Global Stability of an Eco-Epidemiological Model with Time Delay and Saturation Incidence

Abstract

We investigate a delayed eco-epidemiological model with disease in predator and saturation incidence. First, by comparison arguments, the permanence of the model is discussed. Then, we study the local stability of each equilibrium of the model by analyzing the corresponding characteristic equations and find that Hopf bifurcation occurs when the delay 𝜏 passes through a sequence of critical values. Next, by means of an iteration technique, sufficient conditions are derived for the global stability of the disease-free planar equilibrium and the positive equilibrium. Numerical examples are carried out to illustrate the analytical results.

1. Introduction

Recently, more attention has been paid to the eco-epidemiology model which considers both the ecological and epidemiological issues simultaneously due to the fact that most of the ecological populations suffer from various infectious diseases which have a significant role in regulating population sizes (see, e.g., [1–6]). Mukherjee [7] discussed a predator-prey model with disease in prey. The criteria were derived for both local stability and instability involving system parameters. In addition, considering the time required by the susceptible individuals to become infective after their interaction with the infectious individuals, Zhou et al. [8] formulated a delayed eco-epidemiology model and found that the Hopf bifurcation occurs when the delay passes through a sequence of critical values. They also gave an estimation of the length of the time delay to preserve stability. On the other hand, in the predator-prey system, the disease not only can spread in prey but also can spread in predator. Therefore, Zhang et al. [9] studied an eco-epidemiological model with disease in predator and showed that a Hopf bifurcation can occur as the delay increased. The above-mentioned works all used bilinear incidence to model disease transmission.

Note that ecologically the assumption of standard incidence instead of the former bilinear mass action incidence is meaningful for large populations and a low number of infected individuals, a very good justification behind this assumption being found in [10]. Han et al. [11] proposed four modifications of a predator-prey model with standard incidence to include an SIS or SIR parasitic infection. Thresholds were identified, and global stability results were proved. When the disease persists in the prey population and the predators have a sufficient feeding efficiency to survive, the disease also persists in the predator population. Hethcote et al. [12] considered a predator-prey model including an SIS parasitic infection in the prey with infected prey being more vulnerable to predation. Thresholds were identified which determine when the predator population survives and when the disease remains endemic.

However, there are a variety of factors that emphasize the need for a modification of the bilinear incidence and standard incidence. For example, the underlying assumption of homogeneous mixing may not always hold. Incidence rates that increase more gradually than linearly in 𝐼 and 𝑆 may arise from saturation effects. It has been strongly suggested by several authors that the disease transmission process may follow saturation incidence. After studying the cholera epidemic spread in Bari in 1973, Capasso and Serio [13] introduced a saturated incidence rate 𝑔(𝐼)𝑆 into epidemic models with 𝑔(𝐼)=𝛽𝐼/(1+𝛼𝐼). A general saturation incidence rate 𝑔(𝐼)𝑆=𝛽𝐼^𝑝𝑆/(1+𝛼𝐼^𝑝) was proposed by Liu et al. [14] and used by a number of authors; see, for example, Ruan and Wang [15] (𝑝=2), Bhattacharyya and Mukhopadhyay [16] (𝑝=1), and so forth. 𝛽𝐼^𝑝 measures the infection force of the disease, and 1/(1+𝛼𝐼^𝑝) measures the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. This incidence rate seems more reasonable than the bilinear incidence rate 𝛽𝑆𝐼, because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters.

Motivated by the works of Zhang et al. [9] and Capasso and Serio [13], in this paper, we are concerned with the effect of disease in predator and saturated incidence on the dynamics of eco-epidemiological model. To this end, we consider the following delay differential equations: 𝑥̇(𝑡)(𝑡)𝑥(𝑡)=𝑟𝑥1−𝐾(𝑡)𝑆(𝑡),𝑆̇(𝑡)()𝑆()−𝑎𝑥=𝑏𝑥𝑡−𝜏𝑡−𝜏−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝐼(𝑡)(𝑡),𝐼̇(𝑡)=(𝑡)𝐼(𝑡)1+𝛼𝐼𝛽𝑆(𝑡)(𝑡),1+𝛼𝐼−𝑑𝐼 (1.1) with initial conditions 𝑥(𝜃)=𝜙₁(𝜃),𝑆(𝜃)=𝜙₂(𝜃),𝐼(𝜃)=𝜙₃(𝜃),𝜙_𝑖(𝜃)≥0,𝜃∈[−𝜏,0],𝜙_𝑖(0)>0(𝑖=1,2,3), (1.2) where (𝜙₁(𝜃),𝜙₂(𝜃),𝜙₃(𝜃))∈𝐶([−𝜏,0]), ℝ³₊), the Banach space of continuous functions mapping the interval [−𝜏,0] into ℝ³₊₀, here ℝ³₊₀={(𝑥₁,𝑥₂,𝑥₃)∶𝑥_𝑖≥0,𝑖=1,2,3}.

We make the following assumptions for our model (1.1). (A1)The prey population grows logistically with intrinsic growth rate 𝑟 and environmental carrying capacity 𝐾.(A2)There is a spread of disease in predators which are divided solely into susceptible and infectious population. 𝑎 is the capturing rate of susceptible predators, 𝑏 is the growth rate of susceptible predator due to predation of prey. (A3)Susceptible predators become infected when they come in contact with infected predator, and this contact process is assumed to follow the saturation incidence rate 𝛽𝑆(𝑡)𝐼(𝑡)/(1+𝛼𝐼(𝑡)), with 𝛽 measuring the force of infection and 𝛼 the inhibition effect. (A4)𝑐>0 models death rate due to overcrowding, and 𝜏 is the time required for the gestation of susceptible predator. 𝑑 is the death rate of infected predator. All the above-mentioned parameters are assumed to be positive.

The paper is organized as follows. In the next section, the positivity of solutions and the permanence of system are discussed. By analyzing the corresponding characteristic equations, we find conditions for local stability and bifurcation results in Section 3. In Section 4, sufficient conditions are derived for the global stability of the disease-free planar equilibrium and the positive equilibrium of the system. Numerical examples are carried out to illustrate the validity of the main results. The paper ends with a conclusion in the last section.

2. Permanence

To prove the permanence of system (1.1), we need the following lemma, which is a direct application of Theorem 4.9.1 in the study by Kuang [17].

