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

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.


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 p S/ 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: where φ 1 θ , φ 2 θ , φ 3 θ ∈ C −τ, 0 , R 3 , the Banach space of continuous functions mapping the interval −τ, 0 into R 3 0 , here 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.

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: where a, b, c, τ > 0 and x t > 0 for all t ∈ −τ, 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

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 t 1 > 0 such that S t 1 0 and S t > 0 for all t ∈ 0, t 1 .Then Ṡ t 1 ≤ 0. From the second equation of 1.1 , we have 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 Hence, we get lim sup From the second equation of system 1.1 , for t sufficiently large, we have

2.6
Hence, by Lemma 2.1, one can get lim sup

2.7
It follows from the third equation of 1.1 and the above inequality, that for t sufficiently large, we have

2.8
Hence, one can see lim Now, we show that system 1.1 is permanent.
Theorem 2.4.Suppose that where m 2 is defined in 2.13 , then system 1.1 is permanent.
Proof.From the first equation of system 1.1 , we have It then follows that Using the second equation of system 1.1 , for t sufficiently large, we have Hence, by Lemma 2.1 and H 1 , one can derive that lim inf

2.13
From the third equation of system 1.1 and, above inequality, we have

2.15
Therefore, the above calculations and Theorem 2.2 imply that there exist M i , m i i 1, 2, 3 such that

Local Stability
System 1.1 possesses the following equilibria.

3.2
In the following, we discuss the local stability of each equilibrium of system 1.1 by analyzing the corresponding characteristic equations, respectively.

Stability of Equilibrium E 0
The characteristic equation of system 1.1 at the trivial equilibrium E 0 is of the form It is easy to see that 3.3 always has a positive root r.Hence, E 0 is always unstable.

Stability of Equilibrium E 1
The characteristic equation of system 1.1 at the axial equilibrium E 1 is of the form There are two characteristic roots λ 1 −K, λ 2 −d, and another characteristic root is given by the root of λ bKe −λτ .

3.5
It is clear that Re λ > 0. Hence, E 1 is always unstable.

Stability of Equilibrium E 3
The characteristic equation of system 1.1 at the positive equilibrium E 3 is of the form where

3.9
For τ 0, the transcendental 3.8 reduces to the following equation: We can easily get

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 E 3 is asymptotically stable in the absence of delay.
Theorem 3.2.For system 1.1 , if the condition H 2 A 3 < B 3 holds, the positive equilibrium E 3 is conditionally stable.
Proof.Substituting λ iω into 3.8 and separating the real and imaginary parts, one can get

3.12
Squaring and adding 3.12 we get where

3.14
We know that D 3 < 0 provided that the condition H 2 holds.There is at least a positive ω 0 satisfying 3.13 , that is, the characteristic equation 3.8 has a pair of purely imaginary roots of the form ±iω 0 .From 3.12 , we can get the corresponding τ k > 0 such that the characteristic 3.8 has a pair of purely imaginary roots ω 0 , k 0, 1, 2, . . . .

3.17
If the conditions H 2 and H Therefore, the transversality condition holds, hence, the Hopf bifurcation occurs at ω ω 0 and τ τ k .
Theorem 3.3.Suppose that the conditions H 2 and H 3 are satisfied.

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.
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:

4.1
In the following we shall claim that It follows from the first equation of system 1.1 that By comparison, we obtain that Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U 1 ≤ M x 1 , where Hence, for ε > 0 sufficiently small, there is a 10 Discrete Dynamics in Nature and Society We, therefore, derive from the second equation of system 1.1 that, for t > T 1 τ, Hence, by Lemma 2.1, one can get Hence, for ε > 0 sufficiently small, there is a Since H 4 holds, one can see According to Theorem 2.2, we can get lim t → ∞ I t U 3 V 3 0. We derive from the first equation of system 1.1 that, for t > T 2 τ, By comparison we derive that Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V 1 ≥ N x 1 , where Hence, for ε > 0 sufficiently small, there is a T 3 > 0 such that, if T 3 > T 2 τ, x t ≥ N x 1 − ε.We derive from the second equation of system 1.1 that, for t > T 3 , Hence, by Lemma 2.1, one can get Since this is true for arbitrary ε > 0 sufficiently small, we conclude that V 2 ≥ N S 1 , where Hence, for ε > 0 sufficiently small, there is a T 4 > 0 such that, if T 4 > T 3 τ, S t ≥ N S 1 − ε.Again, it follows from the first equation of system 1.2 that, for t > T 4 , A comparison argument yields Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that U 1 ≤ M x 2 , where Hence, for ε > 0 sufficiently small, there is a T 5 > 0 such that, if T 5 > T 4 τ, x t ≤ M x 2 ε.It follows from the second equation of system 1.1 that, for t > T 5 , By Lemma 2.1, one can derive that Since this is true for arbitrary ε > 0 sufficiently small, we conclude that U 2 ≤ M S 2 , where

