Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

Abstract

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any 𝜏, we show that the disease-free equilibrium is globally asymptotically stable; when 𝑅₀<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for any 𝜏=0; when 𝑅₀>1, the disease will persist. However, for any 𝜏≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

1. Introduction

Many scholars have been studying the change law of things by analyzing dynamic behavior of the corresponding system [1–28]. Recently, people established some epidemic models according to many realistic factors such as delay factor, isolation, and population change. People prevent many epidemics from spreading by establishing reasonable mathematics models. In fact, many diseases have different kinds of delays when they spread, such as immunity period delay [2–4], infection period delay [5], and incubation period delay [3, 6–13, 16–18, 25]. The time delay may influence the dynamics of infectious diseases. So it is necessary and useful to discuss delayed epidemic model.

Many scholars have discussed how delay affected the spread of infectious diseases. And some delays do not have an influence on transmission of epidemics [2–12]. Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate were discussed by Kyrychko and Blyuss in [2]; they considered time delay representing temporary immunity period and showed the endemic equilibrium was globally asymptotically stable, which indicated that the delay had no effect on the system with nonlinear incidence rate. And McCluskey studied the global stability for an SIR epidemic model with delay and nonlinear incidence rate [7]; he showed that the endemic equilibrium was globally asymptotically stable by using a Lyapunov function. And global stability of a delayed SIRS epidemic model with a nonmonotonic incidence rate was discussed by Muroya et al. [8], who proved that the endemic equilibrium was globally asymptotically stable. This illustrated that the delay could not affect the dynamics of the system with nonmonotonic incidence rate. At the same time, Xu and Ma studied stability of a delayed SIRS epidemic model with a nonlinear incidence rate [10]; the global asymptotic stability of system at the endemic equilibrium was proved. Besides, Yoshida and Hara discussed global stability of a delayed SIR epidemic model with density dependent birth and death rates in [11]. They investigated the stability of the endemic equilibrium, which was showed to be globally asymptotically stable. These researches above indicate delay has no effect on transmission of epidemics under certain conditions.

However, some delays impact the dynamics of system, which cause Hopf bifurcation to occur [13–20, 24, 27]. And Akkocaoğlu et al. discussed Hopf bifurcation analysis of a general nonlinear differential equation with delay in [13]. They get necessary conditions for the linear stability and Hopf bifurcation. There are also other scholars who analyze the existence of Hopf bifurcation in system. Hopf bifurcation analysis for a model of genetic regulatory system with delay was investigated by Wan and Zou in [14]. They analyzed the direction of the bifurcation and the stability of bifurcated periodic solutions. At the same time, density Hopf bifurcation in two SIRS dependent epidemic models was studied by Greenhalgh et al. [15], who found that Hopf bifurcation was theoretically possible but appeared not to occur for realistic parameter values. Meanwhile, Song et al. studied properties of stability and Hopf bifurcation for a HIV infection model with time delay [18] and showed that Hopf bifurcation would occur under certain conditions. However, Some professors researched how population change affected the dynamic behavior of system [21–25, 28]. Ferrara studied an Ak Solow model with a nonpositive rate of population growth [21] and analyzed how a nonpositive population growth rate hypothesis affected the dynamics of the standard Ak Solow model. At the same time, a note on the Ak Solow-Swan model with bounded population growth rate was researched by him [22]; he examined the effects of assuming a negative lower bound for the population growth rate within the Solow-Swan model with Ak technology and bounded population growth rate introduced by Guerrini. Besides, he researched the Ak Solow model with logistic law technology [23] and analyzed how a logistic law technology hypothesis affected the dynamics of the standard Ak Solow model. Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays was studied by Fang and Jiang [24]; they proved that the system was locally asymptotically stable in a range of the delay and Hopf bifurcation occurred as 𝜏 crossed a critical value. Enatsu et al. analyzed stability of delayed SIR epidemic models with a class of nonlinear incidence rates in [25]. In this paper, on the basis of [23–25, 27], we consider the incidence rate like 𝛽𝑆(𝑡−𝜏)𝐼(𝑡−𝜏)/(1+𝛼𝐼(𝑡−𝜏)) and establish the following models with logistic growth, obtaining conditions for Hopf bifurcation. At the same time, the death rate due to disease and the same natural death rate of 𝑆, 𝐼, and 𝑅 are considered.