Lemma 2.1. Consider the following equation: 𝑥̇(𝑡)=𝑎𝑥(𝑡−𝜏)−𝑏𝑥(𝑡)−𝑐𝑥²(𝑡), (2.1) where 𝑎,𝑏,𝑐,𝜏>0 and 𝑥(𝑡)>0 for all 𝑡∈[−𝜏,0].(1)If 𝑎>𝑏, then limsup_𝑡→+∞𝑥(𝑡)=(𝑎−𝑏)/𝑐.(2)If 𝑎<𝑏, then limsup_𝑡→+∞𝑥(𝑡)=0.

Theorem 2.2. All the solutions of (1.1) with initial conditions (1.2) are all nonnegative.

Proof. Let (𝑥(𝑡),𝑆(𝑡),𝐼(𝑡)) be the solution of system (1.1) satisfying conditions (1.2). From the first and last equations of system (1.1), we have 𝑥(𝑡)=𝑥(0)𝑒^{∫𝑡0[𝑟(1−𝑥(𝜉)/𝐾)−𝑎𝑆(𝜉)]𝑑𝜉},𝐼(𝑡)=𝐼(0)𝑒^{∫𝑡0[𝛽𝑆(𝜉)/(1+𝛼𝐼(𝜉))−𝑑]𝑑𝜉}. (2.2) Hence, 𝑥(𝑡) and 𝐼(𝑡) are positive.
We now claim that 𝑆(𝑡)>0 for all 𝑡>0. Otherwise, there exists a 𝑡₁>0 such that 𝑆(𝑡₁)=0 and 𝑆(𝑡)>0 for all 𝑡∈[0,𝑡₁). Then 𝑆(̇𝑡₁)≤0. From the second equation of (1.1), we have 𝑆̇𝑡₁=𝑏𝑥𝑡₁−𝜏𝑆𝑡₁−𝜏>0, (2.3) which is a contradiction.

Theorem 2.3. All the solutions of (1.1) with initial conditions (1.2) are ultimately bounded.

Proof. From the first equation of (1.1), we have 𝑥̇(𝑡)(𝑡)𝑥(𝑡)≤𝑟𝑥1−𝐾. (2.4) Hence, we get limsup_𝑡→+∞𝑥(𝑡)≤𝐾=̇𝑀₁. (2.5)
From the second equation of system (1.1), for 𝑡 sufficiently large, we have 𝑆̇(𝑡)=𝑏𝑥(𝑡−𝜏)𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝐼(𝑡)(𝑡)()1+𝛼𝐼≤𝑏𝐾𝑆𝑡−𝜏−𝑐𝑆²(𝑡). (2.6) Hence, by Lemma 2.1, one can get limsup_𝑡→+∞𝑆(𝑡)≤𝑏𝐾𝑐=̇𝑀₂. (2.7)
It follows from the third equation of (1.1) and the above inequality, that for 𝑡 sufficiently large, we have 𝐼̇(𝑡)=𝛽𝑆(𝑡)𝐼(𝑡)(𝑡)(𝑡)≤1+𝛼𝐼−𝑑𝐼𝛽𝑀₂𝐼(𝑡)(𝑡)(𝑡).1+𝛼𝐼−𝑑𝐼 (2.8) Hence, one can see limsup_𝑡→+∞𝐼(𝑡)≤(1/(𝑑𝛼))|𝛽𝑀₂−𝑑|=̇𝑀₃.

Now, we show that system (1.1) is permanent.

Theorem 2.4. Suppose that (𝐻₁)𝛽𝑚₂>𝑑, (2.9)where 𝑚₂ is defined in (2.13), then system (1.1) is permanent.

Proof. From the first equation of system (1.1), we have 𝑥̇(𝑡)(𝑡)𝑥(𝑡)≥𝑟𝑥1−𝐾−𝑎𝑀₂𝑟. (2.10) It then follows that liminf_𝑡→+∞𝑥(𝑡)≥𝐾1−𝑎𝑀₂𝑟=̇𝑚₁. (2.11)
Using the second equation of system (1.1), for 𝑡 sufficiently large, we have 𝑆̇(𝑡)≥𝑏𝑚₁𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝑀₃1+𝛼𝑀₃. (2.12) Hence, by Lemma 2.1 and (𝐻₁), one can derive that liminf_𝑡→+∞𝑆(𝑡)≥1𝑐𝑏𝑚₁−𝛽𝑀₃1+𝛼𝑀₃=̇𝑚₂. (2.13)
From the third equation of system (1.1) and, above inequality, we have 𝐼̇(𝑡)≥𝛽𝑚₂𝐼(𝑡)(𝑡)(𝑡)1+𝛼𝐼−𝑑𝐼. (2.14) Since (𝐻₁) holds, then liminf_𝑡→+∞𝐼(𝑡)≥1𝑑𝛼𝛽𝑚₂−𝑑=̇𝑚₃. (2.15)
Therefore, the above calculations and Theorem 2.2 imply that there exist 𝑀_𝑖,𝑚_𝑖(𝑖=1,2,3) such that 0<𝑚₁≤liminf_𝑡→+∞𝑥(𝑡)≤limsup_𝑡→+∞𝑥(𝑡)≤𝑀₁,0<𝑚₂≤liminf_𝑡→+∞𝑆(𝑡)≤limsup_𝑡→+∞𝑆(𝑡)≤𝑀₂,0<𝑚₃≤liminf_𝑡→+∞𝐼(𝑡)≤limsup_𝑡→+∞𝐼(𝑡)≤𝑀₃. (2.16)

3. Local Stability

System (1.1) possesses the following equilibria.(1)The trivial equilibrium 𝐸₀(0,0,0).(2)The axial equilibrium 𝐸₁(𝐾,0,0).(3) The disease-free planar equilibrium 𝐸₂(𝑥₂,𝑆₂,0), where 𝑥₂=𝐾𝑐𝑟𝐾𝑎𝑏+𝑟𝑐,𝑆₂=𝐾𝑏𝑟𝐾𝑎𝑏+𝑟𝑐. (3.1)

(4) The unique positive equilibrium 𝐸₃(𝑥₃,𝑆₃,𝐼₃) exists if 𝛽𝑆₃>𝑑, where 𝑆₃=𝐾𝑏𝛼−𝛽+𝐾𝑏𝛼−𝛽²+4𝑑𝛼(𝐾𝑎𝑏/𝑟+𝑐)(),𝐼2𝛼𝐾𝑎𝑏/𝑟+𝑐₃=𝛽𝑆₃−𝑑𝑑𝛼,𝑥₃=𝐾−𝐾𝑎𝑆₃𝑟. (3.2)

In the following, we discuss the local stability of each equilibrium of system (1.1) by analyzing the corresponding characteristic equations, respectively.

