Dynamical analysis of SIR epidemic models with distributed delay

Journal of Applied Mathematics

Volumn 2013, Issue , 2013, Pages

  Zhao, Wencai ^a   Zhang, Tongqian ^a   Chang, Zhengbo ^a   Meng, Xinzhu ^a   Liu, Yulin ^a  

^a SHANDONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84881398730     PISSN: 1110757X     EISSN: 16870042     Source Type: Journal    
DOI: 10.1155/2013/154387     Document Type: Article

References (43)

1. Kermack W., McKendrick A. G., Contributions to the mathematical theory of epidemics. Proceedings of the Royal Society A 1927 115 700 721
2. Beretta E., Takeuchi Y., Global stability of an SIR epidemic model with time delays. Journal of Mathematical Biology 1995 33 3 250 260 10.1007/BF00169563 MR1331508 ZBL0811.92019
3. Beretta E., Takeuchi Y., Convergence results in SIR epidemic models with varying population sizes. Nonlinear Analysi: Theory, Methods & Applications 1997 28 12 1909 1921 10.1016/S0362-546X(96)00035-1 MR1436361 ZBL0879.34054 (Pubitemid 127417735)
4. Cooke K. L., Stability analysis for a vector disease model. The Rocky Mountain Journal of Mathematics 1979 9 1 31 42 10.1216/RMJ-1979-9-1-31 MR517971 ZBL0423.92029
5. Liu W. M., Levin S. A., Iwasa Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. Journal of Mathematical Biology 1986 23 2 187 204 10.1007/BF00276956 MR829132 ZBL0582.92023 (Pubitemid 16153721)
6. Li M. Y., Graef J. R., Wang L., Karsai J., Global dynamics of a SEIR model with varying total population size. Mathematical Biosciences 1999 160 2 191 213 10.1016/S0025-5564(99)00030-9 MR1706068 ZBL0974.92029 (Pubitemid 29398860)
7. Ma W. B., Takeuchi Y., Hara T., Beretta E., Permanence of an SIR epidemic model with distributed time delays. The Tohoku Mathematical Journal 2002 54 4 581 591 MR1936269 10.2748/tmj/1113247650 ZBL1014.92033 (Pubitemid 36177504)
8. Ma W. B., Song M., Takeuchi Y., Global stability of an SIR epidemic model with time delay. Applied Mathematics Letters 2004 17 10 1141 1145 10.1016/j.aml.2003.11.005 MR2091848 ZBL1071.34082
9. Yongzhen P., Shuping L., Changguo L., Chen S. Z., The effect of constant and pulse vaccination on an SIR epidemic model with infectious period. Applied Mathematical Modelling 2011 35 8 3866 3878 10.1016/j.apm.2011.02.012 MR2793671 ZBL1221.37206
10. Song M., Ma W. B., Takeuchi Y., Permanence of a delayed SIR epidemic model with density dependent birth rate. Journal of Computational and Applied Mathematics 2007 201 2 389 394 10.1016/j.cam.2005.12.039 MR2296865 ZBL1117.34310 (Pubitemid 46176492)
11. Shulgin B., Stone L., Agur Z., Pulse vaccination strategy in the SIR epidemic model. Bulletin of Mathematical Biology 1998 60 6 1123 1148 2-s2.0-0032213158 (Pubitemid 28559961)
12. Takeuchi Y., Ma W. B., Beretta E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Analysis: Theory, Methods & Applications 2000 42 931 947 10.1016/S0362-546X(99)00138-8 MR1780445 ZBL0967.34070
13. Xiao Y. N., Chen L. S., ven den Bosch F., Dynamical behavior for a stage-structured SIR infectious disease model. Nonlinear Analysis: Real World Applications 2002 3 2 175 190 10.1016/S1468-1218(01)00021-9 MR1893972 ZBL1007.92032 (Pubitemid 40030723)
14. Zhang H., Chen L., Nieto J. J., A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Analysis: Real World Applications 2008 9 4 1714 1726 10.1016/j.nonrwa.2007.05.004 MR2422575 ZBL1154.34394 (Pubitemid 351685810)
15. Zhang T., Teng Z., Global behavior and permanence of SIRS epidemic model with time delay. Nonlinear Analysis: Real World Applications 2008 9 4 1409 1424 10.1016/j.nonrwa.2007.03.010 MR2422552 ZBL1154.34390 (Pubitemid 351685787)
16. Zhang T. L., Teng Z. D., Permanence and extinction for a nonautonomous SIRS epidemic model with time delay. Applied Mathematical Modelling 2009 33 2 1058 1071 10.1016/j.apm.2007.12.020 MR2468505 ZBL1168.34358
17. Zhang T. Q., Meng X. Z., Zhang T. H., Song Y., Global dynamics for a new high-dimensional SIR model with distributed delay. Applied Mathematics and Computation 2012 218 24 11806 11819 10.1016/j.amc.2012.04.079 MR2945184
18. Charlton K. M., Nadin-Davis S., Casey G. A., Wandeler A. I., The long incubation period in rabies: delayed progression of infection in muscle at the site of exposure. Acta Neuropathologica 1997 94 1 73 77 2-s2.0-0030742823 10.1007/s004010050674 (Pubitemid 27289105)
19. Fishbein D. B., Baer G. M., Rabies in humans. The Natural History of Rabies 1991 2nd Boca Raton, Fla, USA CRC Press 519 549
20. Fekadu M., Baer G. M., Canine rabies. The Natural History of Rabies 1992 2nd Boca Raton, Fla, USA CRC Press 267 278
21. Grattan-Smith P. J., O'Regan W. J., Ellis P. S. J., O'Flaherty S. J., McIntyre P. B., Barnes C. J., Rabies: a second Australian case, with a long incubation period. Medical Journal of Australia 1992 156 9 651 654 2-s2.0-0026507136
