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Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 979217, 17 pages
http://dx.doi.org/10.1155/2009/979217
Research Article

Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

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

Received 28 April 2009; Accepted 21 August 2009

Academic Editor: Antonia Vecchio

Copyright © 2009 Xiaohong Tian and Rui Xu. 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 the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

1. Introduction

Let denote the number of members of a population susceptible to a disease, the number of infective members, and the number of members who have been removed from the possibility of infection through full immunity, a standard SIR compartmental model is of the form [1]

where the parameters are positive constants in which is the recruitment rate of susceptible population, represents the natural death rate of the population, is the disease-induced death rate of the infectives, and is the recovery rate from the infected compartment. It is assumed further that susceptibles become infectious by contact with infectious individuals. Later they may recover and join the group of immune (or dead) individuals. Based on the previous idea, different types of SIR epidemic models have been investigated (see, e.g., [26]). We note that most of the previous works assume that each species has the same contact and recovery rates ignoring the effect of stage structure. In the real world, any species has a process of growth and development, such as from immature to mature, and growth at various stages of life history showed differences in physiology. In the recent years, there have been a fair amount of work on epidemiological models with stage structure (see, e.g., [710]). In fact, the spread of disease is related to the species stage structure. Some diseases, such as measles, mumps, chickenpox and scarlet fever, only spread or have more opportunities to spread in children, and some other diseases, such as diphtheria, leptospirosis, a variety of sexually transmitted diseases, may spread in adult. By assuming that the mature population does not contract the disease and the immature population is susceptible to the infection in [9], Xiao et al. proposed an SIR disease transmission model with stage structure and bilinear incidence rate as follows:

where denote the densities of the immature population that are susceptible, infectious population and recovered population with immunity, respectively; denotes the density of the mature population which does not contract the disease. The parameters are positive constants. is the birth rate of the immature population. It is assumed that newborn individuals are the recovered population with immunity with probability and are susceptible population with probability . is the rate that the susceptible population become infective, and is the rate that the infective population becomes recovered with immunity. are the death rates of the susceptible, infective, recovered population, respectively, and is reasonable for biological meaning. is the death rate of the mature population. Finally, it is assumed that those immatures born at time that survive to time exit from the immature population and enter the mature population. Xiao et al. [9] proved that if the basic reproduction number is less than unity, the disease-free equilibrium of system (1.2) is globally asymptotically stable; if the basic reproduction number is greater than unity, sufficient conditions were derived for the global stability of an endemic equilibrium.

Incidence rate plays a very important role in the research of epidemiological models; it should generally be written as where is the total population size (see [1]). In classical epidemic models, bilinear incidence rate and standard incidence rate are frequently used. The bilinear incidence rate is based on the law of mass action. This contact law is more appropriate for communicable diseases such as influenza., but not for sexually transmitted diseases. For standard incidence rate, it may be a good approximation if the number of available partners is large enough and everybody could not make more contacts than is practically feasible [11]. After a study of the cholera epidemic spread in Bari in 1973, Capasso and Serio [12] introduced a saturated incidence rate into epidemic models, where tends to a saturation level when gets large, that is,

where measures the force of infection 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 susceptible 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 [13].

Motivated by the work of Capasso and Serio [12] and Xiao et al. [9], in this paper, we are concerned with the effect of stage structure and saturation incidence on the dynamic of an SIR epidemic model. To this end, we study the following delayed differential system

The initial conditions for system (1.4) take the form

where

here .

For continuity of initial conditions, we require

It is easy to show that all solutions of system (1.4) with initial conditions (1.5) and (1.7) are defined on and remain positive for all .

The organization of this paper is as follows. In the next section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system (1.4) is discussed. In Section 3, we study the global stability of the disease-free equilibrium and the endemic equilibrium of system (1.4), respectively. Numerical simulations are carried out in Section 4 to illustrate the main theoretical results. A brief discussion is given in Section 5 to conclude this work.

2. Local Stability

In this section, we discuss the local stability of each of nonnegative equilibria of system (1.4) by analyzing the eigenvalues of the corresponding characteristic equations, respectively.

System (1.4) always has a trivial equilibrium , and a disease free equilibrium , where

The basic reproduction number is given as

It is easy to prove that if system (1.4) has an endemic equilibrium , where

The characteristic equation of system (1.4) at the equilibrium is of the form

Obviously, (2.4) always has three negative real roots and . Noting that and must intersect at a positive value of , hence, the equation has a positive real root. Accordingly, is unstable.

The characteristic equation of system (1.4) at the equilibrium takes the form

Obviously, (2.5) always has three real roots and Clearly, if Other roots are given by the roots of equation

Let Now, we claim that the roots of have only negative real parts. Suppose that then it follows from (2.6) that

which leads to a contradiction. Hence, we have . Therefore, if the disease-free equilibrium is locally asymptotically stable. If (2.5) has a positive root, then the disease-free equilibrium is unstable.