3.1. Stability of Equilibrium 𝐸₀

The characteristic equation of system (1.1) at the trivial equilibrium 𝐸₀ is of the form 𝜆(𝜆−𝑟)(𝜆+𝑑)=0. (3.3) It is easy to see that (3.3) always has a positive root 𝑟. Hence, 𝐸₀ is always unstable.

3.2. Stability of Equilibrium 𝐸₁

The characteristic equation of system (1.1) at the axial equilibrium 𝐸₁ is of the form (𝜆+𝐾)𝜆−𝑏𝐾𝑒^−𝜆𝜏(𝜆+𝑑)=0. (3.4) There are two characteristic roots 𝜆₁=−𝐾, 𝜆₂=−𝑑, and another characteristic root is given by the root of 𝜆=𝑏𝐾𝑒^−𝜆𝜏. (3.5)

It is clear that Re𝜆>0. Hence, 𝐸₁ is always unstable.

3.3. Stability of Equilibrium 𝐸₂

Theorem 3.1. The disease-free planar equilibrium 𝐸₂ is locally asymptotically stable if 𝛽𝑆₂<𝑑, and the equilibrium 𝐸₂ is unstable if 𝛽𝑆₂>𝑑.

Proof. The characteristic equation of system (1.1) at the disease-free planar equilibrium 𝐸₂ is of the form 𝜆+𝑟𝑥₂𝐾𝜆+2𝑐𝑆₂−𝑐𝑆₂𝑒^−𝜆𝜏𝜆+𝑑−𝛽𝑆₂=0. (3.6)
Clearly, 𝜆₁=−𝑟𝑥₂/𝐾 is a negative eigenvalue. The second eigenvalue is given by the root of 𝜆₂=𝑐𝑆₂𝑒^−𝜆2𝜏−2. (3.7) Suppose that Re𝜆₂≥0, then Re𝜆₂=𝑐𝑆₂(𝑒^{−Re𝜆2𝜏}cos(𝜏Im𝜆₂)−2)<0. It is a contradiction, so Re𝜆₂<0. The last eigenvalue is 𝜆₃=𝛽𝑆₂−𝑑. The equilibrium 𝐸₂ is locally asymptotically stable if 𝛽𝑆₂<𝑑, and the equilibrium 𝐸₂ is unstable if 𝛽𝑆₂>𝑑.

3.4. Stability of Equilibrium 𝐸₃

The characteristic equation of system (1.1) at the positive equilibrium 𝐸₃ is of the form 𝜆³+𝐴₁𝜆²+𝐴₂𝜆+𝐴₃+𝑒^−𝜆𝜏𝐵₁𝜆²+𝐵₂𝜆+𝐵₃=0, (3.8) where 𝐴₁=𝑟𝑥₃𝐾+2𝑐𝑆₃+𝛽𝐼₃1+𝛼𝐼₃+𝑑𝛼𝐼₃1+𝛼𝐼₃,𝐴₂=𝑟𝑥₃𝐾+𝑑𝛼𝐼₃1+𝛼𝐼₃×2𝑐𝑆₃+𝛽𝐼₃1+𝛼𝐼₃+𝑑𝛼𝐼₃1+𝛼𝐼₃×𝑟𝑥₃𝐾+𝛽𝐼₃1+𝛼𝐼₃²,𝐴₃=𝑟𝑥₃𝐾×𝑑𝛽𝐼₃1+𝛼𝐼₃+2𝑐𝑆₃+𝛽𝐼₃1+𝛼𝐼₃×𝑑𝛼𝐼₃1+𝛼𝐼₃,𝐵₁=−𝑏𝑥₃,𝐵₂=𝑏𝑥₃×−𝑑𝛼𝐼₃1+𝛼𝐼₃−𝑟𝑥₃𝐾+𝑎𝑆₃,𝐵₃=𝑏𝑥₃×𝑑𝛼𝐼₃1+𝛼𝐼₃×−𝑟𝑥₃𝐾+𝑎𝑆₃. (3.9)

For 𝜏=0, the transcendental (3.8) reduces to the following equation: 𝜆³+𝐴₁+𝐵₁𝜆²+𝐴₂+𝐵₂𝜆+𝐴₃+𝐵₃=0. (3.10) We can easily get 𝐴₁+𝐵₁=𝑟𝑥₃𝐾+𝑐𝑆₃+𝑑𝛼𝐼₃1+𝛼𝐼₃𝐴>0,₂+𝐵₂=𝑟𝑥₃𝐾+𝑑𝛼𝐼₃1+𝛼𝐼₃×𝑐𝑆₃+𝑑𝛼𝐼₃1+𝛼𝐼₃×𝑟𝑥₃𝐾+𝛽𝐼₃1+𝛼𝐼₃²+𝑏𝑥₃𝑎𝑆₃𝐴>0,₃+𝐵₃=𝑟𝑥₃𝐾×𝑑𝛽𝐼₃1+𝛼𝐼₃+𝑐𝑆₃×𝑑𝛼𝐼₃1+𝛼𝐼₃+𝑏𝑥₃×𝑑𝛼𝐼₃1+𝛼𝐼₃×𝑎𝑆₃𝐴>0,₁+𝐵₁×𝐴₂+𝐵₂−𝐴₃+𝐵₃>0. (3.11) Therefore, the Routh-Hurwitz criterion implies that all the roots of (3.8) have negative real parts and we can conclude that the positive equilibrium 𝐸₃ is asymptotically stable in the absence of delay.

Theorem 3.2. For system (1.1), if the condition (𝐻₂)𝐴₃<𝐵₃ holds, the positive equilibrium 𝐸₃ is conditionally stable.