22. Huo H.-F., Ma Z.-P., Dynamics of a delayed epidemic model with non-monotonic incidence rate. Communications in Nonlinear Science and Numerical Simulation 2010 15 2 459 468 10.1016/j.cnsns.2009.04.018 MR2572576 ZBL1221.34197
23. de Quadros C. A., Andrus J. K., Olive J.-M., da Silveira C. M., Eikhof R. M., Carrasco P., Fitzsimmons J. W., Pinheiro F. P., Eradication of poliomyelitis: progress in the Americas. Pediatric Infectious Disease Journal 1991 10 3 222 229 2-s2.0-0025905391 10.1097/00006454-199103000-00011
24. de la Sen M., Ibeas A., Alonso-Quesada S., On vaccination controls for the SEIR epidemic model. Communications in Nonlinear Science and Numerical Simulation 2012 17 6 2637 2658 10.1016/j.cnsns.2011.10.012 MR2877709 ZBL1243.92040
25. Ramsay M., Gay N., Miller E., Rush M., White J., Morgan-Capner P., Brown D., The epidemiology of measles in England and Wales: rationale for the 1994 national vaccination campaign. Communicable Disease Report 1994 4 12 R141 R146 2-s2.0-0028771291
26. Sabin A. B., Measles, killer of millions in developing countries: strategy for rapid elimination and continuing control. European Journal of Epidemiology 1991 7 1 1 22 2-s2.0-0026035347
27. Bainov D., Simeonov P., System With Impulsive Effect: Stability, Theory and Applications 1989 New York, NY, USA John Wiley & Sons
28. Gao S. J., Chen L. S., Nieto J. J., Torres A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 2006 24 35-36 6037 6045 2-s2.0-33746833233 10.1016/j.vaccine.2006.05.018 (Pubitemid 44175617)
29. Liu B., Teng Z. D., Liu W. B., Dynamic behaviors of the periodic Lotka-Volterra competing system with impulsive perturbations. Chaos, Solitons and Fractals 2007 31 2 356 370 10.1016/j.chaos.2005.09.059 MR2259761 ZBL1145.34029 (Pubitemid 44067459)
30. Li Z. X., Zhao Z., Chen L. S., Bifurcation of a three molecular saturated reaction with impulsive input. Nonlinear Analysis: Real World Applications 2011 12 4 2016 2030 10.1016/j.nonrwa.2010.12.018 MR2800997 ZBL1223.34072
31. Lakshmikantham V., Banov D. D., Simeonov P. S., Theory of Impulsive Differential Equations 1989 Singapore, World Scientific MR1082551
32. Nieto J. J., Rodríguez-López R., Hybrid metric dynamical systems with impulses. Nonlinear Analysis: Theory, Methods & Applicationss 2006 64 2 368 380 10.1016/j.na.2005.05.068 MR2188464 ZBL1094.34007 (Pubitemid 41705775)
33. Nieto J. J., Rodríguez-López R., New comparison results for impulsive integro-differential equations and applications. Journal of Mathematical Analysis and Applications 2007 328 2 1343 1368 10.1016/j.jmaa.2006. 06.029 MR2290058 ZBL1113.45007
34. Song X. Y., Li Y. F., Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect. Nonlinear Analysis: Real World Applications 2008 9 1 64 79 10.1016/j.nonrwa. 2006.09.004 MR2370163 ZBL1142.34031 (Pubitemid 47562887)
35. Gao S. J., Teng Z. D., Xie D. H., Analysis of a delayed SIR epidemic model with pulse vaccination. Chaos, Solitons & Fractals 2009 40 2 1004 1011 10.1016/j.chaos.2007.08.056 MR2527842 ZBL1197.34123
36. Hui J., Chen L.-S., Impulsive vaccination of SIR epidemic models with nonlinear incidence rates. Discrete and Continuous Dynamical Systems B 2004 4 3 595 605 10.3934/dcdsb.2004.4.595 MR2073963 ZBL1100.92040 (Pubitemid 38782413)
37. Lu Z. H., Chi X. B., Chen L. S., The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Mathematical and Computer Modelling 2002 36 9-10 1039 1057 10.1016/S0895- 7177(02)00257-1 MR1945059 ZBL1023.92026
38. Meng X. Z., Chen L. S., Cheng H. D., Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination. Applied Mathematics and Computation 2007 186 1 516 529 10.1016/j.amc.2006.07.124 MR2314511 ZBL1111.92049 (Pubitemid 46435636)
39. Meng X. Z., Chen L. S., The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. Applied Mathematics and Computation 2008 197 2 582 597 10.1016/j.amc.2007.07.083 MR2400680 ZBL1131.92056 (Pubitemid 351273773)
40. D'Onofrio A., On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Applied Mathematics Letters 2005 18 7 729 732 10.1016/j.aml.2004.05.012 MR2144719 ZBL1064.92041 (Pubitemid 40612586)
41. Song X. Y., Jiang Y., Wei H. M., Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays. Applied Mathematics and Computation 2009 214 2 381 390 10.1016/j.amc.2009.04.005 MR2541674 ZBL1168.92326
42. Castillo-Chavez C., Song B. J., Dynamical models of tuberculosis and their applications. Mathematical Biosciences and Engineering 2004 1 2 361 404 10.3934/mbe.2004.1.361 MR2130673 ZBL1060.92041
43. Banov D., Simeonov P., Impulsive Differential Equations: Periodic Solutions and Applications 1993 66 Pitman Monographs and Surveys in Pure and Applied Mathematics MR1266625