Global stability of a delayed mosquito-transmitted disease model with stage structure

Electronic Journal of Differential Equations

Volumn 2015, Issue , 2015, Pages 1-19

  Sampath Aruna Pradeep, B G ^a,b   Ma, Wanbiao ^a  

^a UNIVERSITY OF SCIENCE AND TECHNOLOGY BEIJING   (China)

^b UNIVERSITY OF RUHUNA   (Sri Lanka)

Author keywords

Epidemic model; Global stability; Lyapunov functional; Nonlinear response; Time delay

Indexed keywords

EID: 84920547721     PISSN: None     EISSN: 10726691     Source Type: Journal    
DOI: None     Document Type: Article

References (48)

1. A. Abta, A. Kaddar, H. T. Alaoui; Global stability for delay SIR and SEIR epidemic models with saturated incidence rates, Electron. J. Differ. Eq. 2012 (23) (2012), 1-13.
2. W. G. Aiello, H. I. Freedman; A time delay model of single-species growth with stage structure, Math. Biosci. 101 (2) (1990), 139-153.
3. R. M. Anderson, R. M. May; Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1998.
4. S. Busenberg, K. Cook; Vertically Transmitted Diseases, Models and Dynamics, Biomathe- matics 23, Springer-Verlag, Berlin, 1993.
5. L. Chen, X. Song, Z. Lu; Mathematical Models and Methods in Ecology, Sichuan Sci. Tech. Press, Chengdu, 2003.
6. J. Cui, X. Song; Permanence of predator-prey system with stage structure, Discrete Contin. Dyn. Syst. Ser. B. 4 (3) (2004), 547-554.
7. C. Deng, H. Gao; Stability of SVIR system with random perturbations, Int. J. Biomath. 5 (4) (2012), 1250025.
8. M. Derouich, A. Boutayeb; Dengue fever: Mathematical modelling and computer simulation, Appl. Math. Comput. 177 (2) (2006), 528-544.
9. Y. Enatsu, Y. Nakata, Y. Muroya; Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete Contin. Dyn. Syst. Ser. B 15 (1) (2011), 61-74.
10. Y. Enatsu, Y. Nakata, Y. Muroya; Global Stability of SIRS epidemic models with a class of nonlinear incidencerates and distributed delays, Acta Math. Sci. 32B (3) (2012), 851-865.
11. L. Esteva, C. Vargas; Analysis of a dengue disease transmission model, Math. Biosci. 150 (2) (1998), 131-151.
12. L. Esteva, C. Vargas; Infuence of vertical and mechanical transmission on the dynamics of dengue disease, Math. Biosci. 167 (1) (2000), 51-64.
13. L. Esteva, H. M. Yang; Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci. 198 (2) (2005), 132-147.
14. C. Favier, D. Schmit, C. M. Graf, B. Cazelles, N. Degallier, B. Mondet, M. Dubois; Inuence of spatial heterogeneity on an emerging infectious disease: the case of dengue epidemics, Proc. R. Soc. B 272 (1568) (2005), 1171-1177.
15. Z. Feng, J. X. Velasco-Hernandez; Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol. 35 (6) (1997), 523-544.
16. L. Han, Z. Ma, H. W. Hethcote; Four predator prey models with infectious diseases, Math. Comput. Modelling 34 (7-8) (2001), 849-858.
17. Z. Hu, W. Ma, S. Ruan; Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci. 238 (1) (2012), 12-20.
18. G. Huang, Y. Takeuchi and W. Ma; Lyapunov functionals for delay di_erential equations model of viral infections, SIAM J. Appl. Math. 70 (7) (2010), 2693-2708.
19. G. Huang, Y. Takeuchi, W. Ma, D. Wei; Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol. 72 (5) (2010), 1192-1207.
20. C. C. Jansen, N. W. Beebe; The dengue vector Aedes aegypti: what comes next, Microb. Infect. 12 (4) (2010), 272-279.
21. J. Jia, Q. Li; Qualitative analysis of an SIR epidemic model with stage structure, Appl. Math. Comput. 193 (1) (2007), 106-115.
22. Z. Juan, J. Li, Z. Ma; Global dynamics of an SEIR epidemic model with immigration of di_erent compartments, Acta Math. Sci. 26B (3) (2006), 551-567.
23. W. O. Kermack, A. G. McKendrick; A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A 115 (772) (1927), 700-721.