Proof. Substituting 𝜆=𝑖𝜔 into (3.8) and separating the real and imaginary parts, one can get 𝐴₁𝜔²−𝐴₃=𝐵₃−𝐵₁𝜔₂cos(𝜔𝜏)+𝐵₂(),𝜔𝜔sin𝜔𝜏₃−𝐴₂𝜔=𝐵₂𝜔cos(𝜔𝜏)−𝐵₃−𝐵₁𝜔₂sin(𝜔𝜏). (3.12)
Squaring and adding (3.12) we get 𝜔⁶+𝐷₁𝜔⁴+𝐷₂𝜔²+𝐷₃=0, (3.13) where 𝐷₁=𝐴²₁−2𝐴₂−𝐵²₁,𝐷₂=𝐴²₂−𝐵²₂−2𝐴₁𝐴₃+2𝐵₁𝐵₃,𝐷₃=𝐴²₃−𝐵²₃. (3.14)We know that 𝐷₃<0 provided that the condition (𝐻₂) holds. There is at least a positive 𝜔₀ satisfying (3.13), that is, the characteristic equation (3.8) has a pair of purely imaginary roots of the form ±𝑖𝜔₀. From (3.12), we can get the corresponding 𝜏_𝑘>0 such that the characteristic (3.8) has a pair of purely imaginary roots 𝜏_𝑘=1𝜔₀𝐴arccos¹𝜔²₀−𝐴₃𝐵₃−𝐵₁𝜔²₀+𝜔³₀−𝐴₂𝜔₀𝐵₂𝜔₀𝐵₃−𝐵₁𝜔²₀²+𝐵₂𝜔₀²+2𝑘𝜋𝜔₀,(𝑘=0,1,2,…). (3.15) Let 𝜆(𝜏)=𝜈(𝜏)+𝑖𝜔(𝜏) be the roots of (3.8) such that 𝜏=𝜏_𝑘 satisfying 𝜈(𝜏_𝑘)=0 and 𝜔(𝜏_𝑘)=𝜔₀. Differentiating the two sides of (3.8) with respect to 𝜏, we get d𝜆d𝜏⁻¹=2𝜆³+𝐴₁𝜆²−𝐴₂−𝜆²𝜆³+𝐴₁𝜆²+𝐴₂𝜆+𝐴₃+𝐵₁𝜆²−𝐵₃𝜆²𝐵₁𝜆²+𝐵₂𝜆+𝐵₃−𝜏𝜆. (3.16)Therefore, signdRe𝜆d𝜏_{𝜏=𝜏𝑘}=signRed𝜆d𝜏⁻¹_{𝜆=𝑖𝜔0}=1𝜔²₀𝐴signRe³+𝐴₁𝜔²₀+𝑖2𝜔³₀𝐴₁𝜔²₀−𝐴₃+𝑖𝜔³₀−𝐴₂𝜔₀+𝐵₁𝜔²₀+𝐵₃−𝐵₁𝜔²₀+𝑖𝐵₂𝜔₀+𝐵₃=1𝜔²₀sign2𝜔⁶₀+𝐴²₁−2𝐴₂−𝐵²₁𝜔⁴₀+𝐵²₃−𝐴²₃𝐵₃−𝐵₁𝜔²₀²+𝐵₂𝜔₀². (3.17) If the conditions (𝐻₂) and (𝐻₃)𝐴²₁−2𝐴₂>𝐵²₁ hold, one can see signdRe𝜆d𝜏_{𝜏=𝜏𝑘}>0. (3.18)Therefore, the transversality condition holds, hence, the Hopf bifurcation occurs at 𝜔=𝜔₀ and 𝜏=𝜏_𝑘.

Theorem 3.3. Suppose that the conditions (𝐻₂) and (𝐻₃) are satisfied. (1)The positive equilibrium 𝐸₃ of system (1.1) is asymptotically stable for all 𝜏∈[0,𝜏₀) and unstable for 𝜏>𝜏₀. (2)System (1.1) undergoes a Hopf Bifurcation at the positive equilibrium 𝐸₃ when 𝜏=𝜏_𝑘(𝑘=0,1,…).

4. Global Stability

In this section, we study the global stability of equilibriums 𝐄_𝟐 and 𝐄_𝟑. The strategy of proofs is to use an iteration technique and comparison arguments, respectively.

Theorem 4.1. If (𝐻₄)𝛽𝑏𝐾<𝑐𝑑,𝐾𝑎𝑏<𝑟𝑐 holds, then the disease-free planar equilibrium 𝐸₂ is globally asymptotically stable.