The organization of this paper is as follows. In Section 2, SIR epidemic models, their basic reproduction number, and existence of equilibrium are given. In addition, the stability and existence of Hopf bifurcation of system (2) and system (6) are discussed in detail. In Section 3, we present the numerical simulations to verify the conclusions. In Section 4, a brief discussion is given to conclude this work.

2. The SIR Epidemic Model, the Basic Reproduction Number, and Equilibrium

We establish the delayed SIR epidemic model. Here, 𝑆(𝑡) represents the number of individuals who are susceptible to the disease, that is, who are not yet infected at time 𝑡. 𝐼(𝑡) represents the number of infected individuals who are infectious and are able to spread the disease by contacting with susceptible individuals at time 𝑡. 𝑅(𝑡) represents the number of recovered individuals at time 𝑡.

The transfer diagram is depicted in Figure 5.

SIR epidemic model with nonlinear incidence rate is as follows: 𝑑𝑆𝑆𝑑𝑡=𝑟𝑆1−𝐾−𝛽𝑆(𝑡−𝜏)𝐼(𝑡−𝜏)(),1+𝛼𝐼𝑡−𝜏𝑑𝐼=()𝐼()𝑑𝑡𝛽𝑆𝑡−𝜏𝑡−𝜏()−1+𝛼𝐼𝑡−𝜏𝜇+𝛼₁+𝑣𝐼,𝑑𝑅𝑑𝑡=𝑣𝐼−𝜇𝑅. (1) In system (1), it is assumed that the population growth in susceptible host individuals is governed by the logistic growth with a carrying capacity 𝐾>0 as well as intrinsic birth rate constant 𝑟>0, where the parameters 𝜇, 𝛽, 𝛼, 𝛼₁, 𝑘, and 𝜏 are positive constants in which 𝜇 is the natural death rate of the population; 𝛽 is the disease transmission coefficient; 𝛼 is the parameter which measures the psychological or inhibitory effect; 𝛼₁ is the death rate due to disease; 𝑣 is the recovery rate and 𝜏 is the latent period of the epidemic.

For simplicity, define 𝑆(𝑡)=𝑆(𝑡)/𝐾, 𝐼(̃𝑡)=𝐼(𝑡)/𝐾, 𝑅(𝑡)=𝑅(𝑡)/𝐾. 𝑡̃=𝛽𝐾𝑡, 𝑟̃=𝑟/𝛽𝐾, 𝜏=𝛽𝐾𝜏, 𝐺(𝐼(̃𝑡)̃)=𝐺(𝐼(𝑡))/𝐾, 𝜇=𝜇/𝛽𝐾, 𝛼₁=𝛼₁/𝛽𝐾, 𝑣̃=𝑣/𝛽𝐾, 𝛼=𝛼/𝛽𝐾. When dropping the ∼, system (1) becomes the following system: 𝑑𝑆()−𝑆()𝐼()𝑑𝑡=𝑟𝑆1−𝑆𝑡−𝜏𝑡−𝜏(),1+𝛼𝐼𝑡−𝜏𝑑𝐼=𝑆()𝐼()𝑑𝑡𝑡−𝜏𝑡−𝜏()−1+𝛼𝐼𝑡−𝜏𝜇+𝛼₁+𝑣𝐼,𝑑𝑅𝑑𝑡=𝑣𝐼−𝜇𝑅. (2)

The initial conditions for system (2) are 𝑆(𝜃)=𝜑₁(𝜃), 𝐼(𝜃)=𝜑₂(𝜃), 𝑅(𝜃)=𝜑₃(𝜃). (𝜑₁(𝜃),𝜑₂(𝜃),𝜑₃(𝜃))∈𝐶₊=𝐶([−𝜏,0],𝐑³₊), 𝜑_𝑖(0)≥0, 𝑖=1,2,3. 𝐑³₊  =  {𝑥₁,𝑥₂,𝑥₃  ∈  𝐑³∶𝑥_𝑖≥0,  𝑖=1,2,3}. And the feasible region of the model with the initial condition above is 𝛀=(𝑆,𝐼,𝑅)∈ℜ³₊₀∣𝑆+𝐼+𝑅≤1. (3) It is easy to show that 𝛀 is positively invariant with respect to system (1). According to the calculation and the practical significance of the epidemic model, system (2) always has a disease-free equilibrium 𝐸₀(1,0,0). Define the basic reproduction number as follows: 𝑅₀=1𝑎. (4) Let 𝑎=𝜇+𝛼₁+𝑣. If the basic reproduction number 𝑅₀>1, there exists a unique endemic equilibrium 𝐸^∗((𝑟𝛼−1+(1−𝑟𝛼)²+4𝑟𝑎𝛼)/2𝑟𝛼, (𝑆^∗−𝑎)/𝑎𝛼, 𝑣(𝑆^∗−𝑎)/𝜇𝑎𝛼).

