An SIRS Epidemic Model Incorporating Media Coverage with Time Delay

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 number 𝑅₀<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.

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 𝑑𝑆(𝐼)𝑑𝑡=𝑏−𝑑𝑆−𝑔𝑆+𝛾𝑅,𝑑𝐼(𝐼)𝑑𝑡=𝑔𝑆−𝑑+𝜇+𝛿𝐼,𝑑𝑅𝑑𝑡=𝜇𝐼−𝑑+𝛾𝑅, (1) 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 rate 𝑔(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: 𝑔(𝐼)=𝛽₁−𝛽₂𝐼,𝑚+𝐼 (2) 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 𝑑𝑆𝛽𝑑𝑡=𝑏−𝑑𝑆−₁−𝛽₂𝐼𝑚+𝐼𝑆𝐼+𝛾𝑅,𝑑𝐼=𝛽𝑑𝑡¹−𝛽₂𝐼𝑚+𝐼𝑆𝐼−𝑑+𝜇+𝛿𝐼,𝑑𝑅𝑑𝑡=𝜇𝐼−𝑑+𝛾𝑅. (3)

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: 𝑔(𝐼(𝑡−𝜏))=𝛽₁−𝛽₂𝐼(𝑡−𝜏)(),𝑚+𝐼𝑡−𝜏 (4) where 𝜏>0 is a time delay representing the latent period of media coverage. Then model (3) can be modified as 𝑑𝑆(𝑡)(𝑡)−𝛽𝑑𝑡=𝑏−𝑑𝑆₁−𝛽₂𝐼(𝑡−𝜏)()𝑆(𝑡)𝐼(𝑡)(𝑡),(𝑡)𝑚+𝐼𝑡−𝜏+𝛾𝑅𝑑𝐼=𝛽𝑑𝑡¹−𝛽₂𝐼(𝑡−𝜏)()𝑆(𝑡)𝐼(𝑡)−𝐼(𝑡),(𝑡)𝑚+𝐼𝑡−𝜏𝑑+𝜇+𝛿𝑑𝑅(𝑡)−𝑅(𝑡).𝑑𝑡=𝜇𝐼𝑑+𝛾 (5)

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 𝑆(𝜃)=𝜙₁(𝜃),𝐼(𝜃)=𝜙₂(𝜃),𝑅(𝜃)=𝜙₃(𝜃),𝜙_𝑖(𝜃)≥0,𝜃∈[−𝜏,0],𝜙_𝑖(0)>0,𝑖=1,2,3, (6) where (𝜙₁(𝜃),𝜙₂(𝜃),𝜙₃(𝜃))∈𝒞([−𝜏,0],𝐑³₊₀), which is the Banach space of continuous functions mapping the interval [−𝜏,0] into 𝐑³₊₀, where 𝐑³₊₀={(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 Section 2, 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 𝐼(𝑡)=𝐼(0)𝑒^{∫𝑡0((𝛽1−((𝛽2𝐼(𝑠−𝜏))/(𝑚+𝐼(𝑠−𝜏))))𝑆(𝑠)−(𝑑+𝜇+𝛿))𝑑𝑠}. (7) From the initial condition (6), we have 𝐼(𝑡)>0, for 𝑡⩾0. Then, from the third equation of system (5), we have 𝑑𝑅(𝑡)(𝑡)−𝑅(𝑡)𝑅(𝑡).𝑑𝑡=𝜇𝐼𝑑+𝛾>−𝑑+𝛾 (8) A comparison argument shows that 𝑅(𝑡)⩾𝑅(0)𝑒^{∫𝑡0(−(𝑑+𝛾))𝑑𝑠}. (9) From the initial condition (6), we have 𝑅(𝑡)>0, for 𝑡⩾0.
Next, we prove that 𝑆(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 𝑑𝑆(𝑡)𝑑𝑡_𝑡=𝑡1=𝑏+𝛾𝑅𝑡₁>0. (10) This means 𝑆(𝑡)<0 for 𝑡∈(𝑡₁−𝜀,𝑡₁), where 𝜀 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 Theorem 1, 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 𝑑𝑁(𝑡)(𝑡)(𝑡)(𝑡).𝑑𝑡=𝑏−𝑑𝑁−𝛿𝐼<𝑏−𝑑𝑁 (11) Therefore, 𝑁(𝑡)<(𝑏/𝑑)+𝜀 for all large 𝑡, where 𝜀 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 𝑅₀=𝑏𝛽₁𝑑.𝑑+𝜇+𝛿 (12) 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 𝑆^∗=𝑑+𝜇+𝛾𝑚+𝐼^∗𝛽₁(𝑚+𝐼^∗)−𝛽₂𝐼^∗,𝑅^∗=𝜇𝐼^∗,𝐻𝑑+𝛾₁𝐼^∗2+𝐻₂𝐼^∗+𝐻₃𝐻=0,₁1=−𝛽𝑑+𝛾¹−𝛽₂𝛾(),𝐻𝑑+𝛿+𝑑𝑑+𝜇+𝛿₂=−𝑑𝛽₁𝑚𝜇𝑑+𝛾−𝛽₁𝑚(𝑑+𝛿)−𝑏𝛽₂+𝑏𝛽₁11−𝑅₀,𝐻₃=𝑑𝑚𝑑+𝜇+𝛿𝑅₀−1. (13)