Proof. Let (𝑥(𝑡),𝑆(𝑡),𝐼(𝑡)) be any positive solution of system (1.1) with initial conditions (1.2). Let the following hold: 𝑈₁=limsup_𝑡→+∞𝑥(𝑡),𝑈₂=limsup_𝑡→+∞𝑆(𝑡),𝑈₃=limsup_𝑡→+∞𝐼(𝑡),𝑉₁=liminf_𝑡→+∞𝑥(𝑡),𝑉₂=liminf_𝑡→+∞𝑆(𝑡),𝑉₃=liminf_𝑡→+∞𝐼(𝑡). (4.1) In the following we shall claim that 𝑈₁=𝑉₁=𝑥₂,𝑈₂=𝑉₂=𝑆₂,𝑈₃=𝑉₃=0.
It follows from the first equation of system (1.1) that 𝑥̇(𝑡)(𝑡)𝑥(𝑡)≤𝑟𝑥1−𝐾. (4.2) By comparison, we obtain that 𝑈₁=limsup_𝑡→+∞𝑥(𝑡)≤𝐾+𝜀. (4.3) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑈₁≤𝑀^𝑥₁, where 𝑀^𝑥₁=𝐾. (4.4) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₁>0 such that, if 𝑡>𝑇₁, 𝑥(𝑡)≤𝑀^𝑥₁+𝜀.
We, therefore, derive from the second equation of system (1.1) that, for 𝑡>𝑇₁+𝜏, 𝑆̇(𝑡)≤𝑏𝑀^𝑥₁+𝜀𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡). (4.5) Hence, by Lemma 2.1, one can get 𝑈₂=limsup_𝑡→+∞𝑆(𝑡)≤𝑏𝑀^𝑥₁+𝜀𝑐=̇𝑀^𝑆₁. (4.6) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₂>0 such that, if 𝑇₂>𝑇₁+𝜏, 𝑆(𝑡)≤𝑀^𝑆₁+𝜀.
It follows from the third equation of system (1.1) that, for 𝑡>𝑇₂, 𝐼̇(𝑡)≤𝛽𝑀^𝑆₁+𝜀𝐼(𝑡)(𝑡)(𝑡)1+𝛼𝐼−𝑑𝐼. (4.7) Since (𝐻₄) holds, one can see 𝑈₃=limsup_𝑡→+∞𝐼(𝑡)≤0. (4.8) According to Theorem 2.2, we can get lim_𝑡→+∞𝐼(𝑡)=𝑈₃=𝑉₃=0.
We derive from the first equation of system (1.1) that, for 𝑡>𝑇₂+𝜏, 𝑥̇(𝑡)(𝑡𝑥(𝑡)≥𝑟𝑥)1−𝐾−𝑎𝑀^𝑆₁+𝜀𝑟. (4.9) By comparison we derive that 𝑉₁=liminf_𝑡→+∞𝑥(𝑡)𝑎𝑀≥𝐾1−^𝑆1+𝜀𝑟. (4.10) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑉₁≥𝑁^𝑥₁, where 𝑁^𝑥₁=𝐾1−𝑎𝑀^𝑆₁𝑟. (4.11) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₃>0 such that, if 𝑇₃>𝑇₂+𝜏, 𝑥(𝑡)≥𝑁^𝑥₁−𝜀.
We derive from the second equation of system (1.1) that, for 𝑡>𝑇₃, 𝑆̇(𝑡)≥𝑏𝑁^𝑥₁−𝜀𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝜀1+𝛼𝜀. (4.12) Hence, by Lemma 2.1, one can get 𝑉₂=liminf_𝑡→+∞𝑆(𝑡)≥1𝑐𝑏𝑁^𝑥₁−𝜀−𝛽𝜀1+𝛼𝜀. (4.13) Since this is true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑉₂≥𝑁^𝑆₁, where 𝑁^𝑆₁=𝑏𝑁^𝑥₁𝑐. (4.14) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₄>0 such that, if 𝑇₄>𝑇₃+𝜏, 𝑆(𝑡)≥𝑁^𝑆₁−𝜀.
Again, it follows from the first equation of system (1.2) that, for 𝑡>𝑇₄, 𝑥̇(𝑡)(𝑡𝑥(𝑡)≤𝑟𝑥)1−𝐾−𝑎𝑁^𝑆₁−𝜀𝑟. (4.15) A comparison argument yields 𝑈₁=limsup_𝑡→+∞𝑥(𝑡)𝑎𝑁≤𝐾1−^𝑆1−𝜀𝑟. (4.16) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑈₁≤𝑀^𝑥₂, where 𝑀^𝑥₂=𝐾1−𝑎𝑁^𝑆₁𝑟. (4.17) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₅>0 such that, if 𝑇₅>𝑇₄+𝜏, 𝑥(𝑡)≤𝑀^𝑥₂+𝜀.
It follows from the second equation of system (1.1) that, for 𝑡>𝑇₅, 𝑆̇(𝑡)≤𝑏𝑀^𝑥₂+𝜀𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝜀1+𝛼𝜀. (4.18) By Lemma 2.1, one can derive that 𝑉₂=limsup_𝑡→+∞𝑆(𝑡)≤1𝑐𝑏𝑀^𝑥₂+𝜀−𝛽𝜀1+𝛼𝜀. (4.19) Since this is true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑈₂≤𝑀^𝑆₂, where 𝑀^𝑆₂=𝑏𝑀^𝑥₂𝑐. (4.20) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₆>0 such that, if 𝑇₆>𝑇₅+𝜏, 𝑆(𝑡)≤𝑀^𝑆₂+𝜀.
We derive from the first equation of system (1.1) that, for 𝑡>𝑇₆, 𝑥̇(𝑡)(𝑡𝑥(𝑡)≥𝑟𝑥)1−𝐾−𝑎𝑀^𝑆₂+𝜀𝑟. (4.21) By comparison it follows that 𝑉₁=liminf_𝑡→+∞𝑥(𝑡)𝑎𝑀≥𝐾1−^𝑆2+𝜀𝑟. (4.22) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑉₁≥𝑁^𝑥₂, where 𝑁^𝑥₂=𝐾1−𝑎𝑀^𝑆₂𝑟. (4.23) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₇>0 such that, if 𝑇₇>𝑇₆+𝜏, 𝑥(𝑡)≥𝑁^𝑥₂−𝜀.
We derive from the second equation of system (1.1) that, for 𝑡>𝑇₇, 𝑆̇(𝑡)≥𝑏𝑁^𝑥₂−𝜀𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝜀1+𝛼𝜀. (4.24) Hence, by Lemma 2.1, one can get 𝑉₂=liminf_𝑡→+∞𝑆(𝑡)≥1𝑐𝑏𝑁^𝑥₂−𝜀−𝛽𝜀1+𝛼𝜀. (4.25) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑉₂≥𝑁^𝑆₂, where 𝑁^𝑆₂=𝑏𝑁^𝑥₂𝑐. (4.26) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₇>0 such that, if 𝑇₈>𝑇₇+𝜏, 𝑆(𝑡)≥𝑁^𝑆₂−𝜀.
Continuing this process, we get four sequences 𝑀^𝑥_𝑛,𝑀^𝑆_𝑛,𝑁^𝑥_𝑛,𝑁^𝑆_𝑛(𝑛=1,2,…) such that, for 𝑛≥2, 𝑀^𝑥_𝑛=𝐾1−𝑎𝑁^𝑆_𝑛−1𝑟,𝑁^𝑥_𝑛=𝐾1−𝑎𝑀^𝑆_𝑛𝑟,𝑀^𝑆_𝑛=𝑏𝑀^𝑥_𝑛𝑐,𝑁^𝑆_𝑛=𝑏𝑁^𝑥_𝑛𝑐. (4.27) Clearly, we have 𝑁^𝑥_𝑛≤𝑉₁≤𝑈₁≤𝑀^𝑥_𝑛,𝑁^𝑆_𝑛≤𝑉₂≤𝑈₂≤𝑀^𝑆_𝑛. (4.28) It follows from (4.27) that 𝑀^𝑥_𝑛+1=𝐾1−𝐾𝑎𝑏𝑟𝑐+𝑀^𝑥_𝑛𝐾²𝑎²𝑏²𝑟²𝑐². (4.29) Noting that 𝑀^𝑥_𝑛≥𝑆₂ and 𝐾𝑎𝑏<𝑟𝑐, we derive from (4.29) that 𝑀^𝑥_𝑛+1=𝐾1−𝐾𝑎𝑏𝑟𝑐+𝑀^𝑥_𝑛𝐾𝑎𝑏𝑟𝑐−1𝐾𝑎𝑏𝑟𝑐+1≤𝐾1−𝐾𝑎𝑏+𝑟𝑐𝐾𝑐𝑟𝐾𝑎𝑏+𝑟𝑐𝐾𝑎𝑏𝑟𝑐−1𝐾𝑎𝑏𝑟𝑐+1=0. (4.30) Thus, the sequence 𝑀^𝑥_𝑛 is monotonically nonincreasing. Therefore, it follows that lim_𝑛→+∞𝑀^𝑥_𝑛 exists. Taking 𝑛→+∞, we obtain from (4.29) that lim_𝑛→+∞𝑀^𝑥_𝑛+1=𝐾1−𝐾𝑎𝑏𝑟𝑐+lim_𝑛→+∞𝑀^𝑥_𝑛𝐾²𝑎²𝑏²𝑟²𝑐². (4.31) Noting that lim_𝑛→+∞𝑀^𝑥_𝑛+1=lim_𝑛→+∞𝑀^𝑥_𝑛, (4.32) it follows from (4.31) that lim_𝑛→+∞𝑀^𝑥_𝑛+1=lim_𝑛→+∞𝑀^𝑥_𝑛=𝑥₂. (4.33) We derive from (4.33) and the third equation of (4.27) that lim_𝑛→+∞𝑀^𝑆_𝑛+1=lim_𝑛→+∞𝑀^𝑆_𝑛=𝑆₂. (4.34) Similarly, one can derive from (4.27) and (4.34) that lim_𝑛→+∞𝑁^𝑥_𝑛=𝑥₂,lim_𝑛→+∞𝑁^𝑆_𝑛=𝑆₂. (4.35) It follows from (4.28), (4.33), and (4.35) that 𝑉₁=𝑈₁=𝑥₂,𝑉₂=𝑈₂=𝑆₂. (4.36) We, therefore, have lim_𝑡→+∞𝑥(𝑡)=𝑥₂,lim_𝑡→+∞𝑆(𝑡)=𝑆₂,lim_𝑡→+∞𝐼(𝑡)=0. (4.37) Hence, the disease-free planar equilibrium 𝐸₂ is globally asymptotically stable. The proof is complete.