SIR epidemic model with bilinear incidence rate is as follows: when 𝛼=0, system (1) becomes 𝑑𝑆𝑆𝑑𝑡=𝑟𝑆1−𝐾−𝛽𝑆(𝑡−𝜏)𝐼(𝑡−𝜏),𝑑𝐼()𝐼()−𝑑𝑡=𝛽𝑆𝑡−𝜏𝑡−𝜏𝜇+𝛼₁+𝑣𝐼,𝑑𝑅𝑑𝑡=𝑣𝐼−𝜇𝑅. (5) After the same transformation, system (5) becomes 𝑑𝑆()()𝐼(),𝑑𝑡=𝑟𝑆1−𝑆−𝑆𝑡−𝜏𝑡−𝜏𝑑𝐼()𝐼()−𝑑𝑡=𝑆𝑡−𝜏𝑡−𝜏𝜇+𝛼₁+𝑣𝐼,𝑑𝑅𝑑𝑡=𝑣𝐼−𝜇𝑅. (6) System (6) always has a disease-free equilibrium 𝐸₀(1,0,0). Define the basic reproduction number as follows: 𝑅₀=1𝑎. (7) Let 𝑎=𝜇+𝛼₁+𝑣. If the basic reproduction number 𝑅₀>1, there exists a unique endemic equilibrium 𝐸^∗(𝑎,𝑟(1−𝑎),𝑣𝑟(1−𝑎)/𝜇).

First of all, system (2) is studied. We prove that the disease-free equilibrium is globally asymptotically stable. And the endemic equilibrium is locally asymptotically stable when 𝜏∈[0,𝜏₀) and is unstable when 𝜏>𝜏₀. Meanwhile, system (2) undergoes Hopf bifurcation at 𝐸^∗ when 𝜏=𝜏_𝑘. And it is proved that the same conclusions appear in system (6).

2.1. SIR Epidemic Model with Nonlinear Incidence Rate

In this section, we discuss the local stability of the disease-free equilibrium in system (2) by analyzing its corresponding characteristic equation. By defining a reasonable Lyapunov function, we resolve the global dynamics of it without requiring any extra conditions. At the same time, we study the dynamics of system (2) at the endemic equilibrium 𝐸^∗. And the conditions for Hopf bifurcation are get.

Theorem 1. If 𝑅₀<1, the disease-free equilibrium 𝐸₀ of system (2) is locally asymptotically stable for any 𝜏. If 𝑅₀>1, it is unstable for any 𝜏.

Proof. When 𝜏≠0, the characteristic equation at the disease-free equilibrium 𝐸₀ of system (2) takes the form (𝜆+𝑟)𝜆+𝜇𝜆+𝑎−𝑒^−𝜆𝜏=0. (8) Clearly, system (2) always has two negative real roots: 𝜆₁=−𝑟 and 𝜆₂=−𝜇. All other roots are given by the roots of equation 𝜆+𝑎−𝑒^−𝜆𝜏=0,𝜆=𝑒^−𝜆𝜏𝑅−𝑎,_𝑒(𝜆)=𝑒^{−𝑅𝑒(𝜆)}cos(𝜏𝑙𝑚𝜆)−𝑎. (9) Assume 𝑅_𝑒(𝜆)≥0, 𝑅_𝑒(𝜆)𝑅≤1−𝑎,_𝑒(𝜆)≤𝑎𝑅₀−1. (10) Because 𝑅₀<1, 𝑅_𝑒(𝜆)<0. (11) Therefore the disease-free equilibrium 𝐸₀ of system (2) is locally asymptotically stable.
If 𝑅₀>1, let 𝑓(𝜆)=𝜆+𝑎−𝑒^−𝜆𝜏, because 𝑓(0)<0, 𝑓(+∞)>0, there is one positive real root at least. As a result, the disease-free equilibrium 𝐸₀ of system (2) is unstable.
When 𝜏=0, it is easy for us to prove that the disease-free equilibrium 𝐸₀ of system (2) is locally asymptotically stable.

