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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 730783, 22 pages
http://dx.doi.org/10.1155/2011/730783
Research Article

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

Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China

Received 30 June 2011; Accepted 23 August 2011

Academic Editor: Her-Terng Yau

Copyright © 2011 Shuxue Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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., [16]). 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 and 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 into epidemic models with . A general saturation incidence rate was proposed by Liu et al. [14] and used by a number of authors; see, for example, Ruan and Wang [15] , Bhattacharyya and Mukhopadhyay [16] , and so forth. measures the infection force of the disease, and 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 , 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: with initial conditions where , ), the Banach space of continuous functions mapping the interval into , here .

We make the following assumptions for our model (1.1). (A1)The prey population grows logistically with intrinsic growth rate and environmental carrying capacity .(A2)There is a spread of disease in predators which are divided solely into susceptible and infectious population. is the capturing rate of susceptible predators, 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 , with measuring the force of infection and the inhibition effect. (A4) models death rate due to overcrowding, and is the time required for the gestation of susceptible predator. 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: where and for all .(1)If , then .(2)If , then .

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

Proof. Let be the solution of system (1.1) satisfying conditions (1.2). From the first and last equations of system (1.1), we have Hence, and are positive.
We now claim that for all . Otherwise, there exists a such that and for all . Then . 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
From the second equation of system (1.1), for sufficiently large, we have Hence, by Lemma 2.1, one can get
It follows from the third equation of (1.1) and the above inequality, that for sufficiently large, we have Hence, one can see .

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

Theorem 2.4. Suppose that ()where 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 sufficiently large, we have Hence, by Lemma 2.1 and , one can derive that
From the third equation of system (1.1) and, above inequality, we have Since holds, then
Therefore, the above calculations and Theorem 2.2 imply that there exist such that

3. Local Stability

System (1.1) possesses the following equilibria.(1)The trivial equilibrium .(2)The axial equilibrium .(3) The disease-free planar equilibrium , where

(4) The unique positive equilibrium exists if , where

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

The characteristic equation of system (1.1) at the trivial equilibrium is of the form It is easy to see that (3.3) always has a positive root . Hence, is always unstable.

3.2. Stability of Equilibrium

The characteristic equation of system (1.1) at the axial equilibrium is of the form There are two characteristic roots , , and another characteristic root is given by the root of

It is clear that . Hence, is always unstable.

3.3. Stability of Equilibrium

Theorem 3.1. The disease-free planar equilibrium is locally asymptotically stable if , and the equilibrium is unstable if .

Proof. The characteristic equation of system (1.1) at the disease-free planar equilibrium is of the form
Clearly, is a negative eigenvalue. The second eigenvalue is given by the root of Suppose that , then . It is a contradiction, so . The last eigenvalue is . The equilibrium is locally asymptotically stable if , and the equilibrium is unstable if .

3.4. Stability of Equilibrium

The characteristic equation of system (1.1) at the positive equilibrium is of the form where

For , the transcendental (3.8) reduces to the following equation: We can easily get 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 is asymptotically stable in the absence of delay.

Theorem 3.2. For system (1.1), if the condition holds, the positive equilibrium is conditionally stable.

Proof. Substituting into (3.8) and separating the real and imaginary parts, one can get
Squaring and adding (3.12) we get where We know that provided that the condition holds. There is at least a positive satisfying (3.13), that is, the characteristic equation (3.8) has a pair of purely imaginary roots of the form . From (3.12), we can get the corresponding such that the characteristic (3.8) has a pair of purely imaginary roots Let be the roots of (3.8) such that satisfying and . Differentiating the two sides of (3.8) with respect to , we get Therefore, If the conditions and hold, one can see Therefore, the transversality condition holds, hence, the Hopf bifurcation occurs at and .

Theorem 3.3. Suppose that the conditions and are satisfied. (1)The positive equilibrium of system (1.1) is asymptotically stable for all and unstable for . (2)System (1.1) undergoes a Hopf Bifurcation at the positive equilibrium when .

4. Global Stability

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

Theorem 4.1. If () holds, then the disease-free planar equilibrium is globally asymptotically stable.

Proof. Let be any positive solution of system (1.1) with initial conditions (1.2). Let the following hold: 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 sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
We, therefore, derive from the second equation of system (1.1) that, for , Hence, by Lemma 2.1, one can get Hence, for sufficiently small, there is a such that, if , .
It follows from the third equation of system (1.1) that, for , Since holds, one can see According to Theorem 2.2, we can get .
We derive from the first equation of system (1.1) that, for , By comparison we derive that Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
We derive from the second equation of system (1.1) that, for , Hence, by Lemma 2.1, one can get Since this is true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
Again, it follows from the first equation of system (1.2) that, for , A comparison argument yields Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
It follows from the second equation of system (1.1) that, for , By Lemma 2.1, one can derive that Since this is true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
We derive from the first equation of system (1.1) that, for , By comparison it follows that Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
We derive from the second equation of system (1.1) that, for , Hence, by Lemma 2.1, one can get Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
Continuing this process, we get four sequences such that, for , Clearly, we have It follows from (4.27) that Noting that and , we derive from (4.29) that Thus, the sequence is monotonically nonincreasing. Therefore, it follows that exists. Taking , we obtain from (4.29) that Noting that it follows from (4.31) that We derive from (4.33) and the third equation of (4.27) that Similarly, one can derive from (4.27) and (4.34) that It follows from (4.28), (4.33), and (4.35) that We, therefore, have Hence, the disease-free planar equilibrium is globally asymptotically stable. The proof is complete.