Denoting Γ={(𝑆,𝐼,𝑅)∈ℛ³₊∣0<𝑆+𝐼+𝑅⩽𝑏/𝑑}, the following results in [17, 21] are here just recalled.

Lemma 3. For 𝜏=0, we have the following.(i)The disease-free equilibrium 𝐸₀ is globally asymptotically stable if 𝑅₀<1 and unstable if 𝑅₀>1 in the set Γ.(ii)The endemic equilibrium 𝐸^∗ is globally asymptotically stable if 𝑅₀>1 in the set Γ.

3.2. Local Stability at 𝐸₀

The characteristic equation of system (5) at 𝐸₀ is 𝛽det𝜆+𝑑¹𝑏𝑑𝛽−𝛾0𝜆+𝑑+𝜇+𝛿−¹𝑏𝑑00−𝜇𝜆+𝑑+𝛾=0, (14) which is equivalent to ()𝛽𝜆+𝑑𝜆+𝑑+𝛾𝜆+𝑑+𝜇+𝛿−¹𝑏𝑑=0. (15) 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 delay 𝜏⩾0, we have the following:(i)the disease-free equilibrium 𝐸₀ is locally asymptotically stable if 𝑅₀<1.(ii)the disease-free equilibrium 𝐸₀ is unstable if 𝑅₀>1.

3.3. Local Stability and Hopf Bifurcation at 𝐸^∗

In this subsection, we suppose that 𝑅₀>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 det𝜆−𝑎₁−𝑎₂−𝑎₆𝑒^−𝜆𝜏−𝛾−𝑎₃𝜆−𝑎₄+𝑎₆𝑒^−𝜆𝜏00−𝜇𝜆−𝑎₅=0, (16) 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 𝜆³+𝑏₁𝜆²+𝑏₂𝜆+𝑏₃+𝑏₄𝜆²+𝑏₅𝜆+𝑏₆𝑒^−𝜆𝜏=0, (17) 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 −𝜔³𝑖−𝑏₁𝜔²+𝑏₂𝜔𝑖+𝑏₃+−𝑏₄𝜔²+𝑏₅𝜔𝑖+𝑏₆(cos𝜔𝜏−𝑖sin𝜔𝜏)=0. (18) Separating the real and imaginary parts, we have 𝑏₁𝜔²−𝑏₃=𝑏₆−𝑏₄𝜔²cos𝜔𝜏+𝑏₅𝜔sin𝜔𝜏,−𝜔³+𝑏₂𝜔=𝑏₆−𝑏₄𝜔²sin𝜔𝜏−𝑏₅𝜔cos𝜔𝜏, (19) which is equivalent to 𝜔⁶+𝑏²₁−𝑏²₄−2𝑏₂𝜔⁴+𝑏²₂−2𝑏₁𝑏₃−𝑏²₅+2𝑏₄𝑏₆𝜔²+𝑏²₃−𝑏²₆=0. (20) Let 𝑧=𝜔² and denote 𝑝=𝑏²₁−𝑏²₄−2𝑏₂, 𝑞=𝑏²₂−2𝑏₁𝑏₃−𝑏²₅+2𝑏₄𝑏₆, and 𝑟=𝑏²₃−𝑏²₆. Then (20) becomes 𝑧³+𝑝𝑧²+𝑞𝑧+𝑟=0. (21) Next, we need to seek the conditions under which (21) has at least one positive root. Denote ℎ(𝑧)=𝑧³+𝑝𝑧²+𝑞𝑧+𝑟. (22) Since lim_𝑧→+∞ℎ(𝑧)=+∞, we conclude that if 𝑟<0, then (21) has at least one positive root.

