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Threshold dynamics of a periodic SIR model with delay in an infected compartment
1. | School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China |
References:
[1] |
N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.
doi: 10.1007/s00285-006-0015-0. |
[2] |
Z. Bai and Y. Zhou, Threshold dynamics of a delayed SEIRS model with pulse vaccination and general nonlinear incidence,, 2014, ().
|
[3] |
N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762.
doi: 10.1007/s00285-010-0354-8. |
[4] |
N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Math. Biosci., 210 (2007), 647-658.
doi: 10.1016/j.mbs.2007.07.005. |
[5] |
Z. Bai and Y. Zhou, Existence of multiple periodic solutions for an SIR model with seasonality, Nonlinear Anal., 74 (2011), 3548-3555.
doi: 10.1016/j.na.2011.03.008. |
[6] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42.
doi: 10.1216/RMJ-1979-9-1-31. |
[7] |
N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B., 273 (2006), 2541-2550.
doi: 10.1098/rspb.2006.3604. |
[8] |
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.
doi: 10.1007/s11538-009-9487-6. |
[9] |
J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol., 65 (2012), 623-652.
doi: 10.1007/s00285-011-0474-9. |
[10] |
Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044.
doi: 10.1137/080744438. |
[11] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109.
doi: 10.1016/j.nonrwa.2009.11.005. |
[12] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[13] |
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. |
[14] |
P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[15] |
Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363 (2010), 230-237.
doi: 10.1016/j.jmaa.2009.08.027. |
[16] |
C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64 (2012), 933-949.
doi: 10.1007/s00285-011-0440-6. |
[17] |
C. Rebelo, A. Margheri and N. Bacaër, Persistence in some periodic epidemic models with infection age or constant periods of infection, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1155-1170.
doi: 10.3934/dcdsb.2014.19.1155. |
[18] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[19] |
H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277. |
[20] |
H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[21] |
Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model wiht a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.
doi: 10.1137/070700966. |
[22] |
W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717.
doi: 10.1007/s10884-008-9111-8. |
[23] |
R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA, 10 (2009), 3175-3189.
doi: 10.1016/j.nonrwa.2008.10.013. |
[24] |
D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
[25] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[26] |
F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.
doi: 10.1016/j.jmaa.2006.01.085. |
show all references
References:
[1] |
N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.
doi: 10.1007/s00285-006-0015-0. |
[2] |
Z. Bai and Y. Zhou, Threshold dynamics of a delayed SEIRS model with pulse vaccination and general nonlinear incidence,, 2014, ().
|
[3] |
N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762.
doi: 10.1007/s00285-010-0354-8. |
[4] |
N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Math. Biosci., 210 (2007), 647-658.
doi: 10.1016/j.mbs.2007.07.005. |
[5] |
Z. Bai and Y. Zhou, Existence of multiple periodic solutions for an SIR model with seasonality, Nonlinear Anal., 74 (2011), 3548-3555.
doi: 10.1016/j.na.2011.03.008. |
[6] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42.
doi: 10.1216/RMJ-1979-9-1-31. |
[7] |
N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B., 273 (2006), 2541-2550.
doi: 10.1098/rspb.2006.3604. |
[8] |
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.
doi: 10.1007/s11538-009-9487-6. |
[9] |
J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol., 65 (2012), 623-652.
doi: 10.1007/s00285-011-0474-9. |
[10] |
Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044.
doi: 10.1137/080744438. |
[11] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109.
doi: 10.1016/j.nonrwa.2009.11.005. |
[12] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[13] |
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. |
[14] |
P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[15] |
Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363 (2010), 230-237.
doi: 10.1016/j.jmaa.2009.08.027. |
[16] |
C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64 (2012), 933-949.
doi: 10.1007/s00285-011-0440-6. |
[17] |
C. Rebelo, A. Margheri and N. Bacaër, Persistence in some periodic epidemic models with infection age or constant periods of infection, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1155-1170.
doi: 10.3934/dcdsb.2014.19.1155. |
[18] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. |
[19] |
H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277. |
[20] |
H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[21] |
Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model wiht a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.
doi: 10.1137/070700966. |
[22] |
W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717.
doi: 10.1007/s10884-008-9111-8. |
[23] |
R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA, 10 (2009), 3175-3189.
doi: 10.1016/j.nonrwa.2008.10.013. |
[24] |
D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
[25] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[26] |
F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.
doi: 10.1016/j.jmaa.2006.01.085. |
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