Theorem 2. If 𝑅₀<1, the disease-free equilibrium 𝐸₀ of system (2) is globally asymptotically stable for any 𝜏.

Proof. For 𝑡>0, define a differentiable Lyapunov function 𝑉=𝑉₁+𝑉₂. (12) Obviously 𝑉≥0, 𝑉₁𝑉=𝐼,₂=^𝑡_𝑡−𝜏𝑆(𝑡−𝜏)1+𝛼𝐼(𝑡−𝜏)𝑑𝜃. (13) Calculating the derivative of 𝑉(𝑡) along positive solutions of system (2), it follows that 𝑉̇=𝑉̇₁+𝑉̇₂,𝑉̇=𝑆𝐼1+𝛼𝐼−𝑎𝐼. (14) Because 𝑆≤1, 1/(1+𝛼𝐼)<1. And when 𝑅₀<1, 𝑉̇≤(1−𝑎)𝐼, 𝑉̇≤𝑎𝑅₀̇−1𝐼,𝑉≤0. (15) While 𝑉̇=0, if and only if, 𝑆=1, 𝐼=0, 𝑅=0. For all 𝑡, it is easy to show that 𝐸₀ is the largest invariant subset of the set {(𝑆,𝐼,𝑅)∶𝑉̇=0}. Because of LaSalle’s invariance principle, the disease-free equilibrium 𝐸₀ of system (2) is globally asymptotically stable. This completes the proof.

Theorem 3. If 𝑅₀>1, the endemic equilibrium 𝐸^∗ is asymptotically stable when 𝜏∈[0,𝜏₀); Hopf bifurcation arises at 𝐸^∗ when 𝜏 passes through a sequence of critical values 𝜏_𝑘 in system (2).