From (22), we have ℎ^′(𝑧)=3𝑧²+2𝑝𝑧+𝑞. (23) 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 3𝑧²+2𝑝𝑧+𝑞=0 (24) has two real roots 𝑧^∗₁=−𝑝+√Δ3,𝑧^∗₂=−𝑝−√Δ3. (25) It is easy to see that ℎ^′′(𝑧^∗₁)=2√Δ>0 and ℎ^′′(𝑧^∗₂)=−2√Δ<0. It follows that 𝑧^∗₁ and 𝑧^∗₂ are the local minimum and the local maximum of ℎ(𝑧), respectively. Hence, we have the following simple property.

Lemma 5. Suppose that 𝑟⩾0 and Δ>0. Then (21) has positive root if and only if 𝑧^∗₁>0 and ℎ(𝑧^∗₁)⩽0.

From Lemma 5 and the discussion above, we have the following.

Lemma 6. For the polynomial equation (21), we have the following results. (i)If 𝑟<0, then (21) has at least one positive root.(ii)If 𝑟⩾0 and Δ=𝑝²−3𝑞⩽0, then (21) has no positive root.(iii)If 𝑟⩾0 and Δ=𝑝²−3𝑞>0, then (21) has positive roots if and only if 𝑧^∗₁=((−𝑝+√Δ)/3)>0 and ℎ(𝑧^∗₁)⩽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 𝜔₁=√𝑧₁,𝜔₂=√𝑧₂,𝜔₃=√𝑧₃. (26) From (19), we have 𝑏cos𝜔𝜏=⁵𝜔²𝜔²−𝑏₂−𝑏₁𝜔²−𝑏₃𝑏₄𝜔²−𝑏₆𝑏₄𝜔²−𝑏₆²+𝑏²₁𝜔². (27) Thus, if we denote 𝜏_𝑘^(𝑗)=1𝜔_𝑘cos⁻¹𝑏₅𝜔²_𝑘𝜔²_𝑘−𝑏₂−𝑏₁𝜔²_𝑘−𝑏₃𝑏₄𝜔²_𝑘−𝑏₆𝑏₄𝜔²_𝑘−𝑏₆²+𝑏²₁𝜔²_𝑘+2𝑗𝜋, (28) where 𝑘=1,2,3 and 𝑗=0,1,2,…, then ±𝑖𝜔_𝑘 is a pair of purely imaginary roots of (17) with 𝜏=𝜏_𝑘^(𝑗). Define 𝜏₀=𝜏_𝑘⁽⁰⁾₀=min_{𝑘∈{1,2,3}}𝜏_𝑘⁽⁰⁾,𝜔₀=𝜔_𝑘0. (29) 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 𝑃𝜆,𝑒^−𝜆𝜏1,…,𝑒^{−𝜆𝜏𝑚}=𝜆^𝑛+𝑝₁⁽⁰⁾𝜆^𝑛−1+⋯+𝑝⁽⁰⁾_𝑛−1𝜆+𝑝_𝑛⁽⁰⁾+𝑝₁⁽¹⁾𝜆^𝑛−1+⋯+𝑝⁽¹⁾_𝑛−1𝜆+𝑝_𝑛⁽¹⁾𝑒^−𝜆𝜏1+⋯+𝑝₁^(𝑚)𝜆^𝑛−1+⋯+𝑝^(𝑚)_𝑛−1𝜆+𝑝_𝑛^(𝑚)𝑒^{−𝜆𝜏𝑚}, (30) where 𝜏_𝑖⩾0(𝑖=1,2,…,𝑚) and 𝑝_𝑗^(𝑖)(𝑖=0,1,…,𝑚;𝑗=1,2,…,𝑛) are constants. As (𝜏₁,𝜏₂,…,𝜏_𝑚) 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 Lemmas 6 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)if 𝑟⩾0 and Δ=𝑝²−3𝑞⩽0, then all roots of (17) have negative real parts for all 𝜏⩾0;(ii)if either 𝑟<0 or 𝑟⩾0, Δ=𝑝²−3𝑞>0, 𝑧^∗₁=((−𝑝+√Δ)/3)>0, and ℎ(𝑧^∗₁)⩽0, then all roots of (17) have negative real parts for 𝜏∈[0,𝜏₀).

