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Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures
1. | Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501 |
2. | Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555 |
3. | Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914 |
References:
[1] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University, Oxford, 1991. |
[2] |
J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, Y. Zhou and J. Wu), Higher Education Press, 2009, 65-122.
doi: 10.1142/7223. |
[3] |
M. S. Bartlet, Deterministic and stochastic models for recurrent epidemics, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1956, 81-109. |
[4] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
doi: 10.1007/978-1-4612-0873-0. |
[5] |
H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400.
doi: 10.1016/j.amc.2011.10.015. |
[6] |
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, 1st edition, John Wiley and Sons, Chichester, 2000.
doi: 10.1007/978-1-4612-0873-0. |
[7] |
M. J. Faddy, A note on the behavior of deterministic spatial epidemics, Math. Biosci., 80 (1986), 19-22.
doi: 10.1016/0025-5564(86)90064-7. |
[8] |
H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600.
doi: 10.1007/BF02218848. |
[9] |
H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. |
[10] |
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.
doi: 10.1090/S0002-9939-08-09341-6. |
[11] |
J. M. Hyman and T. LaForce, Modeling the spread of influenza among cities, in Bioterrorism (eds. H. T. Banks and C. Castillo-Chavez), SIAM, 2003, 211-236. |
[12] |
Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364.
doi: 10.1016/j.jmaa.2005.01.034. |
[13] |
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic model, Math. Med. Biol., 21 (2004), 75-83.
doi: 10.1007/s11538-008-9352-z. |
[14] |
T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration, Discrete Cont. Dyn. Sys. Series B, 19 (2014), 1105-1118.
doi: 10.3934/dcdsb.2014.19.1105. |
[15] |
J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976.
doi: 10.1007/978-1-4612-0873-0. |
[16] |
M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.
doi: 10.1016/S0025-5564(99)00030-9. |
[17] |
M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canadian Appl. Math. Quart., 17 (2009), 175-187. |
[18] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 284 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[19] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.
doi: 10.1016/j.jmaa.2009.09.017. |
[20] |
Y. Muroya, Y. Enatsu and T. Kuniya, Global stability of extended multi-group SIR epidemic models with patches through migration and cross patch infection, Acta Mathematica Scientia, 33 (2013), 341-361.
doi: 10.1016/S0252-9602(13)60003-X. |
[21] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592.
doi: 10.1016/j.nonrwa.2011.11.016. |
[22] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1007/978-1-4612-0873-0. |
[23] |
R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291.
doi: 10.1016/j.camwa.2010.08.020. |
[24] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[25] |
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.
doi: 10.1007/978-1-4612-0873-0. |
[26] |
W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112.
doi: 10.1016/j.mbs.2002.11.001. |
[27] |
Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004.
doi: 10.1016/j.nonrwa.2009.01.040. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University, Oxford, 1991. |
[2] |
J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, Y. Zhou and J. Wu), Higher Education Press, 2009, 65-122.
doi: 10.1142/7223. |
[3] |
M. S. Bartlet, Deterministic and stochastic models for recurrent epidemics, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1956, 81-109. |
[4] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
doi: 10.1007/978-1-4612-0873-0. |
[5] |
H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400.
doi: 10.1016/j.amc.2011.10.015. |
[6] |
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, 1st edition, John Wiley and Sons, Chichester, 2000.
doi: 10.1007/978-1-4612-0873-0. |
[7] |
M. J. Faddy, A note on the behavior of deterministic spatial epidemics, Math. Biosci., 80 (1986), 19-22.
doi: 10.1016/0025-5564(86)90064-7. |
[8] |
H. I. Freedman, M. X. Tang and S. G. Ruan, Uniform persistence and flows near a closed positively invariant set, J. Dynam. Diff. Equat., 6 (1994), 583-600.
doi: 10.1007/BF02218848. |
[9] |
H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. |
[10] |
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.
doi: 10.1090/S0002-9939-08-09341-6. |
[11] |
J. M. Hyman and T. LaForce, Modeling the spread of influenza among cities, in Bioterrorism (eds. H. T. Banks and C. Castillo-Chavez), SIAM, 2003, 211-236. |
[12] |
Y. Jin and W. Wang, The effect of population dispersal on the spread of a disease, J. Math. Anal. Appl., 308 (2005), 343-364.
doi: 10.1016/j.jmaa.2005.01.034. |
[13] |
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic model, Math. Med. Biol., 21 (2004), 75-83.
doi: 10.1007/s11538-008-9352-z. |
[14] |
T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration, Discrete Cont. Dyn. Sys. Series B, 19 (2014), 1105-1118.
doi: 10.3934/dcdsb.2014.19.1105. |
[15] |
J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976.
doi: 10.1007/978-1-4612-0873-0. |
[16] |
M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191-213.
doi: 10.1016/S0025-5564(99)00030-9. |
[17] |
M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canadian Appl. Math. Quart., 17 (2009), 175-187. |
[18] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 284 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[19] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.
doi: 10.1016/j.jmaa.2009.09.017. |
[20] |
Y. Muroya, Y. Enatsu and T. Kuniya, Global stability of extended multi-group SIR epidemic models with patches through migration and cross patch infection, Acta Mathematica Scientia, 33 (2013), 341-361.
doi: 10.1016/S0252-9602(13)60003-X. |
[21] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592.
doi: 10.1016/j.nonrwa.2011.11.016. |
[22] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1007/978-1-4612-0873-0. |
[23] |
R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010), 2286-2291.
doi: 10.1016/j.camwa.2010.08.020. |
[24] |
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[25] |
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962.
doi: 10.1007/978-1-4612-0873-0. |
[26] |
W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112.
doi: 10.1016/j.mbs.2002.11.001. |
[27] |
Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Anal. RWA, 11 (2010), 995-1004.
doi: 10.1016/j.nonrwa.2009.01.040. |
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