4.20
Hence, for ε > 0 sufficiently small, there is a T 6 > 0 such that, if T 6 > T 5 τ, S t ≤ M S 2 ε.We derive from the first equation of system 1.1 that, for t > T 6 , By comparison it follows that Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that where Hence, for ε > 0 sufficiently small, there is a T 7 > 0 such that, if T 7 > T 6 τ, x t ≥ N x 2 − ε.We derive from the second equation of system 1.1 that, for t > T 7 , Hence, by Lemma 2.1, one can get Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that V 2 ≥ N S 2 , where

4.26
Hence, for ε > 0 sufficiently small, there is a

4.27
Clearly, we have

4.28
It follows from 4.27 that

4.29
Noting that M x n ≥ S 2 and Kab < rc, we derive from 4.29 that

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

4.33
We derive from 4.33 and the third equation of 4.27 that

4.34
Similarly, one can derive from 4.27 and 4.34 that

4.36
We, therefore, have lim Hence, the disease-free planar equilibrium E 2 is globally asymptotically stable.The proof is complete.
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:

4.38
In the following we claim that x x x 3 , S S S 3 , I I I 3 .
It follows from the first equation of system 1.1 that

4.39
By comparison we obtain Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that x ≤ M x 1 , where Hence, for ε > 0 sufficiently small, there is a .We obtain from the second equation of system 1.1 that, for t > T 1 τ, Hence, by Lemma 2.1, we derive that Since it is true for arbitrary ε > 0 sufficiently small, we conclude that S ≤ M S 1 , where Hence, for ε > 0 sufficiently small, there is a

4.45
Since H 5 holds, one can see Since it is true for arbitrary ε > 0 sufficiently small, we conclude that I ≤ M I 1 , where

4.47
We derive from the first equation of system 1.1 that, for t > T 2 ,

4.48
By comparison we derive that Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that x ≥ N x 1 , where Hence, for ε > 0 sufficiently small, there is a T 3 > 0 such that, if T 3 > T 2 τ, x t ≥ N x 1 − ε.We derive from the second equation of system 1.1 that, for t > T 3 ,

4.51
Hence, by Lemma 2.1 and H 5 , one can get

4.52
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that S ≥ N S 1 , where

4.53
Hence, for ε > 0 sufficiently small, we get S t ≥ N S 1 − ε.It follows from the third equation of system 1.1 that

4.54
Provided that βN S 1 > d, one can see

4.55
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that I ≥ N I 1 , where

4.56
It follows from the first equation of system 1.1 that By comparison we derive that Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that x ≤ M x 2 , where Hence, for ε > 0 sufficiently small, there is a We obtain from the second equation of system 1.1 that, for t > T 4 τ,

4.60
Hence, by Lemma 2.1, one can get

4.61
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that S ≤ M S 2 , where

4.62
Hence, for ε > 0 sufficiently small, there is a Hence, by H 5 , one can see

4.64
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that I ≤ M I 2 , where

4.65
We derive from the first equation of system 1.1 that, for t > T 5 , By comparison we derive that

4.67
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that x ≥ N x 2 , where Hence, for ε > 0 sufficiently small, there is a T 6 > 0 such that, if T 6 > T 5 τ, x t ≥ N x 2 − ε.We derive from the second equation of system 1.1 that, for t > T 6 ,

4.69
By Lemma 2.1, one can get

4.70
Since this inequality holds true for arbitrary ε > 0 sufficiently small, we conclude that S ≥ N S 1 , where

4.71
Hence, for ε > 0 sufficiently small, we get S t ≥ N S 2 − ε.It follows from the third equation of system .Hence, the unique positive equilibrium E 3 is globally asymptotically stable.The proof is complete.
In the following we will present two examples to verify our results obtained earlier.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 E 3 1.7881, 0.2119, 0.0596 of system 1.1 is globally stable, as depicted in Figure 2.

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 H 4 holds, the infected predator population becomes extinct and the disease will be eliminated; that is, only sound predator and prey coexist; 2 if H 5 holds, the prey, the sound predator and the infected predator coexist.The disease will not be eliminated, and the system is permanent.
E 2 Theorem 3.1.The disease-free planar equilibrium E 2 is locally asymptotically stable if βS 2 < d, and the equilibrium E 2 is unstable if βS 2 > d.