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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 154387, 15 pages
http://dx.doi.org/10.1155/2013/154387
Research Article

Dynamical Analysis of SIR Epidemic Models with Distributed Delay

1College of Science, Shandong University of Science and Technology, Qingdao 266590, China
2College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China

Received 16 December 2012; Revised 18 June 2013; Accepted 23 June 2013

Academic Editor: Han H. Choi

Copyright © 2013 Wencai Zhao 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

SIR epidemic models with distributed delay are proposed. Firstly, the dynamical behaviors of the model without vaccination are studied. Using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed. The basic reproduction number is got. In order to study the important role of vaccination to prevent diseases, the model with distributed delay under impulsive vaccination is formulated. And the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model are obtained by using Floquet’s theorem, small-amplitude perturbation skills, and comparison theorem. Lastly, numerical simulation is presented to illustrate our main conclusions that vaccination has significant effects on the dynamical behaviors of the model. The results can provide effective tactic basis for the practical infectious disease prevention.

1. Introduction

Infectious diseases have always been the problem that people have to face. Emerging diseases pose a continual threat to public health such as SARS and avian influenza. It is very necessary to establish and study mathematical models which can reflect the spread of the infectious diseases. A famous model in which the population is partitioned into three classes, the susceptible, infectious and recovered, with sizes denoted by , , and , respectively, that could be used to describe an influenza epidemic was developed early in the 20th century by Kermack and McKendrick [1]. This model known as the susceptible-infectious-recovered (SIR) model is as follows: where denotes the number of members of a population susceptible to the disease, denotes the total population of infectives with some virus at time denotes the number of members who have been removed from the possibility of infection through a temporal immunity. In the model, parameters , and are positive constants, where represents the death rates of susceptibles, infectives, and recovered. is the contact rate and is the recovery rate from the infected compartment. is the recruitment rate of the susceptible population. The SIR infectious disease model is a basic but important biologic model and has been studied by many authors [117]. Many diseases have the incubation period, such as rabies; the incubation period in rabies ranges from about two weeks to several months, and rarely even to years [1821]. Diseases with incubation period always lead to time delay in the epidemic models, so delay differential equation is widely used in the epidemic mathematical model, such as [2, 7, 8, 10, 1216, 22]. Based on model (1), a new epidemic model with distributed delay is as follows: where the function .

As is known to all, one of strategies to control infectious diseases is vaccination. Then a number of epidemic models in ecology can be formulated as dynamical systems of differential equations with vaccination [2326]. Based on ODE, systems with sudden perturbations lead to impulsive differential equations. The theory of impulsive differential equations has been studied intensively and systematically in [2734]. Compared to continuous constant vaccination, pulse vaccination seems more reasonable in the real world. Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as a strategy for the elimination of childhood viral infections such as measles hepatitis, parotitis, smallpox, and phthisis. Under the pulse vaccination strategy (PVS) [9, 11, 17, 28, 3541], what we are interested is how large a fraction of the population should we keep vaccinated in order to prevent the agent from establishing; that is, it is very important for us to investigate the conditions under which a given agent can invade a partially vaccinated population. Thus, we also need to consider the following epidemic model with distributed delay and pulse vaccination strategy at fixed moments, which is more realistic as where (with ) is the proportion of those vaccinated successfully to all of the susceptibles and is a fixed positive constant and denotes the period of the impulsive effect, .

The organization of this paper is as follows. In Section 2, we will show the boundedness of the SIR system and give the transformation of models and some lemmas. In Section 3, we will analyze the local stability of equilibrium of system (7) and give the basic reproduction number. In Section 4.1, we will prove the existence and globally asymptotical stability of the periodic solution of the “infection-free” model. In Section 4.2, we obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. Finally, we give numerical analysis and biological conclusions to show our main results.

2. Transformation of Models and Prerequisites

For system (2), the total population size satisfies and . Hence it is sufficient to consider system (2) with respect to . Note that the variable does not appear in the first and second equations of system (2); hence, we only need to consider the subsystem of (2) as follows: In order to study the system, we can use the chain transformation . Since and is convergent, then Hence, system (4) becomes We understand the relationship between the two systems as follows. If is the solution of system (4) corresponding to continuous and bounded initial function , then is a solution of system (7) with , and . Conversely, if is any solution of system (7) defined on the entire real line and bounded on , then is given by , and so satisfies system (4).

System (7) will be analyzed with the following initial conditions , , .