24. Y. Kuang; Delay Differential Equations with Applications in Population Dynamics, Aca- demic Press, New York, 1993.
25. V. Lakshmikantham, S. Leela; Differential and Integral Inequalities, Vol.1-2, Academic Press, New York, 1969.
26. F. Li, H. Li; Holf bifurcation of a predator-prey model with time delay and stage structure for the prey, Math. Comput. Modelling. 55 (3-4) (2012), 672-679.
27. W. Ma, Y. Takeuchi, T. Hara, E. Beretta; Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54 (4) (2002), 581-591.
28. Z. Ma, Y. Zhou, W. Wang, Z. Jin; Mathematical Models and Dynamics of Infectious Diseases, Science Press, Beijing, 2004.
29. Z. Ma, Z. Li, S. Wang, T. Li, F. Zhang; Permanence of a predator-prey system with stage structure and time delay, Appl. Math. Comput. 201 (1-2) (2008), 65-71.
30. S. Murrell, S. Wu, M. Butler; Review of dengue virus and the development of a vaccine, Biotech. Adv. 29 (2) (2011), 239-247.
31. B. G. S. A. Pradeep, W. Ma; Stability properties of a delayed HIV dynamics model with Beddington-Deangelis functional response and absorption effect, Dyn. Contin. Discret. I. Ser. A 21 (5) (2014), 421-434.
32. H. L. Smith; An Introduction to Delay Di_erential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, Springer, 2011.
33. X. Song, L. Chen; Modelling and analysis of a single-species system with stage structure and harvesting, Math. Comput. Modelling. 36 (1-2) (2002), 67-82.
34. X. Song, H. Guo; Globally stability of a stage-structered predator-prey system, Int. J. Biomath. 1 (3) (2008), 313-326.
35. X. Song, M. Hao, X. Meng; A stage structured predator prey model with disturbing pulse and time delays, Appl. Math. Modelling. 33 (1) (2009), 211-223.
36. L. T. Takahashi, N. A. Maidana, W. C. Ferreira Jr, P. Pulino, H. M. Yang; Mathematical models for the Aedes aegypi dispersal dynamics: travelling waves by wing and wind, Bull. Math. Biol. 67 (3) (2005), 509-528.
37. J. Wang, Q. Lu, Z. Feng; A nonautonomous predator-prey system with stage structure and double time delays, J. Comput. Appl. Math. 230 (1) (2009), 283-299.
38. J. Wang, Y. Takeuchi, S. Liu; A multi group epidemic model with distributed delay and vaccination, Int. J. Biomath. 5 (3) (2012), 1260001.
39. H. Wei, X. Li, M. Martcheva; An epidemic model of a vector-borne disease with direct trans- mission and time delay, J. Math. Anal. Appl. 342 (2) (2008), 895-908.
40. M. J. Wonham, T. de-Camino-Beck, M. Lewis; An epidemiological model for West Nile virus: invation analysis and control applications, Proc. Roy. Soc. Landon, Ser. B 271 (1538) (2004), 501-507.
41. R. Xu; Global dynamics of a predator-prey model with time delay and stage structure for the prey, Nonl. Anal. RWA. 12 (4) (2011), 2151-2162.
42. R. Xu, Z. Ma; Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Soliton. Fract. 41 (5) (2009), 2319-2325.
43. C. Yu, J. Wei, X. Zou; Bifurcation analysis in an age-structured model of a single species livingin two identical patches, Appl. Math. Modelling. 34 (4) (2010), 1068-1077.
44. L. Zha, J. Cui, X. Zhou; Ratio-dependent predator-prey model with stage structure and time delay, Int. J. Biomath. 5 (4) (2012), 1250014.
45. X. Zhang, X. Liu; Backward bifurcation of an epidemic model with saturated treatment func- tion, J. Math. Anal. Appl. 348 (1) (2008), 433-443.
46. X. Zhang, X. Liu; Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonl. Anal. RWA. 10 (2) (2009), 565-575.
47. X. Zhao; Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
48. X. Zhou, J. Cui; Stability and Holf bifurcation analysis of an eco-epidemiological model with delay, J. Franklin Inst. 347 (9) (2010), 1654-1680.