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New results of the ultimate bound on the trajectories of the family of the Lorenz systems
The threshold of a stochastic SIRS epidemic model in a population with varying size
1. | School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China |
2. | School of Mathematics and Statistics, Northeast Normal University, Changchun 130024 |
3. | Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, United Kingdom |
References:
[1] |
R. M. Anderson and R. M. May, Population biology of infectious diseases I,, Nature, 280 (1979), 361.
doi: 10.1038/280361a0. |
[2] |
R. M. Anderson and R. M. May, Population biology of infectious diseases II,, Nature, 280 (1979), 455. Google Scholar |
[3] |
R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts,, Philos. Trans. R. Soc. Lond. B, 291 (1981), 451.
doi: 10.1098/rstb.1981.0005. |
[4] |
M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability,, Nonlinearity, 18 (2005), 913.
doi: 10.1088/0951-7715/18/2/022. |
[5] |
S. Busenberg, K. L. Cooke and M. A. Pozio, Analysis of a model of a vertically transmitted disease,, J. Math. Biol., 17 (1983), 305.
doi: 10.1007/BF00276519. |
[6] |
S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size,, J. Math. Biol., 28 (1990), 257.
doi: 10.1007/BF00178776. |
[7] |
S. Busenberg, K. L. Cooke and H. Thieme, Demographic change and persistence of HIV/AIDS in a heterogeneous population,, SIAM J. Appl. Math., 51 (1991), 1030.
doi: 10.1137/0151052. |
[8] |
M. Carletti, K. Burrage and P. M. Burrage, Numerical simulation of stochastic ordinary differential equations in biomathematical modelling,, Math. Comput. Simulation, 64 (2004), 271.
doi: 10.1016/j.matcom.2003.09.022. |
[9] |
N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use,, J. Math. Anal. Appl., 325 (2007), 36.
doi: 10.1016/j.jmaa.2006.01.055. |
[10] |
M. Fan, M. Y. Li and K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size,, Math. Biosciences, 170 (2001), 199.
doi: 10.1016/S0025-5564(00)00067-5. |
[11] |
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.
doi: 10.1137/10081856X. |
[12] |
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Review, 43 (2001), 525.
doi: 10.1137/S0036144500378302. |
[13] |
C. Y. Ji, D. Q. Jiang and N. Z. Shi, The behavior of an SIR epidemic model with stochastic perturbation,, Stochastic Anal. Appl., 30 (2012), 755.
doi: 10.1080/07362994.2012.684319. |
[14] |
X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008).
doi: 10.1533/9780857099402. |
[15] |
M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size,, Math. Biosciences, 182 (2003), 1.
doi: 10.1016/S0025-5564(02)00184-0. |
[16] |
R. M. May, R. M. Anderson and A. R. Mclean, Possible demographic consequences of HIV/AIDS epidemics. I. assuming HIV infection always leads to AIDS,, Math. Biosciences, 90 (1988), 475.
doi: 10.1016/0025-5564(88)90079-X. |
[17] |
C. J. Sun and Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination,, Applied Mathematical Modelling, 34 (2010), 2685.
doi: 10.1016/j.apm.2009.12.005. |
[18] |
E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system,, Physica A, 354 (2005), 111.
doi: 10.1016/j.physa.2005.02.057. |
[19] |
C. Vargas-De-Leon, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence,, Chaos, 44 (2011), 1106.
doi: 10.1016/j.chaos.2011.09.002. |
[20] |
Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations,, Nonlinear Anal. RWA, 14 (2013), 1434.
doi: 10.1016/j.nonrwa.2012.10.007. |
[21] |
Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence,, Applied Mathematical Letter, 34 (2014), 90.
doi: 10.1016/j.aml.2013.11.002. |
[22] |
Y. Zhao, D. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination,, Physica A, 392 (2013), 4916.
doi: 10.1016/j.physa.2013.06.009. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, Population biology of infectious diseases I,, Nature, 280 (1979), 361.
doi: 10.1038/280361a0. |
[2] |
R. M. Anderson and R. M. May, Population biology of infectious diseases II,, Nature, 280 (1979), 455. Google Scholar |
[3] |
R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts,, Philos. Trans. R. Soc. Lond. B, 291 (1981), 451.
doi: 10.1098/rstb.1981.0005. |
[4] |
M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability,, Nonlinearity, 18 (2005), 913.
doi: 10.1088/0951-7715/18/2/022. |
[5] |
S. Busenberg, K. L. Cooke and M. A. Pozio, Analysis of a model of a vertically transmitted disease,, J. Math. Biol., 17 (1983), 305.
doi: 10.1007/BF00276519. |
[6] |
S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size,, J. Math. Biol., 28 (1990), 257.
doi: 10.1007/BF00178776. |
[7] |
S. Busenberg, K. L. Cooke and H. Thieme, Demographic change and persistence of HIV/AIDS in a heterogeneous population,, SIAM J. Appl. Math., 51 (1991), 1030.
doi: 10.1137/0151052. |
[8] |
M. Carletti, K. Burrage and P. M. Burrage, Numerical simulation of stochastic ordinary differential equations in biomathematical modelling,, Math. Comput. Simulation, 64 (2004), 271.
doi: 10.1016/j.matcom.2003.09.022. |
[9] |
N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use,, J. Math. Anal. Appl., 325 (2007), 36.
doi: 10.1016/j.jmaa.2006.01.055. |
[10] |
M. Fan, M. Y. Li and K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size,, Math. Biosciences, 170 (2001), 199.
doi: 10.1016/S0025-5564(00)00067-5. |
[11] |
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.
doi: 10.1137/10081856X. |
[12] |
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations,, SIAM Review, 43 (2001), 525.
doi: 10.1137/S0036144500378302. |
[13] |
C. Y. Ji, D. Q. Jiang and N. Z. Shi, The behavior of an SIR epidemic model with stochastic perturbation,, Stochastic Anal. Appl., 30 (2012), 755.
doi: 10.1080/07362994.2012.684319. |
[14] |
X. Mao, Stochastic Differential Equations and Applications,, $2^{nd}$ edition, (2008).
doi: 10.1533/9780857099402. |
[15] |
M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size,, Math. Biosciences, 182 (2003), 1.
doi: 10.1016/S0025-5564(02)00184-0. |
[16] |
R. M. May, R. M. Anderson and A. R. Mclean, Possible demographic consequences of HIV/AIDS epidemics. I. assuming HIV infection always leads to AIDS,, Math. Biosciences, 90 (1988), 475.
doi: 10.1016/0025-5564(88)90079-X. |
[17] |
C. J. Sun and Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination,, Applied Mathematical Modelling, 34 (2010), 2685.
doi: 10.1016/j.apm.2009.12.005. |
[18] |
E. Tornatore, S. M. Buccellato and P. Vetro, Stability of a stochastic SIR system,, Physica A, 354 (2005), 111.
doi: 10.1016/j.physa.2005.02.057. |
[19] |
C. Vargas-De-Leon, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence,, Chaos, 44 (2011), 1106.
doi: 10.1016/j.chaos.2011.09.002. |
[20] |
Q. Yang and X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations,, Nonlinear Anal. RWA, 14 (2013), 1434.
doi: 10.1016/j.nonrwa.2012.10.007. |
[21] |
Y. Zhao and D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence,, Applied Mathematical Letter, 34 (2014), 90.
doi: 10.1016/j.aml.2013.11.002. |
[22] |
Y. Zhao, D. Jiang and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination,, Physica A, 392 (2013), 4916.
doi: 10.1016/j.physa.2013.06.009. |
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