Proof. The characteristic equation at the endemic equilibrium 𝐸^∗ is 𝜆²+𝑎𝑝−𝑞2𝑟𝑆^∗−𝑟−𝑞𝜆𝑒^−𝜆𝜏+𝑎+2𝑟𝑆^∗−𝑟+𝑝𝜆+𝑎2𝑟𝑆^∗−𝑟=0. (16) Here 𝐼^∗/(1+𝛼𝐼^∗)=𝑝, 𝑆^∗/(1+𝛼𝐼^∗)²=𝑞.
When 𝜏=0, the characteristic equation at the endemic equilibrium 𝐸^∗ is 𝜆²+𝑎+2𝑟𝑆^∗−𝑟+𝑝−𝑞𝜆+𝑎𝑝−𝑞2𝑟𝑆^∗−𝑟+𝑎2𝑟𝑆^∗−𝑟=0. (17) Here let 𝑚₁=𝑎+2𝑟𝑆^∗𝑚−𝑟+𝑝−𝑞,₂=𝑎𝑝−𝑞2𝑟𝑆^∗−𝑟+𝑎2𝑟𝑆^∗−𝑟. (18) Because 𝑟−𝑟𝑆^∗=𝑝, 𝑎>𝑞, and 𝑟>𝑝, order 𝑔=𝑟𝑆^∗. One has 𝑚₁𝑚=𝑎+𝑟−𝑝−𝑞>0,₂=𝑎𝑔−𝑞𝑔+𝑞𝑟>0. (19) According to Hurwitz criterion, the endemic equilibrium 𝐸^∗ of system (2) is locally asymptotically stable.
When 𝜏≠0, let ℎ₁=𝑎𝑝−𝑞2𝑟𝑆^∗−𝑟,ℎ₂ℎ=𝑞,₃=𝑎+2𝑟𝑆^∗−𝑟+𝑝,ℎ₄=𝑎2𝑟𝑆^∗−𝑟. (20) The characteristic equation at the endemic equilibrium 𝐸^∗ of system (2) takes the form 𝜆²+ℎ₁−ℎ₂𝜆𝑒^−𝜆𝜏+ℎ₃𝜆+ℎ₄=0. (21) Suppose that 𝜆=𝑖𝜔 is a root of (21). On substituting 𝜆=𝑖𝜔, we derive that −𝜔²+ℎ₁−ℎ₂𝜔𝑖((cos(𝜔𝜏)−𝑖sin(𝜔𝜏)))+ℎ₃𝜔𝑖+ℎ₄=0. (22) Separating real and imaginary parts, it follows that −𝜔²+ℎ₁cos(𝜔𝜏)−ℎ₂𝜔sin(𝜔𝜏)+ℎ₄=0,−ℎ₁sin(𝜔𝜏)−ℎ₂𝜔cos(𝜔𝜏)+ℎ₃𝜔=0. (23) Squaring and adding equations the both, then we have 𝜔⁴+ℎ²₃−2ℎ²₄−ℎ²₂𝜔²+ℎ²₄−ℎ²₁=0. (24) Letting 𝑧=𝜔², then (24) becomes 𝑧²+ℎ²₃−2ℎ²₄−ℎ²₂𝑧+ℎ²₄−ℎ²₁=0. (25) Here ℎ²₄−ℎ²₃=((𝑎−𝑞)(2𝑟𝑆^∗−𝑟)+𝑎𝑝)((𝑎+𝑞)(2𝑟𝑆^∗−𝑟)−𝑎𝑝). It is easy to show that (𝑎−𝑞)(2𝑟𝑆^∗−𝑟)+𝑎𝑝>0. One has 𝑎+𝑞2𝑟𝑆^∗−𝑟−𝑎𝑝+𝑞2𝑟𝑆^∗.−𝑟−𝑎𝑝=𝑎3𝑔−2𝑟+𝑞2𝑔−𝑟 (26) So when 𝑟>2𝑔, 𝑎(3𝑔−2𝑟)+𝑞(2𝑔−𝑟)<0. Then there is a unique positive 𝑧 satisfying (25). That is, there is a single pair of purely imaginary roots ±𝑖𝜔₀ of (21).