By the mean value theorem of integrals, there exists , such that We know that and ; hence we get .

By the same transformation, from system (3), we have

Motivated by the application of systems (9) to population dynamics (refer to [27]), we assume that solutions of systems (9) satisfy the initial conditions , and lie in , where is positive, bounded, and continuous function for .

We would like to have the following definitions first.

Definition 1. Let . The map is said to belong to class if (i)is continuous in , and for each , exist, (ii) is locally Lipschitzian in .

Also we have the following lemmas.

Lemma 2 (see [31]). Let , and . Assume that where is continuous in and for each , , exist; is nondecreasing. Let be the maximal (minimal) solution of the scalar impulsive differential equation existing on . Then implies that , where is any solution of (9) existing on .

Lemma 3. Consider the following system: Then system (12) has a unique positive -periodic solution as and for each solution, as .

Proof. Solving the first equation of system (12), we have Substituting into the second equation of (12), we integrate both sides in interval , and we get By using stroboscopic map of difference equation, we have The fixed point of the above mapping is Therefore, we can get the following -periodic solution of system (12): Next, we will prove the attractivity of periodic solutions. Let be an any solution of the system (12); then, for , we have On the one hand, by the recurrence formula, we have Thus and then we have .
On the other hand, and , for large enough we have the following approximate recursive formula: Thus, for any , we get Then we have , thus . The proof is completed.

3. The Stability of Equilibrium of System (7) and the Basic Reproduction Number

In this section, we will consider the local stability of equilibrium of system (7) and give the basic reproduction number. Obviously, the system (7) has a disease-free equilibrium and an endemic equilibrium , where , , . Let ; we have the following theorem.

Theorem 4. System (7) always has a disease-free equilibrium . If and only if , system (7) has an endemic equilibrium .

3.1. Local Stability of Disease-Free Equilibrium

We calculate the Jacobian matrix of system (7) evaluated at ; one gets the following matrix: Obviously, is locally asymptotically stable if which implies that , and unstable if . Then can be used as the basic reproductive number. Thus, we obtain the following result.

Theorem 5. If , then disease-free equilibrium of system (7) is locally asymptotically stable and unstable if .

3.2. Local Stability of Endemic Equilibrium

About the local stability of endemic equilibrium , we have the following theorem.

Theorem 6. If and or , the equilibrium of system (7) is locally asymptotically stable.

Proof. The Jacobian matrix of system (7) evaluated at is Let be its eigenvalues with . After a simple calculation, it follows that For , there are two cases as follows: (i) for ;(ii).
Now we prove that the case (ii) is not true. Note that ; one gets , that is, . If the case (ii) is true, we have that and . The second additive compound matrix [6] of (see the Appendix) is as follows: where , and is used. Notice that , and two cases will happen as follows: (a)if , then ;(b)if , but , then .
According to the property of the second additive compound matrix [6], the eigenvalues of are . Then, we have Notice that and , then we get , which contradicts with case (ii). Therefore, for . So is locally asymptotically stable for and or . This completes the proof.

3.3. Analysis at

In this section, we consider the stability of system (7) under using the center manifold theory, as described in [42, Theorem 4.1]. To apply this method, the following simplification and change of variables are made first. Let , and the system (7) becomes with corresponding to . The virus-free equilibrium is . The linearization matrix of system (7) around the infection-free equilibrium when is The matrix has eigenvalues , which meets the requirement of a simple zero eigenvalue and others having negative real part. A right eigenvector corresponding to the zero eigenvalue is and the left eigenvector satisfying is . For the system (7), we can get Thus, , by item (iv) of Theorem 4.1 in [42]; we can give the following result.

Theorem 7. The disease-free equilibrium for system (7) is locally asymptotically stable for near .

4. Disease Impulsive Control for System (9)

4.1. The Existence and Globally Asymptotical Stability of the “Infection-Free” Periodic Solution of System (9)

We demonstrate the expression of the infectives-free solution of the system (9) firstly, in which the infectives are entirely absent from the population permanently. Consider the infectives-free subsystem of system (9) in the form By Lemma 3, system (33) has a unique positive -periodic solution given by and for each solution, as . Hence, we have Theorem 8.

Theorem 8. The system (9) has an “infection-free” periodic solution for .

In next section, we will prove that “infection-free” periodic solution is globally asymptotically stable.

Theorem 9. Let be any solution of (9), and then is globally asymptotically stable provided , where .

