+(b4λ2+b5λ+b6)e-λτ=0,
where b1, b2, b3, b4, b5, and   b6 are defined as in Section 3. From the discussion in Section 3, each of (38) has no purely imaginary root provided that R0<1. Thus, we conclude that (33) has no center of the form as (E0,τ,p) and (E*,τ,p). On the other hand, from the discussion in Section 3 about the local Hopf bifurcation, it is easy to verify that (E*,τj,2π/ω0) is an isolated center, and there exist ϵ>0, δ>0, and a smooth curve λ:(τj-δ,τj+δ)→𝒞 such that det⁡(Δ(λ(τ)))=0, |λ(τ)-ω0|<ϵ for all τ∈[τj-δ,τj+δ] and
(39)λ(τj)=ω0i,dReλ(τ)dτ|τ=τj>0.
Let(40)Ωϵ,2π/ω0={(η,p);0<η<ϵ,|p-2πω0|<ϵ}.
It is easy to verify that on [τj-δ,τj+δ]×∂Ωϵ,2π/ω0,
(41)det⁡⁡(Δ(E*,τ,p)(η+2πpi))=0iffη=0,τ=τj,p=2πω0.
Therefore, the hypotheses(A1)–(A4) in [27] are satisfied. Moreover, if we define
(42)H±(E*,τj,2πω0)(η,p)=det⁡(Δ(E*,τj±δ,p)(η+2πpi)),
then we have the crossing number of isolated center (E*,τj,2π/ω0) as follows:
(43)γ(E*,τj,2πω0)=degB(H-(E*,τj,2πω0),Ωϵ,2π/ω0)-degB(H+(E*,τj,2πω0),Ωϵ,2π/ω0)=-1.
Thus, we have
(44)∑(z¯,τ¯,p¯)∈𝒞(E*,τj,2π/ω0)‍γ(z¯,τ¯,p¯)<0,
where (z¯,τ¯,p¯) has all or parts of the form (E*,τk,2π/ω0)(k=0,1,…).
It follows from Lemma11 that the connected component ℓ(E*,τj,2π/ω0) through (E*,τj,2π/ω0) in Γ is unbounded. From (28), we can know that if R0>1 holds, for j≥1,
(45)τj=1ω0{cos⁡-1(b4ω2-b2b4-b1b3ω2b42+b32ω2)+2jπ}>2πω0.
Now we prove that the projection ofℓ(E*,τj,2π/ω0) onto τ-space is [τ¯,+∞), where τ¯≤τj. Clearly, it follows from the proof of Lemma 14 that system (5) with τ=0 has no nontrivial periodic solution. Hence, the projection of ℓ(E*,τj,2π/ω0) onto τ-space is away from zero.
For a contradiction, we suppose that the projection ofℓ(E*,τj,2π/ω0) onto τ-space is bounded; this means that the projection of ℓ(E*,τj,2π/ω0) onto τ-space is included in an interval (0,τ*). Noticing 2π/ω0<τj and applying Lemma 14, we have 0<p<τ* for (z(t),τ,p) belonging to ℓ(E*,τj,2π/ω0). Applying Lemma 12, we know that the projection of ℓ(E*,τj,2π/ω0) onto z-space is bounded. So the component of ℓ(E*,τj,2π/ω0) is bounded. It contradicts our conclusion that ℓ(E*,τj,2π/ω0) is unbounded. The contradiction implies that the projection of ℓ(E*,τj,2π/ω0) onto τ-space is unbounded above.
Hence, system (5) has at least j-1 periodic solution for every τ>τj, (j=1,2,…). This completes the proof.


## 5. Numerical Simulation

Example 1.
In this case, we setb=10, d=0.02, β1=0.0002, β2=0.00018, m=30, δ=0.1, μ=0.05, and γ=0.01. From (12), we compute R0=0.5882<1. Furthermore, from (13), system (5) has only a disease-free equilibrium E0=(500,0,0). From Theorem 4, we know that the disease-free equilibrium E0 is locally asymptotically stable for any time delay τ⩾0.
Figure1 shows that E0 is locally asymptotically stable, and the trajectories of I(t) always converge to zero for τ taking some different values.Figure 1
The trajectories ofI(t) with τ=5,15,25,35,45,55, respectively. E0 is always stable.Example 2.
In this case, we setb=10, d=0.02, β1=0.002, β2=0.0018, m=30, δ=0.1, μ=0.05, and γ=0.01. From (12), we compute R0=5.8824>1. Furthermore, from (13), we get a disease-free equilibrium E0=(500,0,0) and an endemic equilibrium E*=(178.7543,41.9016,69.8360) of system (5). From the algorithm of Section 3.3, we can compute τ0≈20.4343 and h′(zk)=13.2438>0. Thus, from Theorems 4 and 10, we know that the disease-free equilibrium E0 is unstable for all τ⩾0 and that the endemic equilibrium E* is stable for τ∈[0,20.4343). When τ crosses τ0, a family of periodic orbits bifurcate from E*.
Figure2 shows that the endemic equilibrium E* is stable with τ=19. Figure 3 shows that the endemic equilibrium E* is unstable and a periodic orbit bifurcate from E* with τ=20.5. Figure 4 shows that the endemic equilibrium E* is still unstable and a periodic orbit bifurcate from E* with τ=30. We can see from Figures 3 and 4 that the period and amplitude of the oscillation are increasing with the increasing of time delay. Furthermore, Figure 5 shows that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of τ1≈146.4764.The trajectories and phase graphs of system (5) with τ=19. E* is stable.
(a)
(b)The trajectories and phase graphs of system (5) with τ=20.5, E* is unstable and a periodic orbit bifurcate from E*.
(a)
(b)The trajectories and phase graphs of system (5) with τ=30. E* is unstable and a periodic orbit bifurcate from E*.
(a)
(b)The trajectories and phase graphs of system (5) with τ=160. E* is unstable and a periodic orbit bifurcate from E*.
(a)
(b)


## 6. Discussion

In this paper, we proposed an SIRS epidemic model incorporating media coverage with time delay. We first investigated the positivity and boundedness of the solution of system (5) and show that the solution of system (5) with the initial condition (6) is positive and bounded.Secondly, we studied the stability of the disease-free equilibrium. Our results show that the disease-free equilibrium is globally stable for allτ⩾0 when the basic reproduction number R0<1. This is to say, the time delay in media coverage cannot influence the stability of the disease-free equilibrium. In other words, we can ignore the effect of time delay for R0<1.However, whenR0>1, the stability of the endemic equilibrium will be affected by the time delay in media coverage. We found that there existed a critical value of time delay τ, such that the stability of the endemic equilibrium changed and periodic oscillations occurred when the time delay passes through this critical value. Furthermore, we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay.These results mean that, whenR0>1 and the time delay is small enough, the epidemic will eventually become endemic disease. However, if the delay of information about and appraisal of an epidemic on media coverage is too large, it will lead to repeated episodes of epidemic, and then it is unfavourable for the containment of the epidemic. We suggest that it is helpful for controlling epidemic to communicate about the outbreak of an epidemic as soon as possible.



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*Source: 680743-2014-03-03.xml*