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Complete classification of global dynamics of a virus model with immune responses
Global stability of a multi-group SIS epidemic model for population migration
1. | Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan |
2. | Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555 |
References:
[1] |
J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, Y. Zhou and J. Wu), Higher Education Press, (2009), 65-123. |
[2] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. |
[3] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, 1967. |
[4] |
S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.
doi: 10.1137/0522069. |
[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, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[7] |
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester, 2000. |
[8] |
Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Diff. Equat., 218 (2005), 292-324.
doi: 10.1016/j.jde.2004.10.009. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.
doi: 10.1007/s00285-010-0368-2. |
[13] |
T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Analysis RWA., 12 (2011), 2640-2655.
doi: 10.1016/j.nonrwa.2011.03.011. |
[14] |
A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236.
doi: 10.1016/0025-5564(76)90125-5. |
[15] |
J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. |
[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 problem for coupled systems of differential equations on networks, J. Diff. Equat., 284 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[18] |
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. |
[19] |
J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with transport-related infection, Chaos Solitons and Fractals, 40 (2009), 145-158.
doi: 10.1016/j.chaos.2007.07.047. |
[20] |
X. Liu and Y. Takeuchi, Spread of disease with transport-related infection and entry screening, J. Theoret. Biol., 242 (2006), 517-528.
doi: 10.1016/j.jtbi.2006.03.018. |
[21] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay (distributed or discrete), Nonlinear Analysis RWA., 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[22] |
K. Mischaikow, H. L. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
doi: 10.1090/S0002-9947-1995-1290727-7. |
[23] |
Y. Muroya, A. Bellen, Y. Enatsu and Y. Nakata, Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population, Nonlinear Analysis RWA., 13 (2012), 258-274.
doi: 10.1016/j.nonrwa.2011.07.031. |
[24] |
Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Analysis RWA., 14 (2013), 1693-1704.
doi: 10.1016/j.nonrwa.2012.11.005. |
[25] |
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. |
[26] |
Y. Nakata, On the global stability of a delayed epidemic model with transport-related infection, Nonlinear Analysis RWA., 12 (2011), 3028-3034.
doi: 10.1016/j.nonrwa.2011.05.004. |
[27] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043. |
[28] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Analysis RWA., 13 (2012), 1581-1592.
doi: 10.1016/j.nonrwa.2011.11.016. |
[29] |
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. |
[30] |
H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[31] |
H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
doi: 10.1137/080732870. |
[32] |
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. |
[33] |
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, 1962. |
[34] |
C. Vargas-De-León, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos Solitons and Fractals, 44 (2011), 1106-1110.
doi: 10.1016/j.chaos.2011.09.002. |
[35] |
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. |
[36] |
L. Wang and G. Z. Dai, Global stability of virus spreading in complex heterogeneous networks, SIAM J. Appl. Math., 68 (2008), 1495-1502.
doi: 10.1137/070694582. |
[37] |
Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Analysis RWA., 11 (2010), 995-1004.
doi: 10.1016/j.nonrwa.2009.01.040. |
[38] |
Z. Yuan and X. Zou, Global threshold property in an epidemic models for disease with latency spreading in a heterogeneous host population, Nonlinear Analysis RWA., 11 (2010), 3479-3490.
doi: 10.1016/j.nonrwa.2009.12.008. |
show all references
References:
[1] |
J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, Y. Zhou and J. Wu), Higher Education Press, (2009), 65-123. |
[2] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. |
[3] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, 1967. |
[4] |
S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.
doi: 10.1137/0522069. |
[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, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[7] |
O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Wiley, Chichester, 2000. |
[8] |
Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multi-group SIS epidemic model with age structure, J. Diff. Equat., 218 (2005), 292-324.
doi: 10.1016/j.jde.2004.10.009. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.
doi: 10.1007/s00285-010-0368-2. |
[13] |
T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Analysis RWA., 12 (2011), 2640-2655.
doi: 10.1016/j.nonrwa.2011.03.011. |
[14] |
A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236.
doi: 10.1016/0025-5564(76)90125-5. |
[15] |
J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. |
[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 problem for coupled systems of differential equations on networks, J. Diff. Equat., 284 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[18] |
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. |
[19] |
J. Liu and Y. Zhou, Global stability of an SIRS epidemic model with transport-related infection, Chaos Solitons and Fractals, 40 (2009), 145-158.
doi: 10.1016/j.chaos.2007.07.047. |
[20] |
X. Liu and Y. Takeuchi, Spread of disease with transport-related infection and entry screening, J. Theoret. Biol., 242 (2006), 517-528.
doi: 10.1016/j.jtbi.2006.03.018. |
[21] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay (distributed or discrete), Nonlinear Analysis RWA., 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[22] |
K. Mischaikow, H. L. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347 (1995), 1669-1685.
doi: 10.1090/S0002-9947-1995-1290727-7. |
[23] |
Y. Muroya, A. Bellen, Y. Enatsu and Y. Nakata, Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population, Nonlinear Analysis RWA., 13 (2012), 258-274.
doi: 10.1016/j.nonrwa.2011.07.031. |
[24] |
Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Analysis RWA., 14 (2013), 1693-1704.
doi: 10.1016/j.nonrwa.2012.11.005. |
[25] |
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. |
[26] |
Y. Nakata, On the global stability of a delayed epidemic model with transport-related infection, Nonlinear Analysis RWA., 12 (2011), 3028-3034.
doi: 10.1016/j.nonrwa.2011.05.004. |
[27] |
H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511530043. |
[28] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Analysis RWA., 13 (2012), 1581-1592.
doi: 10.1016/j.nonrwa.2011.11.016. |
[29] |
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. |
[30] |
H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[31] |
H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
doi: 10.1137/080732870. |
[32] |
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. |
[33] |
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, 1962. |
[34] |
C. Vargas-De-León, On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos Solitons and Fractals, 44 (2011), 1106-1110.
doi: 10.1016/j.chaos.2011.09.002. |
[35] |
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. |
[36] |
L. Wang and G. Z. Dai, Global stability of virus spreading in complex heterogeneous networks, SIAM J. Appl. Math., 68 (2008), 1495-1502.
doi: 10.1137/070694582. |
[37] |
Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Analysis RWA., 11 (2010), 995-1004.
doi: 10.1016/j.nonrwa.2009.01.040. |
[38] |
Z. Yuan and X. Zou, Global threshold property in an epidemic models for disease with latency spreading in a heterogeneous host population, Nonlinear Analysis RWA., 11 (2010), 3479-3490.
doi: 10.1016/j.nonrwa.2009.12.008. |
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