- Previous Article
- MBE Home
- This Issue
-
Next Article
A note on global stability for malaria infections model with latencies
Stochastic dynamics of SIRS epidemic models with random perturbation
1. | School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin, 130024, China |
2. | Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH |
References:
[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.
doi: 10.1080/07362990701857129. |
[2] |
R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature., 280 (1979), 361-367, doi: 10.1038/280361a0.
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.
doi: 10.1214/aop/1176989928. |
[5] |
P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Biol., 63 (2011), 433-457.
doi: 10.1007/s00285-010-0376-2. |
[6] |
S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Springer, Berlin, 1993.
doi: 10.1007/978-3-642-75301-5. |
[7] |
G. Chen and T. Li, Stability of stochastic delayed SIR model, Stoch Dynam., 9 (2009), 231-252.
doi: 10.1142/S0219493709002658. |
[8] |
Y. S. Chow, Local convergence of martingales and the law of large numbers, Ann. Math. Statist., 36 (1965), 552-558.
doi: 10.1214/aoms/1177700166. |
[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.
doi: 10.1137/10081856X. |
[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.
doi: 10.1007/BF00276034. |
[12] |
L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations., 217 (2005), 26-53.
doi: 10.1016/j.jde.2005.06.017. |
[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.
doi: 10.1016/S0893-9659(02)00069-1. |
[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.
doi: 10.1007/BF00276956. |
[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.
doi: 10.1007/BF00277162. |
[17] |
Q. Lu, Stability of SIRS system with random perturbations, Phys. A., 388 (2009), 3677-3686.
doi: 10.1016/j.physa.2009.05.036. |
[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.
doi: 10.1016/j.aml.2003.11.005. |
[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.
doi: 10.1007/b98869. |
[23] |
I. Nasell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19.
doi: 10.1016/S0025-5564(02)00098-6. |
[24] |
E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A., 354 (2005), 111-126.
doi: 10.1016/j.physa.2005.02.057. |
[25] |
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM. J. Control. Optim., 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
show all references
References:
[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.
doi: 10.1080/07362990701857129. |
[2] |
R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature., 280 (1979), 361-367, doi: 10.1038/280361a0.
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.
doi: 10.1214/aop/1176989928. |
[5] |
P. H. Baxendale and P. E. Greenwood, Sustained oscillations for density dependent Markov processes, J. Math. Biol., 63 (2011), 433-457.
doi: 10.1007/s00285-010-0376-2. |
[6] |
S. Busenberg and K. Cooke, Vertically Transmitted Diseases: Models and Dynamics, Springer, Berlin, 1993.
doi: 10.1007/978-3-642-75301-5. |
[7] |
G. Chen and T. Li, Stability of stochastic delayed SIR model, Stoch Dynam., 9 (2009), 231-252.
doi: 10.1142/S0219493709002658. |
[8] |
Y. S. Chow, Local convergence of martingales and the law of large numbers, Ann. Math. Statist., 36 (1965), 552-558.
doi: 10.1214/aoms/1177700166. |
[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.
doi: 10.1137/10081856X. |
[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.
doi: 10.1007/BF00276034. |
[12] |
L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations., 217 (2005), 26-53.
doi: 10.1016/j.jde.2005.06.017. |
[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.
doi: 10.1016/S0893-9659(02)00069-1. |
[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.
doi: 10.1007/BF00276956. |
[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.
doi: 10.1007/BF00277162. |
[17] |
Q. Lu, Stability of SIRS system with random perturbations, Phys. A., 388 (2009), 3677-3686.
doi: 10.1016/j.physa.2009.05.036. |
[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.
doi: 10.1016/j.aml.2003.11.005. |
[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.
doi: 10.1007/b98869. |
[23] |
I. Nasell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19.
doi: 10.1016/S0025-5564(02)00098-6. |
[24] |
E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system, Phys. A., 354 (2005), 111-126.
doi: 10.1016/j.physa.2005.02.057. |
[25] |
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, SIAM. J. Control. Optim., 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
[1] |
Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265 |
[2] |
Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581 |
[3] |
Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137 |
[4] |
Serge Troubetzkoy. Recurrence in generic staircases. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047 |
[5] |
Michael Blank. Recurrence for measurable semigroup actions. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1649-1665. doi: 10.3934/dcds.2020335 |
[6] |
Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1 |
[7] |
Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1 |
[8] |
Milton Ko. Rényi entropy and recurrence. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403 |
[9] |
Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729 |
[10] |
Oliver Jenkinson. Ergodic Optimization. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197 |
[11] |
Jie Li, Kesong Yan, Xiangdong Ye. Recurrence properties and disjointness on the induced spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1059-1073. doi: 10.3934/dcds.2015.35.1059 |
[12] |
Rafael De La Llave, A. Windsor. An application of topological multiple recurrence to tiling. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 315-324. doi: 10.3934/dcdss.2009.2.315 |
[13] |
A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587 |
[14] |
Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175 |
[15] |
Vincent Penné, Benoît Saussol, Sandro Vaienti. Dimensions for recurrence times: topological and dynamical properties. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 783-798. doi: 10.3934/dcds.1999.5.783 |
[16] |
Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039 |
[17] |
Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259 |
[18] |
Ethan M. Ackelsberg. Rigidity, weak mixing, and recurrence in abelian groups. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1669-1705. doi: 10.3934/dcds.2021168 |
[19] |
Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757 |
[20] |
Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]