Theorem 4.2. If (𝐻₅)𝛽𝑏𝐾>𝑐𝑑and𝐾𝑎𝑏<𝑟𝑐,𝛽>𝐾𝑏𝛼 holds, then the positive equilibrium 𝐸₃ is globally asymptotically stable.

Proof. Let (𝑥(𝑡),𝑆(𝑡),𝐼(𝑡)) be any positive solution of system (1.1) with initial conditions (1.2). Let the following hold: 𝑥=limsup_𝑡→+∞𝑥(𝑡),𝑆=limsup_𝑡→+∞𝑆(𝑡),𝐼=limsup_𝑡→+∞𝐼(𝑡),𝑥=liminf_𝑡→+∞𝑥(𝑡),𝑆=liminf_𝑡→+∞𝑆(𝑡),𝐼=liminf_𝑡→+∞𝐼(𝑡). (4.38) In the following we claim that 𝑥=𝑥=𝑥₃,𝑆=𝑆=𝑆₃,𝐼=𝐼=𝐼₃.
It follows from the first equation of system (1.1) that 𝑥̇(𝑡)(𝑡)𝑥(𝑡)≤𝑟𝑥1−𝐾. (4.39) By comparison we obtain 𝑥=limsup_𝑡→+∞𝑥(𝑡)≤𝐾+𝜀. (4.40) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑥≤𝑀^𝑥₁, where 𝑀^𝑥₁=𝐾. (4.41) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₁>0 such that, if 𝑡>𝑇₁, 𝑥(𝑡)≤𝑀^𝑥₁+𝜀.We obtain from the second equation of system (1.1) that, for 𝑡>𝑇₁+𝜏, 𝑆̇(𝑡)≤𝑏𝑀^𝑥₁+𝜀𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡). (4.42) Hence, by Lemma 2.1, we derive that 𝑆=limsup_𝑡→+∞𝑆(𝑡)≤𝑏𝑀^𝑥₁+𝜀𝑐. (4.43) Since it is true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑆≤𝑀^𝑆₁, where 𝑀^𝑆₁=𝑏𝑀^𝑥₁𝑐. (4.44) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₂>0 such that, if 𝑇₂>𝑇₁+𝜏, 𝑆(𝑡)≤𝑀^𝑆₁+𝜀.
It follows from the third equation of system (1.1) that 𝐼̇(𝑡)≤𝛽𝑀^𝑆₁+𝜀_{(𝑡)1+𝛼𝐼}𝐼(𝑡)−𝑑𝐼(𝑡). (4.45) Since (𝐻₅) holds, one can see 𝐼=limsup_𝑡→+∞𝐼(𝑡)≤𝛽𝑀^𝑆₁+𝜀−𝑑𝑑𝛼. (4.46) Since it is true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝐼≤𝑀^𝐼₁, where 𝑀^𝐼₁=𝛽𝑀^𝑆₁−𝑑𝑑𝛼. (4.47)
We derive from the first equation of system (1.1) that, for 𝑡>𝑇₂, 𝑥̇(𝑡)(𝑡𝑥(𝑡)≥𝑟𝑥)1−𝐾−𝑎𝑀^𝑆₁+𝜀𝑟. (4.48) By comparison we derive that 𝑥=liminf_𝑡→+∞𝑥(𝑡)𝑎𝑀≥𝐾1−^𝑆1+𝜀𝑟. (4.49) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑥≥𝑁^𝑥₁, where 𝑁^𝑥₁=𝐾1−𝑎𝑀^𝑆₁𝑟. (4.50) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₃>0 such that, if 𝑇₃>𝑇₂+𝜏, 𝑥(𝑡)≥𝑁^𝑥₁−𝜀.
We derive from the second equation of system (1.1) that, for 𝑡>𝑇₃, 𝑆̇(𝑡)≥𝑏𝑁^𝑥₁𝑆𝜏−𝜀𝑡−−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝑀^𝐼₁1+𝛼𝑀^𝐼₁. (4.51) Hence, by Lemma 2.1 and (𝐻₅), one can get 𝑆=liminf_𝑡→+∞𝑆(𝑡)≥1𝑐𝑏𝑁^𝑥₁−𝜀−𝛽𝑀^𝐼₁1+𝛼𝑀^𝐼₁. (4.52) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑆≥𝑁^𝑆₁, where 𝑁^𝑆₁=1𝑐𝑏𝑁^𝑥₁−𝛽𝑀^𝐼₁1+𝛼𝑀^𝐼₁. (4.53) Hence, for 𝜀>0 sufficiently small, we get 𝑆(𝑡)≥𝑁^𝑆₁−𝜀.
It follows from the third equation of system (1.1) that 𝐼̇(𝑡)≥𝛽𝑁^𝑆₁−𝜀𝐼(𝑡)(𝑡)(𝑡)1+𝛼𝐼−𝑑𝐼. (4.54) Provided that 𝛽𝑁^𝑆₁>𝑑, one can see 𝐼=liminf_𝑡→+∞𝐼(𝑡)≥𝛽𝑁^𝑆₁−𝜀−𝑑𝑑𝛼. (4.55) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝐼≥𝑁^𝐼₁, where 𝑁^𝐼₁=𝛽𝑁^𝑆₁−𝑑𝑑𝛼. (4.56)
It follows from the first equation of system (1.1) that 𝑥̇(𝑡)(𝑡𝑥(𝑡)≤𝑟𝑥)1−𝐾−𝑎𝑁^𝑆₁−𝜀𝑟. (4.57) By comparison we derive that 𝑥=limsup_𝑡→+∞𝑥(𝑡)𝑎𝑁≤𝐾1−^𝑆1−𝜀𝑟. (4.58) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑥≤𝑀^𝑥₂, where 𝑀^𝑥₂=𝐾1−𝑎𝑁^𝑆₁𝑟. (4.59) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₄>0 such that, if 𝑡>𝑇₄, 𝑥(𝑡)≤𝑀^𝑥₂+𝜀.