Theorem 4.2. If holds, then the positive equilibrium is globally asymptotically stable.

Proof. Let be any positive solution of system (1.1) with initial conditions (1.2). Let the following hold: In the following we claim that .
It follows from the first equation of system (1.1) that By comparison we obtain Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .We obtain from the second equation of system (1.1) that, for , Hence, by Lemma 2.1, we derive that Since it is true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
It follows from the third equation of system (1.1) that Since holds, one can see Since it is true for arbitrary sufficiently small, we conclude that , where
We derive from the first equation of system (1.1) that, for , By comparison we derive that Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
We derive from the second equation of system (1.1) that, for , Hence, by Lemma 2.1 and , one can get Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, we get .
It follows from the third equation of system (1.1) that Provided that , one can see Since this inequality holds true for arbitrary sufficiently small, we conclude that , where
It follows from the first equation of system (1.1) that By comparison we derive that Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
We obtain from the second equation of system (1.1) that, for , Hence, by Lemma 2.1, one can get Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
It follows from the third equation of system (1.1) that Hence, by , one can see Since this inequality holds true for arbitrary sufficiently small, we conclude that , where
We derive from the first equation of system (1.1) that, for , By comparison we derive that Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , .
We derive from the second equation of system (1.1) that, for , By Lemma 2.1, one can get Since this inequality holds true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, we get .
It follows from the third equation of system (1.1) that Since holds, one can see Since this inequality holds true for arbitrary sufficiently small, we conclude that , where
Continuing this process, we obtain six sequences such that, for , A direct calculation shows that By induction, we can show that , . Therefore, the sequence is decreasing, and the sequence is increasing. Clearly, we have Hence, the limits of the sequences and exist. Denote We derive from (4.75) that Since holds, . It, therefore, follows from (4.79) that . Accordingly, we derive from (4.75) that By a simple calculation, we obtain It follows from (4.75) and (4.81) that , . Hence, the unique positive equilibrium 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 . It is easy to show that . By Theorem 4.1 we see that the equilibrium of system (1.1) is globally stable (see Figure 1).

730783.fig.001
Figure 1: The temporal solution found by numerical integration of system (1.1) with ,,,,,,,,, and .

Example 4.4. In system (1.1), we let . It is easy to show that . By Theorem 4.2 we see that the equilibrium of system (1.1) is globally stable, as depicted in Figure 2.

730783.fig.002
Figure 2: The temporal solution found by numerical integration of system (1.1) with ,,,,,,,,, and .

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 holds, the infected predator population becomes extinct and the disease will be eliminated; that is, only sound predator and prey coexist; (2) if 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).

References

  1. K. Kundu and J. Chattopadhyay, “A ratio-dependent eco-epidemiological model of the Salton Sea,” Mathematical Methods in the Applied Sciences, vol. 29, no. 2, pp. 191–207, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. K. P. Das, S. Roy, and J. Chattopadhyay, “Effect of disease-selective predation on prey infected by contact and external sources,” BioSystems, vol. 95, no. 3, pp. 188–199, 2009. View at Publisher · View at Google Scholar
  3. X. Zhou and J. Cui, “Stability and Hopf bifurcation analysis of an eco-epidemiological model with delay,” Journal of the Franklin Institute, vol. 347, no. 9, pp. 1654–1680, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. B. Mukhopadhyay and R. Bhattacharyya, “Role of predator switching in an eco-epidemiological model with disease in the prey,” Ecological Modelling, vol. 220, no. 7, pp. 931–939, 2009. View at Publisher · View at Google Scholar
  5. X. Zhou, X. Shi, and X. Song, “The dynamics of an eco-epidemiological model with distributed delay,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 4, pp. 685–699, 2009. View at Publisher · View at Google Scholar
  6. N. Bairagi, R. R. Sarkar, and J. Chattopadhyay, “Impacts of incubation delay on the dynamics of an eco-epidemiological system—a theoretical study,” Bulletin of Mathematical Biology, vol. 70, no. 7, pp. 2017–2038, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. D. Mukherjee, “Hopf bifurcation in an eco-epidemic model,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2118–2124, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. X. Zhou, X. Shi, and X. Song, “Analysis of a delay prey-predator model with disease in the prey species only,” Journal of the Korean Mathematical Society, vol. 46, no. 4, pp. 713–731, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J.-F. Zhang, W.-T. Li, and X.-P. Yan, “Hopf bifurcation and stability of periodic solutions in a delayed eco-epidemiological system,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 865–876, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Haque and D. Greenhalgh, “A predator-prey model with disease in the prey species only,” Mathematical Methods in the Applied Sciences, vol. 30, no. 8, pp. 911–929, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. L. Han, Z. Ma, and H. W. Hethcote, “Four predator prey models with infectious diseases,” Mathematical and Computer Modelling, vol. 34, no. 7-8, pp. 849–858, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. H. W. Hethcote, W. Wang, L. Han, and Z. Ma, “A predator—prey model with infected prey,” Theoretical Population Biology, vol. 66, no. 3, pp. 259–268, 2004. View at Publisher · View at Google Scholar
  13. V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Mathematical Biosciences, vol. 42, no. 1-2, pp. 43–61, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. W. M. Liu, S. A. Levin, and Y. Iwasa, “Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,” Journal of Mathematical Biology, vol. 23, no. 2, pp. 187–204, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135–163, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. R. Bhattacharyya and B. Mukhopadhyay, “On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3824–3833, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, London, UK, 1993.