Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations

Nonlinear Analysis: Real World Applications

Volumn 14, Issue 3, 2013, Pages 1434-1456

  Yang, Qingshan ^a   Mao, Xuerong ^b  

^a NORTHEAST NORMAL UNIVERSITY   (China)

^b UNIVERSITY OF STRATHCLYDE   (United Kingdom)

Author keywords

Ergodic property; Extinction; Positive recurrence; Stochastic Lyapunov functions

Indexed keywords

ANALYTICAL RESULTS; ASYMPTOTIC BEHAVIORS; ERGODIC PROPERTY; MULTI-GROUP; POSITIVE RECURRENCE; REPRODUCTIVE NUMBER; SEIR EPIDEMIC MODELS; STATIONARY DISTRIBUTION; STOCHASTIC PERTURBATIONS;

COMPUTER SIMULATION; EPIDEMIOLOGY; LIGHT EXTINCTION; LYAPUNOV FUNCTIONS;

STOCHASTIC SYSTEMS;

EID: 84872020070     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2012.10.007     Document Type: Article

References (55)

1. E. Beretta, and Y. Takeuchi Global stability of a SIR epidemic model with time delay J. Math. Biol. 33 1995 250 260
2. E. Beretta, and Y. Takeuchi Convergence results in SIR epidemic model with varying population sizes Nonlinear Anal. 28 1997 1909 1921
3. E. Beretta, T. Hara, W. Ma, and Y. Takeuchi Global asymptotic stability of a SIR epidemic model with distributed time delay Nonlinear Anal. 47 2001 4107 4115
4. D. Bernoulli Essai d'une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l'inoculation pour la prévenir, Mém Math. Phys. Acad. Roy. Sci. Paris 1760 1 45
5. V. Capasso, and G. Serio A generalization of the Kermack-McKendrick deterministic epidemic model Math. Biosci. 42 1978 43 61
6. W.R. Derrick, and P. van den Driessche A disease transmission model in nonconstant population J. Math. Biol. 31 1993 495 512
7. A. D'Onofrio, P. Manfredi, and E. Salinelli Bifurcation thresholds in an SIR model with information-dependent vaccination Math. Model. Nat. Phenom. 2 2007 23 38
8. W.H. Hamer Epidemic disease in England The Lancet 1 1960 733 739
9. H.W. Hethcote, and D.W. Tudor Integral equation models for endemic infectious diseases J. Math. Biol. 9 1980 37 47
10. H.W. Hethcote, M.A. Lewis, and P. van den Driessche An epidemiological model with delay and a nonlinear incidence rate J. Math. Biol. 27 1989 49 64
11. W.O. Kermack, and A.G. Mckendrick A contributions to the mathematical theory of epidemics (part I) Proc. R. Soc. Lond. Ser. A 115 1927 700 721
12. W.M. Liu, S.A. Levin, and Y. Iwasa Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models J. Math. Biol. 23 1986 187 204
13. W.M. Liu, H.W. Hethcote, and S.A. Levin Dynamical behaviour of epidemiological models with nonlinear incidence rates J. Math. Biol. 25 1987 359 380
14. W. Ma, Y. Takeuchi, T. Hara, and E. Beretta Permanence of an SIR epidemic model with distributed time delays Tohoku Math. J. 54 2002 581 591
15. W. Ma, M. Song, and Y. Takeuchi Global stability of an SIR epidemic model with time-delay Appl. Math. Lett. 17 2004 1141 1145
16. R. Ross The Prevention of Malaria second ed. 1911 John Murray London
17. R. Ross An application of the theory of probabilities to the study of a priori pathometry, part I Proc. R. Soc. Lond. A 92 1916 204 230
18. R. Ross, and H.P. Hudson An application of the theory of probabilities to the study of a priori pathometry, part II Proc. R. Soc. Lond. A 93 1917 212 225
19. R. Ross, and H.P. Hudson An application of the theory of probabilities to the study of a priori pathometry, III Proc. R. Soc. Lond. A 93 1917 225 240
20. S. Ruan, and W. Wang Dynamical behaviour of an epidemic model with nonlinear incidence rate J. Differential Equations 188 2003 135 163
21. M. Song, and W. Ma Asymptotic properties of a revised SIR epidemic model with density dependent birth rate and tie delay Dyn. Contin. Discret. Impuls. 13 2006 199 208
22. M. Song, W. Ma, and Y. Takeuchi Permanence of a delayed SIR epidemic model with density dependent birth rate J. Comput. Appl. Math. 201 2007 389 394
23. R. Xu, and Z. Ma Global stability of a delayed SEIRS epidemic model with saturation incidence rate Nonlinear Dynam. 61 2010 229 239