The characteristic equation of system (1.4) at the endemic equilibrium takes the form

where

Clearly, (2.8) always has a negative real root . Noting that , roots of equation have only negative real parts. In addition, from the discussion above, we see that roots of the (2.6) have only negative real parts. By the general theory on characteristic equations of delay differential equations from [14], we see that if the endemic equilibrium is locally asymptotically stable.

Based on the discussions above, we have the following result.

Theorem 2.1. For system (1.4), one has the following: (i)if , the endemic equilibrium is locally asymptotically stable, (ii)if , the disease-free equilibrium is locally asymptotically stable.

3. Global Stability

In this section, we discuss the global stability of the disease-free equilibrium and the endemic equilibrium of system (1.4), respectively. The technique of proofs is to use a comparison argument and an iteration scheme.

We first consider the subsystem of (1.4)

Letting system (3.1) becomes

The initial conditions for system (3.2) take the form

Clearly, system (3.2) has a nonnegative equilibrium , where ; when system (3.2) has a positive equilibrium , where

Moreover, from Theorem 2.1, we see that is locally asymptotically stable if , and is locally asymptotically stable if .

To study the global dynamics of system (1.4), we need only to discuss the global behavior of solutions of system (3.2). In the following, we investigate the global asymptotic stability of the equilibria and by using the comparison arguments and the iteration scheme [15], respectively. To this end, we need the following result developed by Song and Chen in [16].

Lemma 3.1. Consider the following equation: where for . One has the following: (i)if , then ; (ii)if , then

Lemma 3.2. Let . If , then is globally asymptotically stable.

Proof. Let be any positive solution of system (3.2) with initial condition (3.3). Let Now we claim that .
From Lemma 3.1, it is easy to show that
Hence, we know that for there exists a such that, if , We derive from the first equation of system (3.2) that By comparison, we have Since this is true for arbitrary sufficiently small, it follows that where Hence, for sufficiently small, there is a such that, if ,
For sufficiently small, we derive from the second equation of system (3.2) that, for ,
Consider the following auxiliary system: By Lemma 3.1 it follows from (3.13) that By comparison, we obtain that Since the inequality is true for arbitrary sufficiently small, it follows that , where Hence, for sufficiently small, there is a such that, if
For sufficiently small, we derive from the first equation of system (3.2) that, for ,
By comparison and by Lemma 3.1, we have Since the inequality is true for arbitrary sufficiently small, it follows that , where Hence, for sufficiently small, there is a such that, if ,
For sufficiently small, we derive from the second equation of system (3.2) that, for ,
By comparison and by Lemma 3.1, we have Since the inequality holds for arbitrary sufficiently small, it follows that , where Therefore, for sufficiently small, there is a such that if
For sufficiently small, we derive from the first equation of system (3.2) that, for ,
By comparison and by Lemma 3.1, we have Since the inequality holds for arbitrary sufficiently small, it follows that , where Hence, for sufficiently small, there is a such that, if
For sufficiently small, we derive from the second equation of system (3.2) that, for ,
By comparison and by Lemma 3.1, we have Since the inequality holds for arbitrary sufficiently small, we conclude that , where Therefore, for sufficiently small, there is a such that, if
For sufficiently small, we derive from the first equation of system (3.2) that, for ,
By comparison and by Lemma 3.1 it follows that Since this is true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if
For sufficiently small, we derive from the second equation of system (3.2) that, for ,
By comparison and by Lemma 3.1 it follows that Since this is true for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if
Continuing this process, we derive four sequences such that for ,
Clearly, we have It follows from (3.36) that Noting that and , we derive from (3.37) that Hence, the sequence is monotonically nonincreasing. Therefore, exists. Taking it follows from (3.37) that We therefore obtain from (3.35) and (3.39) that It follows from (3.36), (3.39), and (3.40) that We therefore have Noting that if and hold, the positive equilibrium is locally asymptotically stable, we conclude that is globally asymptotically stable. The proof is complete.

Theorem 3.3. If and hold, then the endemic equilibrium of system (1.4) is globally asymptotically stable; that is, the disease remains endemic.

Proof. From (3.7), we know that According to the results of Lemma 3.2, we prove that
In the following, we show the existence of .
By Lemma 3.2, it follows from (3.43) that for sufficiently small, there exists a , such that, if ,

Therefore, we derive from the third equation of system (1.4) that, for ,
By comparison, we have Since the inequality holds for arbitrary sufficiently small, we have , where Hence, for sufficiently small, there is a such that, if ,
Again, for sufficiently small, it follows from the third equation of system (1.4) that, for ,
By comparison, we have Since the inequality holds for arbitrary sufficiently small, we conclude that , where Hence, for sufficiently small, there is a such that, if , It follows from (3.48) and (3.52) that Noting that if and hold, the endemic equilibrium is locally asymptotically stable, we see that is globally asymptotically stable. This completes the proof.

Theorem 3.4. If holds, the disease-free equilibrium of system (1.4) is globally asymptotically stable; that is, the disease fades out.