Proof. Firstly, we will prove the local stability. The local stability of periodic solution may be determined by considering the behavior of small amplitude perturbations of the solution. This may be written as where satisfies and , the identity matrix. Hence, the fundamental solution matrix is There is no need to calculate the exact form of as it is not required in the analysis that follows. The linearization of the fourth, fifth, and sixth equations of system (9) becomes The stability of the periodic solution is determined by the eigenvalues of which are According to Floquet theory (see [43]), is locally stable if .
Denote For , we have which leads to . Thus the periodic solution is locally stable.
In the following, we prove the global attractivity. Since holds, we have We can choose a small enough such that Note that , by impulsive differential inequalities, we have for all large enough. For simplification, we may assume that (45) holds for all . From the third equation of system (9) and (45), we get By impulsive differential inequalities, we have where is the periodic solution of the following equation:
From the second and fifth equations of the system (9), we have which leads to Hence and as . Therefore, as , since for .
Next, we prove that as . For all , there must exist a such that for . Without loss of generality, we may assume that for all , and then from system (9), we have Then we have and as , where is the solution of Therefore, for any , there exists a such that . Let ; we have for large enough, which implies as .
Similarly, as can be analyzed by the same method as the above, so we omit it. This completes the proof.

4.2. Permanence of System (9)

Theorem 10. If , then system (9) is permanent, that is, there exist three positive constants ,  , and such that , and for large enough.

Proof. Suppose that the is any positive solution of system (9). From system (9), we can get Consider the following impulsive differential equation: Then, we have , where So we have for large enough. From (56) and the third and sixth equations of the system (9), we have Consider the following system: Obviously, as , Then there exists such that
Next, we prove that there exists a constant such that for large enough. We will do it in the following two steps.
Step (I). Since , we can choose small enough such that We will prove that cannot hold for all . Otherwise, So we have for large enough, where is the solution of From the third and sixth equations of the system (9), we have and , where So we have Integrating (66) on , we have Then as , which is a contradiction to the boundedness of . Hence, there exists a such that . For the sake of simplification, we let be .
Step (II). If for all , and we let , then our aim is obtained. Otherwise, let there are two possible case for .
Case (I). . Then for and . Choose , such that Let , we claim that there exists a such that . Otherwise, consider (62) with for and . We have for . So as in the above step (I), we have From system (9), we get for . Integrating (71) on , we have Thus, we have which is a contradiction.
Let , and then for and , since is continuous and when . For , suppose ; from (71), we have Let , hence; we have for . For , the same arguments can be continued since .
Case (II). . Then for and ; suppose . There are two possible cases for .
Case (IIa). for all . We claim that there must be a such that . Otherwise, consider (62) with ; we have for and . By a similar argument as in step II case (I), we get Since for , (71) holds on , so we have Thus which is a contradiction. Let and then for and . For , suppose , we have Let , so for . For , the same arguments can be continued since .
Case (IIb). There exists a such that . Let and then for and . For , integrating (71) on , we have So, for . Since , for , the same arguments can be continued.
Hence, for all . The proof is completed.

5. Numerical Analysis and Conclusion

To verify the theoretical results obtained in this paper, we will give some numerical simulations. We consider the hypothetical set of parameter values as , , , , and with . By calculation, we know that , and according to Theorem 5, and we know that the disease-free equilibrium of system (7) is locally asymptotically stable for this case (see Figure 1). We set the hypothetical set of parameter values as , , , , with . By calculation, we know that , and according to Theorem 6, we know that the endemic equilibrium of system (7) is locally asymptotically stable for this case (see Figure 2).

fig1
Figure 1: Illustration of basic behavior of system (7) on the threshold values , where , the initial value is , and .
fig2
Figure 2: Illustration of basic behavior of system (7) on the threshold values , where , the initial value is , and .

On the other hand, we consider the hypothetical set of parameter values as , and with . If , by calculation, we know that . According to Theorem 9, we know that the “infection-free” periodic solution of system (9) is globally asymptotically stable for this case (see Figure 3). We can explain this in the epidemiology that if we take such a strategy by improving vaccination proportion of susceptible persons in practice, as a result, the infectious population vanishes; that is, diseases eliminate. Figures 3(b), 3(c), 3(d), and 3(e) show the “infection-free” periodic solution under . In contrast, if we decrease vaccination proportion of susceptible persons to , the diseases will be permanent (in this case .) (see Figure 4). Comparing Figure 1 with Figure 3 and Figure 2 with Figure 4, we find that the impulse vaccination proportion of susceptible persons has played a very important role in the actual epidemic prevention.

fig3
Figure 3: Illustration of basic behavior of system (9) on the threshold values , where , and .
fig4
Figure 4: Illustration of basic behavior of system (9) on the threshold values , where , and .