24. R.M. Anderson, and R.M. May Infectious Diseases of Humans: Dynamics and Control 1998 Oxford University Press Oxford
25. N.T.J. Bailey The Mathematical Theory of Infectious Diseases 1975 Griffin London
26. J.D. Murray Mathematical Biology 1993 Springer-Verlag New York
27. E. Beretta, and V. Capasso Global stability results for a multigroup SIR epidemic model T.G. Hallam, L.J. Gross, S.A. Levin, Mathematical Ecology 1986 World Scientific Singapore 317 342
28. Z.L. Feng, W.Z. Huang, and C. Castillo-Chavez Global behavior of a multi-group SIS epidemic model with age structure J. Differential Equations 218 2005 292 324
29. S. Gao, Z. Teng, J.J. Nieto, and A. Torres Analysis of an SIR epidemic model with pulse vaccination and distributed time delay J. Biomed. Biotechnol. 2007 10 Article ID 64870
30. S. Gao, L. Chen, and Z. Teng Pulse vaccination of an SEIR epidemic model with time delay Nonlinear Anal. RWA 9 2008 599 607
31. H. Guo, M.Y. Li, and Z. Shuai Global stability of the endemic equilibrium of multigroup SIR epidemic models Can. Appl. Math. Q. 14 2006 259 284
32. L.K. Hotta Bayesian melding estimation of a stochastic SEIR model Math. Popul. Stud. 17 2010 101 111
33. W. Huang, K.L. Cooke, and C. Castillo-Chavez Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission J. Appl. Math. 52 1992 835 854
34. C. Koide, and H. Seno Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease Math. Comput. Modelling 23 1996 67 91
35. A. Korobeinikov Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages Bull. Math. Biol. 71 2009 75 83
36. G. Li, and Z. Jin Global stability of an SEIR epidemic model with infectious force in latent, infected and immune period Chaos Solitons Fractals 25 2005 1177 1184
37. E.P. Paul, and E. Michael Outbreak properties of epidemic models: the roles of temporal forcing and stochasticity on pathogen invasion dynamics J. Theoret. Biol. 271 2011 1 9
38. H. Guo, M.Y. Li, and Z. Shuai A graph-theoretic approach to the method of global Lyapunov functions Proc. Amer. Math. Soc. 136 2008 2793 2802
39. R 0 in models for infectious diseases in heterogeneous populations J. Math. Biol. 28 1990 365 382
40. J. Beddington, and R. May Harvesting natural populations in a randomly fluctuating environment Science 197 1977 463 465
41. N. Dalal, D. Greenhalgh, and X. Mao A stochastic model of AIDS and condom use J. Math. Anal. Appl. 325 2007 36 53
42. N. Dalal, D. Greenhalgh, and X. Mao A stochastic model for internal HIV dynamics J. Math. Anal. Appl. 341 2008 1084 1101
43. L. Imhof, and S. Walcher Exclusion and persistence in deterministic and stochastic chemostat models J. Differential Equations 217 2005 26 53
44. M.Y. Li, and Z.S. Shuai Global-stability problem for coupled systems of differential equations on networks J. Differential Equations 248 2010 1 20
45. D.B. West Introduction to Graph Theory 1996 Prentice Hall Upper Saddle River
46. A. Berman, and R.J. Plemmons Nonnegative Matrices in the Mathematical Sciences 1979 Academic Press New York
47. J.W. Moon, Counting Labelled Trees, in: Canadian Mathematical Congress, Montreal, 1970.
48. R.Z. Hasminskii Stochastic Stability of Differential Equations 1980 Alphen aan den Rijn The Netherlands
49. C. Zhu, and G. Yin Asymptotic properties of hybrid diffusion systems SIAM J. Control Optim. 46 4 2007 1155 1179
50. X. Mao Stochastic Differential Equations and Applications 1997 Horwood Chichester
51. G. Strang Linear Algebra and Its Applications 1988 Thomson Leanning, Inc. The United States
52. D. Higham An algorithmic introduction to numerical simulation of stochastic differential equations SIAM Rev. 43 2001 525 546
53. N. Ikeda, and S. Watanabe Stochastic Differential Equations and Diffusion Processes 1981 North-Holland New York
54. K.B. Gopal, and N.B. Rabi Stability in distribution for a class of singular diffusions Ann. Probab. 20 1992 312 321
55. Yury A. Kutoyants Statistical Inference for Ergodic Diffusion Processes 2003 Springer London