We obtain from the second equation of system (1.1) that, for 𝑡>𝑇₄+𝜏, 𝑆̇(𝑡)≤𝑏𝑀^𝑥₂+𝜀𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝑁^𝐼₁1+𝛼𝑁^𝐼₁. (4.60) Hence, by Lemma 2.1, one can get 𝑆=limsup_𝑡→+∞𝑆(𝑡)≤1𝑐𝑏𝑀^𝑥₂+𝜀−𝛽𝑁^𝐼₁1+𝛼𝑁^𝐼₁. (4.61) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑆≤𝑀^𝑆₂, where 𝑀^𝑆₂=1𝑐𝑏𝑀^𝑥₂−𝛽𝑁^𝐼₁1+𝛼𝑁^𝐼₁. (4.62) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₅>0 such that, if 𝑇₅>𝑇₄+𝜏, 𝑆(𝑡)≤𝑀^𝑆₂+𝜀.
It follows from the third equation of system (1.1) that 𝐼̇(𝑡)≤𝛽𝑀^𝑆₂+𝜀𝐼(𝑡)(𝑡)(𝑡)1+𝛼𝐼−𝑑𝐼. (4.63) Hence, by (𝐻₅), one can see 𝐼=limsup_𝑡→+∞𝐼(𝑡)≤𝛽𝑀^𝑆₂+𝜀−𝑑𝑑𝛼. (4.64) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝐼≤𝑀^𝐼₂, where 𝑀^𝐼₂=𝛽𝑀^𝑆₂−𝑑𝑑𝛼. (4.65)
We derive from the first equation of system (1.1) that, for 𝑡>𝑇₅, 𝑥̇(𝑡)(𝑡𝑥(𝑡)≥𝑟𝑥)1−𝐾−𝑎𝑀^𝑆₂+𝜀𝑟. (4.66) By comparison we derive that 𝑥=liminf_𝑡→+∞𝑥(𝑡)𝑎𝑀≥𝐾1−^𝑆2+𝜀𝑟. (4.67) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑥≥𝑁^𝑥₂, where 𝑁^𝑥₂=𝐾1−𝑎𝑀^𝑆₂𝑟. (4.68) Hence, for 𝜀>0 sufficiently small, there is a 𝑇₆>0 such that, if 𝑇₆>𝑇₅+𝜏, 𝑥(𝑡)≥𝑁^𝑥₂−𝜀.
We derive from the second equation of system (1.1) that, for 𝑡>𝑇₆, 𝑆̇(𝑡)≥𝑏𝑁^𝑥₂−𝜀𝑆(𝑡−𝜏)−𝑐𝑆²(𝑡)−𝛽𝑆(𝑡)𝑀^𝐼₂1+𝛼𝑀^𝐼₂. (4.69) By Lemma 2.1, one can get 𝑆=liminf_𝑡→+∞𝑆(𝑡)≥1𝑐𝑏𝑁^𝑥₂−𝜀−𝛽𝑀^𝐼₂1+𝛼𝑀^𝐼₂. (4.70) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝑆≥𝑁^𝑆₁, where 𝑁^𝑆₂=1𝑐𝑏𝑁^𝑥₂−𝛽𝑀^𝐼₂1+𝛼𝑀^𝐼₂. (4.71) Hence, for 𝜀>0 sufficiently small, we get 𝑆(𝑡)≥𝑁^𝑆₂−𝜀.
It follows from the third equation of system (1.1) that 𝐼̇(𝑡)≥𝛽𝑁^𝑆₂−𝜀𝐼(𝑡)(𝑡)(𝑡)1+𝛼𝐼−𝑑𝐼. (4.72) Since (𝐻₅) holds, one can see 𝐼=liminf_𝑡→+∞𝐼(𝑡)≥𝛽𝑁^𝑆₂−𝜀−𝑑𝑑𝛼. (4.73) Since this inequality holds true for arbitrary 𝜀>0 sufficiently small, we conclude that 𝐼≥𝑁^𝐼₂, where 𝑁^𝐼₂=𝛽𝑁^𝑆₂−𝑑𝑑𝛼. (4.74)
Continuing this process, we obtain six sequences 𝑀^𝑥_𝑛,𝑀^𝑆_𝑛,𝑀^𝐼_𝑛,𝑁^𝑥_𝑛,𝑁^𝑆_𝑛,𝑁^𝐼_𝑛(𝑛=1,2,…) such that, for 𝑛≥2, 𝑀^𝑥_𝑛=𝐾1−𝑎𝑁^𝑆_𝑛−1𝑟,𝑀^𝑆_𝑛=1𝑐𝑏𝑀^𝑥_𝑛−𝛽𝑁^𝐼_𝑛−11+𝛼𝑁^𝐼_𝑛−1,𝑀^𝐼_𝑛=𝛽𝑀^𝑆_𝑛−𝑑,𝑁𝑑𝛼_𝑥𝑛=𝐾1−𝑎𝑀^𝑆_𝑛𝑟,𝑁^𝑆_𝑛=1𝑐𝑏𝑁^𝑥_𝑛−𝛽𝑀^𝐼_𝑛1+𝛼𝑀^𝐼_𝑛,𝑁^𝐼_𝑛=𝛽𝑁^𝑆_𝑛−𝑑.𝑑𝛼 (4.75) A direct calculation shows that 𝑀^𝑥₂−𝑀^𝑥₁=−𝐾𝑎𝑁^𝑆₁𝑟𝑀<0,_𝑆2−𝑀^𝑆₁=1𝑐𝑏𝑀^𝑥₂−𝑀^𝑥₁−𝛽𝑁^𝐼₁1+𝛼𝑁^𝐼₁𝑀<0,_𝐼2−𝑀^𝐼₁=𝛽𝑀^𝑆₂−𝑀^𝑆₁𝑁𝑑𝛼<0,_𝑥2−𝑁^𝑥₁=𝐾𝑎𝑀^𝑆₁−𝑀^𝑠₂𝑟𝑁>0,_𝑆2−𝑁^𝑆₁=1𝑐𝑏𝑁^𝑥₂−𝑁^𝑥₁+𝛽𝑀^𝐼₁1+𝛼𝑀^𝐼₁−𝛽𝑀^𝐼₂1+𝛼𝑀^𝐼₂𝑁>0,_𝐼2−𝑁^𝐼₁=𝛽𝑁^𝑆₂−𝑁^𝑆₁𝑑𝛼>0. (4.76) By induction, we can show that 𝑀^𝑆_𝑛+1<𝑀^𝑆_𝑛, 𝑁^𝑆_𝑛+1>𝑁^𝑆_𝑛. Therefore, the sequence 𝑀^𝑆_𝑛 is decreasing, and the sequence 𝑁^𝑆_𝑛 is increasing. Clearly, we have 𝑁^𝑥_𝑛≤𝑥≤𝑥≤𝑀^𝑥_𝑛,𝑁^𝑆_𝑛≤𝑆≤𝑆≤𝑀^𝑆_𝑛,𝑁^𝐼_𝑛≤𝐼≤𝐼≤𝑀^𝐼_𝑛. (4.77) Hence, the limits of the sequences 𝑀^𝑆_𝑛 and 𝑁^𝑆_𝑛 exist. Denote 𝑆=lim_𝑡→+∞𝑀^𝑆_𝑛,𝑆=lim_𝑡→+∞𝑁^𝑆_𝑛. (4.78) We derive from (4.75) that 𝑆−𝑆𝛽−𝐾𝑏𝛼+𝐾𝑏𝑎𝑑𝑟𝑆+𝑆=0. (4.79) Since (𝐻₅) holds, 𝛽−𝐾𝑏𝛼+(𝐾𝑏𝑎/𝑑𝑟)(𝑆+𝑆)>0. It, therefore, follows from (4.79) that 𝑆=𝑆. Accordingly, we derive from (4.75) that 1𝑆=𝑐𝑎𝑏𝐾1−𝑆𝑟−𝛽𝛼+𝑑𝑆. (4.80) By a simple calculation, we obtain 𝑆=𝑆=𝑆₃. (4.81) It follows from (4.75) and (4.81) that 𝐼=𝐼=𝐼₃, 𝑥=𝑥=𝑥₃. Hence, the unique positive equilibrium 𝐸₃ is globally asymptotically stable. The proof is complete.