In this paper, a SIR epidemic model with distributed delay is proposed. The dynamics behavior of the model without vaccination or under impulsive vaccination is studied, respectively. By using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed and by using Floquet's theorem, small-amplitude perturbation skills and comparison theorem the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model under impulsive vaccination are obtained. Lastly, we give some numerical simulation to illustrate our main conclusions. We think our mathematical results would be helpful in diseases control.

Appendix

A. The Second Additive Compound Matrix

Let be a matrix. Then its second additive compound matrix is as follows:

Proposition 5. Let be the spectrum of . Then the spectrum of is .

Acknowledgments

The authors would like to thank the referees and the editor for their careful reading of the paper and many valuable comments and suggestions that greatly improved the presentation of this paper. This work is supported by Shandong Provincial Natural Science Foundation, China (no. ZR2012AM012), a Project of Shandong Province Higher Educational Science and Technology Program, China (no. J13LI05), and the SDUST Research Fund (no. 2011KYTD105).

References

  1. W. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics,” Proceedings of the Royal Society A, vol. 115, pp. 700–721, 1927.
  2. E. Beretta and Y. Takeuchi, “Global stability of an SIR epidemic model with time delays,” Journal of Mathematical Biology, vol. 33, no. 3, pp. 250–260, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. E. Beretta and Y. Takeuchi, “Convergence results in SIR epidemic models with varying population sizes,” Nonlinear Analysi: Theory, Methods & Applications, vol. 28, no. 12, pp. 1909–1921, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. K. L. Cooke, “Stability analysis for a vector disease model,” The Rocky Mountain Journal of Mathematics, vol. 9, no. 1, pp. 31–42, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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 · View at MathSciNet
  6. M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, “Global dynamics of a SEIR model with varying total population size,” Mathematical Biosciences, vol. 160, no. 2, pp. 191–213, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. W. B. Ma, Y. Takeuchi, T. Hara, and E. Beretta, “Permanence of an SIR epidemic model with distributed time delays,” The Tohoku Mathematical Journal, vol. 54, no. 4, pp. 581–591, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. B. Ma, M. Song, and Y. Takeuchi, “Global stability of an SIR epidemic model with time delay,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1141–1145, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. P. Yongzhen, L. Shuping, L. Changguo, and S. Z. Chen, “The effect of constant and pulse vaccination on an SIR epidemic model with infectious period,” Applied Mathematical Modelling, vol. 35, no. 8, pp. 3866–3878, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Song, W. B. Ma, and Y. Takeuchi, “Permanence of a delayed SIR epidemic model with density dependent birth rate,” Journal of Computational and Applied Mathematics, vol. 201, no. 2, pp. 389–394, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. B. Shulgin, L. Stone, and Z. Agur, “Pulse vaccination strategy in the SIR epidemic model,” Bulletin of Mathematical Biology, vol. 60, no. 6, pp. 1123–1148, 1998. View at Scopus
  12. Y. Takeuchi, W. B. Ma, and E. Beretta, “Global asymptotic properties of a delay SIR epidemic model with finite incubation times,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, pp. 931–947, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Y. N. Xiao, L. S. 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
  14. H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714–1726, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. T. Zhang and Z. Teng, “Global behavior and permanence of SIRS epidemic model with time delay,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1409–1424, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. T. L. Zhang and Z. D. Teng, “Permanence and extinction for a nonautonomous SIRS epidemic model with time delay,” Applied Mathematical Modelling, vol. 33, no. 2, pp. 1058–1071, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. T. Q. Zhang, X. Z. Meng, T. H. Zhang, and Y. Song, “Global dynamics for a new high-dimensional SIR model with distributed delay,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11806–11819, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  18. K. M. Charlton, S. Nadin-Davis, G. A. Casey, and A. I. Wandeler, “The long incubation period in rabies: delayed progression of infection in muscle at the site of exposure,” Acta Neuropathologica, vol. 94, no. 1, pp. 73–77, 1997. View at Publisher · View at Google Scholar · View at Scopus