Let 𝜆(𝜏)=𝛼(𝜏)+𝑖𝜔(𝜏) (31) be the root of (17) near 𝜏=𝜏_𝑘^(𝑗) satisfying 𝛼(𝜏_𝑘^(𝑗))=0 and 𝜔(𝜏_𝑘^(𝑗))=𝜔_𝑘. Then, from Lemma  8 in [26], we have the following transversality condition.

Lemma 9. Suppose that 𝑧_𝑘=𝜔²_𝑘 and ℎ^′(𝑧_𝑘)≠0, where ℎ(𝑧) is defined by (22). Then 𝑑Re𝜆𝜏_𝑘^(𝑗)𝑑𝜏≠0, (32) and 𝑑(Re𝜆(𝜏_𝑘^(𝑗)))/𝑑𝜏 has the same sign with ℎ^′(𝑧_𝑘).

The proof of Lemma 9 is similar to that in the proof of Lemma  8 in [26], and here we omit it.

Then, from the above discussion and Lemmas 8 and 9, we have the following theorem.

Theorem 10. Suppose 𝑅₀>1 holds, and 𝜏_𝑘^(𝑗), 𝜔₀, and 𝜏₀ are defined by (28) and (29), respectively. Then (i)if 𝑟⩾0 and Δ=𝑝²−3𝑞⩽0, the endemic equilibrium 𝐸^∗ of system (5) is locally asymptotically stable for all 𝜏⩾0;(ii)if either 𝑟<0 or 𝑟⩾0, Δ=𝑝²−3𝑞>0, 𝑧^∗₁=((−𝑝+√Δ)/3)>0, and ℎ(𝑧^∗₁)⩽0, the endemic equilibrium 𝐸^∗ of system (5) is locally asymptotically stable for 𝜏∈[0,𝜏₀);(iii)if the conditions of (ii) are satisfied and ℎ^′(𝑧_𝑘)≠0, then system (5) exhibits Hopf bifurcation at the endemic equilibrium 𝐸^∗ when 𝜏 pass through 𝜏=𝜏_𝑘^(𝑗).