In the following we will present two examples to verify our results obtained earlier.

Example 4.3. In system (1.1), we let 𝑟=2,𝐾=2,𝑎=1,𝑐=2,𝛽=2,𝛼=1,𝑑=0.8,𝑏=0.3,𝜏=1. It is easy to show that 𝐾𝑏𝛽−𝑐𝑑=−0.4<0,𝑟𝑐−𝐾𝑎𝑏=3.4>0. By Theorem 4.1 we see that the equilibrium 𝐸₂(1.7391,0.2609,0) of system (1.1) is globally stable (see Figure 1).

Figure 1 

The temporal solution found by numerical integration of system (1.1) with 𝑟=2,𝐾=2,𝑎=2,𝑐=2,𝛽=2,𝛼=1,𝑑=0.8,𝑏=0.3,𝜏=1, and (𝑥0,𝑆0,𝐼0)=(1,1,1).

Example 4.4. In system (1.1), we let 𝑟=2,𝐾=2,𝑎=1,𝑐=2,𝛽=2,𝛼=1,𝑑=0.4,𝑏=0.3,𝜏=1. It is easy to show that 𝐾𝑏𝛽−𝑐𝑑=0.4>0,𝐾𝑎𝑏−𝑟𝑐=−3.4<0,𝛽−𝐾𝑏𝛼=1.400>0. By Theorem 4.2 we see that the equilibrium 𝐸₃(1.7881,0.2119,0.0596) of system (1.1) is globally stable, as depicted in Figure 2.

Figure 2 

The temporal solution found by numerical integration of system (1.1) with 𝑟=2,𝐾=2,𝑎=1,𝑐=2,𝛽=2,𝛼=1,𝑑=0.4,𝑏=0.3,𝜏=1, and (𝑥0,𝑆0,𝐼0)=(1,1,1).

5. Conclusion

In this paper, we have incorporated the disease for the predator and the time delay into an eco-epidemiology model. A saturation incidence function was used to model the behavioral change of the susceptible predator when their number increases or due to the crowding effect of the infected predator. First, by comparison arguments, the permanence of system (1.1) was studied. Then, by analyzing the corresponding characteristic equations, sufficient conditions were derived for the local stability of each equilibrium of system (1.1). From Theorem 3.3, we showed that system (1.1) undergoes a Hopf bifurcation when the delay passes through a sequence of critical values. Next, by using the iteration technique and comparison arguments, we derived sufficient conditions for the global stability of the disease-free planer equilibrium and positive equilibrium of system (1.1). By Theorems 4.1 and 4.2, we showed that (1) if (𝐻₄) holds, the infected predator population becomes extinct and the disease will be eliminated; that is, only sound predator and prey coexist; (2) if (𝐻₅) holds, the prey, the sound predator and the infected predator coexist. The disease will not be eliminated, and the system is permanent.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 11071254).