  19. D. B. Fishbein, “Rabies in humans,” in The Natural History of Rabies, G. M. Baer, Ed., pp. 519–549, CRC Press, Boca Raton, Fla, USA, 2nd edition, 1991.
  20. M. Fekadu, “Canine rabies,” in The Natural History of Rabies, G. M. Baer, Ed., pp. 267–278, CRC Press, Boca Raton, Fla, USA, 2nd edition, 1992.
  21. P. J. Grattan-Smith, W. J. O'Regan, P. S. J. Ellis, S. J. O'Flaherty, P. B. McIntyre, and C. J. Barnes, “Rabies: a second Australian case, with a long incubation period,” Medical Journal of Australia, vol. 156, no. 9, pp. 651–654, 1992. View at Scopus
  22. H.-F. Huo and Z.-P. Ma, “Dynamics of a delayed epidemic model with non-monotonic incidence rate,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 2, pp. 459–468, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. C. A. de Quadros, J. K. Andrus, J.-M. Olive et al., “Eradication of poliomyelitis: progress in the Americas,” Pediatric Infectious Disease Journal, vol. 10, no. 3, pp. 222–229, 1991. View at Publisher · View at Google Scholar · View at Scopus
  24. M. de la Sen, A. Ibeas, and S. Alonso-Quesada, “On vaccination controls for the SEIR epidemic model,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2637–2658, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Ramsay, N. Gay, E. Miller et al., “The epidemiology of measles in England and Wales: rationale for the 1994 national vaccination campaign,” Communicable Disease Report, vol. 4, no. 12, pp. R141–R146, 1994. View at Scopus
  26. A. B. Sabin, “Measles, killer of millions in developing countries: strategy for rapid elimination and continuing control,” European Journal of Epidemiology, vol. 7, no. 1, pp. 1–22, 1991. View at Scopus
  27. D. Bainov and P. Simeonov, System With Impulsive Effect: Stability, Theory and Applications, John Wiley & Sons, New York, NY, USA, 1989.
  28. S. J. Gao, L. S. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,” Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006. View at Publisher · View at Google Scholar · View at Scopus
  29. B. Liu, Z. D. Teng, and W. B. Liu, “Dynamic behaviors of the periodic Lotka-Volterra competing system with impulsive perturbations,” Chaos, Solitons and Fractals, vol. 31, no. 2, pp. 356–370, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Z. X. Li, Z. Zhao, and L. S. Chen, “Bifurcation of a three molecular saturated reaction with impulsive input,” Nonlinear Analysis: Real World Applications, vol. 12, no. 4, pp. 2016–2030, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore,, 1989. View at MathSciNet
  32. J. J. Nieto and R. Rodríguez-López, “Hybrid metric dynamical systems with impulses,” Nonlinear Analysis: Theory, Methods & Applicationss, vol. 64, no. 2, pp. 368–380, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. J. J. Nieto and R. Rodríguez-López, “New comparison results for impulsive integro-differential equations and applications,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1343–1368, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. X. Y. Song and Y. F. Li, “Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect,” Nonlinear Analysis: Real World Applications, vol. 9, no. 1, pp. 64–79, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. S. J. Gao, Z. D. Teng, and D. H. Xie, “Analysis of a delayed SIR epidemic model with pulse vaccination,” Chaos, Solitons & Fractals, vol. 40, no. 2, pp. 1004–1011, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. J. Hui and L.-S. Chen, “Impulsive vaccination of SIR epidemic models with nonlinear incidence rates,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 3, pp. 595–605, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. Z. H. Lu, X. B. Chi, and L. S. Chen, “The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission,” Mathematical and Computer Modelling, vol. 36, no. 9-10, pp. 1039–1057, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. X. Z. Meng, L. S. Chen, and H. D. Cheng, “Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 516–529, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. X. Z. Meng and L. S. Chen, “The dynamics of a new SIR epidemic model concerning pulse vaccination strategy,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 582–597, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. A. D'Onofrio, “On pulse vaccination strategy in the SIR epidemic model with vertical transmission,” Applied Mathematics Letters, vol. 18, no. 7, pp. 729–732, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. X. Y. Song, Y. Jiang, and H. M. Wei, “Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays,” Applied Mathematics and Computation, vol. 214, no. 2, pp. 381–390, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. C. Castillo-Chavez and B. J. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 361–404, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, 1993. View at MathSciNet