4. Global Continuation of Local Hopf Bifurcations

In this section, we study the global continuation of periodic solutions bifurcating from the positive equilibrium 𝐸^∗ 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: 𝑧(̇𝑡)=ℱ𝑧_𝑡,𝜏,𝑝, (33) where 𝑧_𝑡(𝜃)=(𝑧_1𝑡(𝜃),𝑧_2𝑡(𝜃),𝑧_3𝑡(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 𝐗=𝒞[−𝜏,0],𝐑²,Γ=Cl𝑧,𝜏,𝑝∈𝐗×𝐑×𝐑⁺;𝑧isanonconstantperiodicsolutionof(33),𝒩=𝑧,𝜏,𝑝;ℱ𝑧,𝜏,𝑝=0. (34) Let ℓ_(𝐸^∗_{,𝜏𝑗,2𝜋/𝜔0)} denote the connected component passing through (𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) in Γ, where 𝜏_𝑗 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 _{(𝑧,𝜏,𝑝)∈ℓ∗(𝑧,𝜏,𝑝)∩𝒩}𝛾_𝑚𝑧_∗,𝜏,𝑝=0, (35)for all 𝑚=1,2,…, where 𝛾_𝑚(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 Lemma 11 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 Theorems 1 and 2, it is easy to have the following.

Lemma 12. If the condition 𝑅₀>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 condition 𝑅₀>1 holds, then when 𝜏=0, the positive equilibrium 𝐸^∗ is globally stable in 𝐑³₊.

Lemma 14. If 𝑅₀>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: 𝑑𝑆(𝑡)(𝑡)−𝛽𝑑𝑡=𝑏−𝑑𝑆₁−𝛽₂𝐼(𝑡)(𝑡)𝑆(𝑡)𝐼(𝑡)(𝑡),(𝑡)𝑚+𝐼+𝛾𝑅𝑑𝐼=𝛽𝑑𝑡¹−𝛽₂𝐼(𝑡)(𝑡)𝑆(𝑡)𝐼(𝑡)−𝐼(𝑡),(𝑡)𝑚+𝐼𝑑+𝜇+𝛿𝑑𝑅(𝑡)−𝑅(𝑡).𝑑𝑡=𝜇𝐼𝑑+𝛾 (36) 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 𝐑³₊; thus, there is no periodic orbit in the first quadrant. This ends the proof.

Theorem 15. Let 𝜔₀ and 𝜏_𝑗(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>𝜏_𝑗(𝑗=1,2,…).