From (23), we get the corresponding 𝜏_𝑘>0 such that the characteristic equation (21) has a pair of purely imaginary roots: 𝜏_𝑘=1𝜔₀𝜔arccos⁰²ℎ₁−ℎ₁ℎ₄+ℎ₂ℎ₃𝜔₀²ℎ²₁+ℎ₂𝜔²₀+2𝑘𝜋𝜔₀,𝑘=1,2,3,…. (27) Next we show 𝑑(Re𝜆)𝑑𝜏_{𝜏=𝜏𝑘}>0. (28) This will signify that there exists at least one eigenvalue with positive real part for 𝜏>𝜏_𝑘. Differentiating (21) with respect to 𝜏, we will obtain 2𝜆𝑑𝜆𝑑𝜏−ℎ₂𝑑𝜆𝑒𝑑𝜏^−𝜆𝜏+ℎ₁−ℎ₂𝜆𝑒^−𝜆𝜏×−𝑑𝜆𝑑𝜏𝜏−𝜆+ℎ₃𝑑𝜆𝑑𝜏=0. (29) Therefore 𝑑𝜆𝑑𝜏⁻¹=2𝜆+ℎ₃−ℎ₂𝑒^−𝜆𝜏−𝜏ℎ₁−ℎ₂𝜆𝑒^−𝜆𝜏ℎ₁−ℎ₂𝜆𝜆𝑒^−𝜆𝜏=2𝜆+ℎ₃𝜆ℎ₁−ℎ₂𝜆𝑒^−𝜆𝜏−ℎ₂𝑒^−𝜆𝜏ℎ₁−ℎ₂𝜆𝑒^−𝜆𝜏𝜆−𝜏𝜆=2𝜆+ℎ₃−𝜆𝜆²+ℎ₃𝜆+ℎ₄−ℎ₂𝜆ℎ₁−ℎ₂𝜆−𝜏𝜆,𝑑()signRe𝜆𝑑𝜏_{𝜏=𝜏𝑘}=signRe𝑑𝜆𝑑𝜏⁻¹_{𝜆=𝑖𝜔0}=signRe2𝜆+ℎ₃−𝜆𝜆²+ℎ₃𝜆+ℎ_{4𝜆=𝑖𝜔0}ℎ−Re²𝜆ℎ₁−ℎ₂𝜆_{𝜆=𝑖𝜔0}=signRe2𝜔²₀−ℎ₃𝜔₀𝑖𝜔²₀𝜔²₀−ℎ₄−ℎ₃𝜔₀𝑖+ℎ₂𝜔₀𝑖𝜔²₀ℎ₁−ℎ₂𝜔₀𝑖2𝜔=sign²⁰−ℎ₄+ℎ²₃𝜔²₀−ℎ₄²+ℎ²₃𝜔²₀−ℎ²₂ℎ²₁+ℎ²₂𝜔²₀ℎ=sign²²𝜔⁴₀+2ℎ²₁𝜔²₀+ℎ²₃ℎ²₁−2ℎ₄ℎ²₁−ℎ²₂ℎ²₄𝜔²₀−ℎ₄²+ℎ²₃𝜔²₀ℎ²₁+ℎ²₂𝜔²₀. (30) Here, we will show ℎ²₃ℎ²₁−2ℎ₄ℎ²₁−ℎ²₂ℎ²₄>0. Let ℎ²₃ℎ²₁−2ℎ₄ℎ²₁−ℎ²₂ℎ²₄=𝐻.
One has 𝐻=𝑔−𝑟²𝑎⁴+2𝑟−𝑔³+2𝑞2𝑔−𝑟𝑔−𝑟𝑎³+𝑔²𝑔−𝑟²+𝑞4𝑟²𝑟−4𝑔5𝑟−2𝑔𝑎²+2𝑔²𝑞2𝑔−𝑟𝑔−𝑟+𝑞²8𝑔²2𝑟−𝑔+2𝑟²𝑟−5𝑔𝑎+𝑔²𝑞²2𝑔−𝑟². (31) According to the formula above, if 𝑟≥5𝑔, 𝐻>0, that is, when 𝑟𝛼−1+(1−𝑟𝛼)²+4𝑟𝑎𝛼<12𝑟𝛼5. (32)
We can derive 𝑑(Re𝜆)/𝑑𝜏|_{𝜏=𝜏𝑘}>0 when 0<𝑎<0.2. Therefore, the transversality condition holds and the conditions for Hopf bifurcation are satisfied at 𝜏=𝜏_𝑘 in system (2).