Proof. Choose sufficiently small satisfying From (3.7), we know that for sufficiently small, there exists a such that if We derive from the first equation of system (3.2) that By Lemma 3.1 and by a comparison argument, we get Since this inequality holds for arbitrary sufficiently small, we conclude that Hence, for sufficiently small, there is a such that, for , For sufficiently small satisfying (3.54), it follows from (3.59) and the second equation of system (3.2) that Noting that (3.54) holds, we conclude that Hence, for sufficiently small satisfying (3.54), there is a such that, if .
On the other hand, we derive from the first equation of system (3.2) that, for ,
By Lemma 3.1 and by a comparison argument, we have Since this inequality is true for arbitrary sufficiently small, we conclude that which, together with (3.58), yields According to (3.61) and (3.65), we can easily prove that Using a similar argument as in the proof of Theorem 3.3, we can show that if , then Noting that if , the disease-free equilibrium is locally stable, we conclude that is globally asymptotically stable. This completes the proof.

4. Numerical Examples

In this section, we give two examples to illustrate the main theoretical results above.

Example 4.1. In system (1.4), let Computation gives the following value for the basic reproduction number and system (1.4) has a unique endemic equilibrium . Clearly, By Theorem 3.3, we see that the endemic equilibrium of system (1.4) is globally asymptotically stable. Numerical simulation illustrates the previous result (see Figure 1).

979217.fig.001
Figure 1: The numerical solution of system (1.4) with ; .

Example 4.2. In system (1.4), let Computation gives the following value for the basic reproduction number system (1.4) has only a disease-free equilibrium . By Theorem 3.4, we see that the disease-free equilibrium of system (1.4) is globally asymptotically stable. Numerical simulation illustrates this fact (see Figure 2).

979217.fig.002
Figure 2: The numerical solution of system (1.4) with

5. Discussion

In this paper, we have discussed the effect of stage structure and saturation incidence rate on an SIR epidemic model with time delay. The basic reproduction number was found. The local stability of each of feasible equilibria of system (1.4) was investigated. When the basic reproduction number is greater than unity, by using the iteration scheme, we have established sufficient conditions for the global stability of the endemic equilibrium of system (1.4). By Theorem 3.3, we see that when and , the endemic equilibrium is globally stable. Biologically, these indicate that when the proportionality (infection) constant and /or the birth rate of the immature population is sufficiently large and the death rates of susceptible population and the mature population are sufficiently small such that , then the disease remains endemic. On the other hand, by Theorem 3.4, we see that, if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. Biologically, if the proportionality (infection) constant and /or the birth rate of the immature population is small enough and the death rates of susceptible population and the mature population are large enough such that , then the disease fades out. We would like to point out here that Theorem 3.3 has room for improvement, we leave this for future work.

Acknowledgments

The authors wish to thank the reviewers for their valuable comments and suggestions that greatly improved the presentation of this work. This work was supported by the National Natural Science Foundation of China (No. 10671209) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References

  1. Z. Ma, Y. Zhou, W. Wang, and Z. Jin, Mathematical Modeling and Research of Epidemic Dynamical System, Science Press, Beijing, China, 2004.
  2. E. Beretta, T. Hara, W. Ma, and Y. Takeuchi, “Global asymptotic stability of an SIR epidemic model with distributed time delay,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 6, pp. 4107–4115, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Z. Jin, Z. E. Ma, and S. L. Yuan, “A SIR epidemic model with varying population size,” Journal of Engineering Mathematics, vol. 20, no. 3, pp. 93–98, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. G. Pang and L. Chen, “A delayed SIRS epidemic model with pulse vaccination,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1629–1635, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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 · View at MathSciNet
  6. Y. Takeuchi, W. Ma, and E. Beretta, “Global asymptotic properties of a delay SIR epidemic model with finite incubation times,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 6, pp. 931–947, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. W. Wang, G. Mulone, F. Salemi, and V. Salone, “Permanence and stability of a stage-structured predator-prey model,” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 499–528, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Xiao and L. Chen, “Modeling and analysis of a predator-prey model with disease in the prey,” Mathematical Biosciences, vol. 171, no. 1, pp. 59–82, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. Xiao, L. Chen, and F. ven den Bosch, “Dynamical behavior for a stage-structured SIR infectious disease model,” Nonlinear Analysis: Real World Applications, vol. 3, no. 2, pp. 175–190, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. D. Yuan and B. A. Hu, “A SI epidemic model with two-stage structure,” Acta Mathematicae Applicatae Sinica, vol. 25, no. 2, pp. 193–203, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. T. Zhang and Z. Teng, “Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence,” Chaos, Solitons & Fractals, vol. 37, no. 5, pp. 1456–1468, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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 · View at MathSciNet
  13. D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Mathematical Biosciences, vol. 208, no. 2, pp. 419–429, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  15. R. Xu and Z. Ma, “The effect of dispersal on the permanence of a predator-prey system with time delay,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 354–369, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. X. Song and L. Chen, “Optimal harvesting and stability for a two-species competitive system with stage structure,” Mathematical Biosciences, vol. 170, no. 2, pp. 173–186, 2001. View at Publisher · View at Google Scholar · View at MathSciNet