Proof. It is sufficient to prove that the projection of ℓ_(𝐸^∗_{,𝜏𝑗,2𝜋/𝜔0)} onto 𝜏-space is [𝜏,+∞) for each 𝑗>0, where 𝜏≤𝜏_𝑗.
The characteristic matrix of (33) at an equilibrium 𝑧=(𝑧⁽¹⁾,𝑧⁽²⁾,𝑧⁽³⁾)∈𝐑³ takes the following form: Δ𝑧,𝜏,𝑝(𝜆)=𝜆𝐼𝑑−𝐷ℱ𝑧,𝜏,𝑝𝑒^𝜆𝐼𝑑, (37) 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 detΔ𝐸₀(𝜆)=(𝛽,𝜏,𝑝𝜆+𝑑)𝜆+𝑑+𝛾𝜆+𝑑+𝜇+𝛿−¹𝑏𝑑Δ𝐸=0,det^∗,𝜏,𝑝(𝜆)=𝜆³+𝑏₁𝜆²+𝑏₂𝜆+𝑏₃+𝑏₄𝜆²+𝑏₅𝜆+𝑏₆𝑒^−𝜆𝜏=0, (38) 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^∗,𝜏_𝑗,2𝜋/𝜔₀) is an isolated center, and there exist 𝜖>0, 𝛿>0, and a smooth curve 𝜆∶(𝜏_𝑗−𝛿,𝜏_𝑗+𝛿)→𝒞 such that det(Δ(𝜆(𝜏)))=0, |𝜆(𝜏)−𝜔₀|<𝜖 for all 𝜏∈[𝜏_𝑗−𝛿,𝜏_𝑗+𝛿] and 𝜆𝜏_𝑗=𝜔₀(𝜏)𝑖,𝑑Re𝜆𝑑𝜏_{𝜏=𝜏𝑗}>0. (39)
Let Ω_{𝜖,2𝜋/𝜔0}=𝜂,𝑝;0<𝜂<𝜖,𝑝−2𝜋𝜔₀<𝜖. (40) It is easy to verify that on [𝜏_𝑗−𝛿,𝜏_𝑗+𝛿]×𝜕Ω_{𝜖,2𝜋/𝜔0}, detΔ𝐸^∗,𝜏,𝑝𝜂+2𝜋𝑝𝑖=0iff𝜂=0,𝜏=𝜏_𝑗,𝑝=2𝜋𝜔₀. (41)
Therefore, the hypotheses (𝐴₁)--(𝐴₄) in [27] are satisfied. Moreover, if we define 𝐻^±𝐸^∗,𝜏_𝑗,2𝜋𝜔₀Δ𝐸𝜂,𝑝=det^∗,𝜏_𝑗±𝛿,𝑝𝜂+2𝜋𝑝𝑖, (42) then we have the crossing number of isolated center (𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) as follows: 𝛾𝐸^∗,𝜏_𝑗,2𝜋𝜔₀=deg_𝐵𝐻⁻𝐸^∗,𝜏_𝑗,2𝜋𝜔₀,Ω_{𝜖,2𝜋/𝜔0}−deg_𝐵𝐻⁺𝐸^∗,𝜏_𝑗,2𝜋𝜔₀,Ω_{𝜖,2𝜋/𝜔0}=−1. (43) Thus, we have _{(𝑧,𝜏,𝑝)∈𝒞∗𝑗0)(𝐸,𝜏,2𝜋/𝜔}𝛾𝑧,𝜏,𝑝<0, (44) where (𝑧,𝜏,𝑝) has all or parts of the form (𝐸^∗,𝜏_𝑘,2𝜋/𝜔₀)(𝑘=0,1,…).
It follows from Lemma 11 that the connected component ℓ_(𝐸^∗_{,𝜏𝑗,2𝜋/𝜔0)} through (𝐸^∗,𝜏_𝑗,2𝜋/𝜔₀) in Γ is unbounded. From (28), we can know that if 𝑅₀>1 holds, for 𝑗≥1, 𝜏_𝑗=1𝜔₀cos⁻¹𝑏₄𝜔²−𝑏₂𝑏₄−𝑏₁𝑏₃𝜔²𝑏²₄+𝑏²₃𝜔²+2𝑗𝜋>2𝜋𝜔₀. (45)
Now we prove that the projection of ℓ_(𝐸^∗_{,𝜏𝑗,2𝜋/𝜔0)} onto 𝜏-space is [𝜏,+∞), where 𝜏≤𝜏_𝑗. 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 of ℓ_(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<𝜏_𝑗 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>𝜏_𝑗, (𝑗=1,2,…). This completes the proof.

5. Numerical Simulation

Example 1. In this case, we set 𝑏=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.
Figure 1 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 of 𝐼(𝑡) with 𝜏=5,15,25,35,45,55, respectively. 𝐸0 is always stable.

Example 2. In this case, we set 𝑏=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^′(𝑧_𝑘)=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^∗.
Figure 2 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.

(a)

(b)

Figure 2 

The trajectories and phase graphs of system (5) with 𝜏=19. 𝐸∗ is stable.

(a)

(b)

Figure 3 

The trajectories and phase graphs of system (5) with 𝜏=20.5, 𝐸∗ is unstable and a periodic orbit bifurcate from 𝐸∗.

(a)

(b)

Figure 4 

The trajectories and phase graphs of system (5) with 𝜏=30. 𝐸∗ is unstable and a periodic orbit bifurcate from 𝐸∗.

(a)

(b)

Figure 5 

The trajectories and phase graphs of system (5) with 𝜏=160. 𝐸∗ is unstable and a periodic orbit bifurcate from 𝐸∗.

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 all 𝜏⩾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, when 𝑅₀>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, when 𝑅₀>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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous referee for the very helpful suggestions and comments which led to improvements to their original paper. This work is supported by the National Natural Science Foundation of China (no.11061016), Science and Technology Department of Henan Province (no. 122300410417), and Education Department of Henan Province (no. 13A110108).