2.2. SIR Epidemic Model with Bilinear Incidence Rate

Using the same methods in system (2), we can prove that the disease-free equilibrium in system (6) is globally asymptotically stable for any 𝜏. Here the proof is ignored. And we only study the dynamics of system (6) at the endemic equilibrium 𝐸^∗.

Theorem 4. If 𝑅₀>1, the endemic equilibrium 𝐸^∗ is asymptotically stable when 𝜏∈[0,𝜏₀); system (6) undergoes Hopf bifurcation at 𝐸^∗ when 𝜏=𝜏_𝑘.

Proof. The characteristic equation at the endemic equilibrium 𝐸^∗ of system (6) is 𝜆²+𝑟𝑆^∗+𝑎𝐼^∗−2𝑟𝑆^∗2+𝐼^∗−𝑆^∗𝜆𝑒^−𝜆𝜏+𝑎+2𝑟𝑆^∗−𝑟𝜆+2𝑟𝑎𝑆^∗−𝑟𝑎=0. (33) When 𝜏=0, the characteristic equation at the endemic equilibrium 𝐸^∗ is 𝜆²+𝑎+2𝑟𝑆^∗−𝑟+𝐼^∗−𝑆^∗𝜆+2𝑟𝑎𝑆^∗−𝑟𝑎+𝑟𝑆^∗+𝑎𝐼^∗−2𝑟𝑆^∗2=0. (34) Because 𝑎=𝑆^∗,𝐼^∗=𝑟(1−𝑎),𝑎+2𝑟𝑆^∗−𝑟+𝐼^∗−𝑆^∗=𝑟𝑎>0,2𝑟𝑎𝑆^∗−𝑟𝑎+𝑟𝑆^∗+𝑎𝐼^∗−2𝑟𝑆^∗2=𝑎𝑟(1−𝑎)𝑟𝑎²𝑅₀−1>0, (35) the endemic equilibrium 𝐸^∗ of system (6) is locally asymptotically stable. When 𝜏≠0, let ℎ₁=𝑟𝑆^∗+𝑎𝐼^∗−2𝑟𝑆^∗2=2𝑎𝑟−3𝑟𝑎²,ℎ₂=𝐼^∗−𝑆^∗ℎ=𝑟−𝑟𝑎−𝑎,₃=𝑎+2𝑟𝑆^∗ℎ−𝑟=𝑎+2𝑟𝑎−𝑟,₄=2𝑟𝑎𝑆^∗−𝑟𝑎=2𝑟𝑎²−𝑟𝑎. (36) The characteristic equation at the endemic equilibrium 𝐸^∗ of system (6) will become 𝜆²+ℎ₁+ℎ₂𝜆𝑒^−𝜆𝜏+ℎ₃𝜆+ℎ₄=0. (37) Let 𝜆=𝑖𝜔; using the same methods in system (2), we can obtain the equation with 𝜔, as follows: 𝜔⁴+ℎ²₃−2ℎ₄−ℎ²₂𝜔²+ℎ²₄−ℎ²₁=0. (38) Order 𝑧=𝜔²; (38) becomes 𝑧²+ℎ²₃−2ℎ₄−ℎ²₂𝑧+ℎ²₄−ℎ²₁ℎ=0,₂₄−ℎ²₁=−𝑟²𝑎²(5𝑎−3)(𝑎−1). (39) We know 𝑎<1. When 𝑎<3/5, ℎ²₄−ℎ²₁<0. That is, there is a single pair of purely imaginary roots ±𝑖𝜔₀ of (39). At the same time, 𝜏_𝑘 can be gotten: 𝜏_𝑘=1𝜔₀𝜔arccos²⁰−ℎ₁−ℎ₁ℎ₄−ℎ₂ℎ₃𝜔²₀ℎ²₁+ℎ²₂𝜔²₀+2𝑘𝜋𝜔₀,𝑘=1,2,3,…. (40) And then 𝑑𝜆𝑑𝜏⁻¹=2𝜆+ℎ₃𝜆ℎ₁+ℎ₂𝑒^−𝜆𝜏+ℎ₂𝜆ℎ₁+ℎ₂𝜆−𝜏𝜆,𝑑()signRe𝜆𝑑𝜏_{𝜏=𝜏𝑘}=signRe𝑑𝜆𝑑𝜏⁻¹_{𝜆=𝑖𝜔0},Re𝑑𝜆𝑑𝜏⁻¹_{𝜆=𝑖𝜔0}=ℎ²₂𝜔⁴+2ℎ²₁𝜔²+ℎ²₁ℎ²₃−2ℎ²₁ℎ₄−ℎ²₂ℎ²₄ℎ²₁+ℎ²₂𝜔². (41)
Let 𝐺=ℎ²₁ℎ²₃−2ℎ²₁ℎ₄−ℎ²₂ℎ²₄, 𝐺  =  (36𝑎⁶−72𝑎⁵+60𝑎⁴−22𝑎³+3𝑎²)𝑟⁴  +  (−8𝑎⁶+16𝑎⁵−10𝑎⁴+2𝑎³)𝑟³  +  (5𝑎⁶−8𝑎⁵+3𝑎⁴)𝑟². Order 𝑓₁=𝑎²36𝑎⁴−72𝑎³+60𝑎²,𝑓−22𝑎+3₂=𝑎³−8𝑎³+16𝑎²,𝑓−10𝑎+2₃=5𝑎⁶−8𝑎⁵+3𝑎⁴,𝑓₃=𝑎⁴(5𝑎−3)(𝑎−1). (42) Obviously, 𝑟>0. When 𝑎<3/5,  𝑓₃>0. At the same time, if 0≤𝑎≤0.5, 𝑓₁>0 and 𝑓₂>0 in theory. In addition, the conclusion is also illustrated by the graphics which show the trends of the functions. See Figures 1(a) and 1(b).
In a word, when 0≤𝑎≤0.5, 𝐺>0. Therefor, 𝑑(Re𝜆)/𝑑𝜏|_{𝜏=𝜏𝑘}>0. That is, the transversality condition holds and the conditions for Hopf bifurcation are satisfied at 𝜏=𝜏_𝑘 in system (6). Here, we can find that conditions for Hopf bifurcation become weaker in system (6) than that in system (2). It means that system (6) undergoes Hopf bifurcation at 𝐸^∗ when 𝜏=𝜏_𝑘 easily.

(a)

(b)

Figure 1 

The graphics of 𝑓1 and 𝑓2.

3. The Numerical Simulations

In this section, we study system (1) and system (5) numerically. According to the different dates that can reflect the actual situation, we get the different simulations to prove our conclusions obviously.

3.1. The Numerical Simulations of System (1)

The disease-free equilibrium 𝐸₀ of system (1) is globally asymptotically stable.

Let 𝐾=90, 𝛽=0.005, 𝜇=0.035, 𝛼₁=0.23, 𝑣=0.2, 𝑟=0.15, 𝜏=1, 𝛼=0.13; here 𝑅₀=0.9677; see Figure 2(a).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 2 

The numerical simulation graphics of system (1).

