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Mathematical Biosciences and Engineering (MBE)
 

Stochastic dynamics of SIRS epidemic models with random perturbation
Pages: 1003 - 1025, Issue 4, August 2014

doi:10.3934/mbe.2014.11.1003      Abstract        References        Full text (430.4K)           Related Articles

Qingshan Yang - School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China (email)
Xuerong Mao - Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom (email)

1 E. J. Allen, L. J. S. Allen, A. Arciniega and P. E. Greenwood, Construction of equivalent stochastic differential equation models, Stoch Anal Appl., 26 (2008), 274-297.       
2 R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature., 280 (1979), 361-367, doi: 10.1038/280361a0.
3 N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, Second edition. Hafner Press [Macmillan Publishing Co., Inc.] New York, 1975.       
4 G. K. Basak and R. N. Bhattacharya, Stability in distribution for a class of singular diffusions, Ann Probab., 20 (1992), 312-321.       
5 P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Biol., 63 (2011), 433-457.       
6 S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Springer, Berlin, 1993.       
7 G. Chen and T. Li, Stability of stochastic delayed SIR model, Stoch Dynam., 9 (2009), 231-252.       
8 Y. S. Chow, Local convergence of martingales and the law of large numbers, Ann. Math. Statist., 36 (1965), 552-558.       
9 A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.       
10 R. Z. Hasminskii, Stochastic Stability of Differential Equations, Alphen aan den Rijn, The Netherlands, 1980.       
11 H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47.       
12 L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations., 217 (2005), 26-53.       
13 A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models, Appl. Math. Lett., 15 (2002), 955-960.       
14 Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer, London, 2004.       
15 W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.       
16 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.       
17 Q. Lu, Stability of SIRS system with random perturbations, Phys. A., 388 (2009), 3677-3686.       
18 W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time-delay, Appl. Math. Lett., 17 (2004), 1141-1145.       
19 X. Mao, Stablity of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific & Technical Harlow, UK, 1991.
20 X. Mao, Exponentially Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.       
21 X. Mao, Stochastic Differential Equations and Their Applications, 2nd ed., Horwood Publishing, Chichester, 1997.       
22 J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.       
23 I. Nasell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19.       
24 E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A., 354 (2005), 111-126.
25 C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM. J. Control. Optim., 46 (2007), 1155-1179.       

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