When 𝑅₀>1, the dynamics of system (1) are shown by the following simulations.

Let 𝐾=20, 𝛽=0.054, 𝜇=0.0540, 𝛼₁=0.0162, 𝛼=0.013,𝑣=0.0324, and𝑟=0.0216; here 𝑅₀=10.5263. When 𝜏=0, the endemic equilibrium 𝐸^∗ is asymptotically stable; see Figure 2(b).

When 𝜏=0.0561<𝜏₀=0.6461, the endemic equilibrium 𝐸^∗ is also asymptotically stable; see Figures 2(c) and 2(d).

When 𝜏=0.6651>𝜏₀=0.6461, the endemic equilibrium 𝐸^∗ is unstable, and the Hopf bifurcation arises at 𝐸^∗; see Figures 2(e)–2(h).

3.2. The Numerical Simulations of System (5)

When 𝑅₀<1, the disease-free equilibrium 𝐸₀ of system (5) is globally asymptotically stable.

Let 𝐾=100, 𝛽=0.0015, 𝜇=0.0175, 𝛼₁=0.055, 𝑣=0.08, and 𝑟=0.125; here 𝑅₀=0.9836; see Figure 3(a).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3 

The numerical simulation graphics of system (5).

When 𝑅₀>1, the dynamics of system (5) are showed by the following simulations.

Let 𝐾=15, 𝛽=0.05, 𝜇=0.0375, 𝛼₁=0.0450,𝑣=0.0750, and 𝑟=0.112; here 𝑅₀=4.7619.

When 𝜏=0, the endemic equilibrium 𝐸^∗ is asymptotically stable; see Figure 3(b).

When 𝜏=0.6811<𝜏₀=0.9411, the endemic equilibrium 𝐸^∗ is also asymptotically stable; see Figures 3(c) and 3(d).

When 𝜏=1.0011>𝜏₀=0.9411, the endemic equilibrium 𝐸^∗ is unstable, and the Hopf bifurcation arises at 𝐸^∗; see Figures 3(e)–3(h).

Besides, when 𝛼₁ increases, 𝑅₀ will decrease. Let 𝐾=20, 𝛽=0.054, 𝜇=0.0540, 𝛼=0.013, 𝑣=0.0324, 𝑟=0.0216, 𝜏=0.6651. We can get the changing trends of 𝑅₀ as 𝛼₁ changes in system (1). Obviously, there exists 𝛼^∗₁ such that 𝑅₀=1. That is, if 𝛼₁>𝛼^∗₁, the disease will die out.

At the same time, let 𝐾=15,𝛽=0.05, 𝜇=0.0375, 𝑣=0.0750, 𝑟=0.1125, 𝜏=1.0011. We can get the changing trends of 𝑅₀ as 𝛼₁ changes in system (5). Obviously, there exists 𝛼^∗₁ such that 𝑅₀=1. That is, if 𝛼₁>𝛼^∗₁, the disease will die out.

See Figures 4(a) and 4(b).

(a)

(b)

Figure 4 

The numerical simulation graphics of 𝑅0.

Figure 5 

4. Discussions

In this paper, we investigate the dynamics of SIR epidemic models with the logistic growth of population of susceptible individuals. The incidence rate like 𝛽𝑆(𝑡−𝜏)𝐼(𝑡−𝜏)/(1+𝛼𝐼(𝑡−𝜏)) is considered here. In addition, the death rate due to disease and the same natural death rate of 𝑆, 𝐼, and 𝑅 are considered in two systems. We prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is less than unity for any delay 𝜏 and the endemic equilibrium is locally asymptotically stable when the basic reproduction number is greater than unity for 𝜏=0 in system (2) and system (6), respectively. At the same time, when 𝜏≠0, the sufficient conditions for the Hopf bifurcations are obtained, and the periodic solution will occur in the two systems. From the biological point of view, the delay here has influence on the transmission of the infectious diseases. That is, the delay is harmful. Besides, the conditions for Hopf bifurcations in system (2) are different from those in system (6). The conditions for Hopf bifurcation become weaker in system (6) than those in system (2). Therefore, according to the different situations, people should take different measures to control diseases.

Acknowledgments

This work is supported by the National Science Foundation of China (10471040) and the National Sciences Foundation of Shanxi Province (2009011005-1).