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Construction of a finite-time Lyapunov function by meshless collocation
Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations
1. | Centre for Disease Modeling, Department of Mathematics and Statistics, York University, 4700 Keele Street Toronto, ON, M3J 1P3 |
2. | Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada |
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1992. |
[2] |
N. Bame, S. Bowong, J. Mbang, G. Sallet and J. Tewa, Global stability analysis for SEIS models with n latent classes, Math. Biosci. Eng., 5 (2008), 20-33. |
[3] |
F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154.
doi: 10.1016/S0025-5564(01)00057-8. |
[4] |
A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73. |
[5] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. |
[6] |
H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. |
[7] |
H. Guo, M. Y. Li and Z. Shuai, A graph-theoretical approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
doi: 10.1090/S0002-9939-08-09341-6. |
[8] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model with amelioration for infectious diseases, J. Biol. Dyn., 2 (2008), 154-168. |
[9] |
H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol., 14 (1978), 338-349.
doi: 10.1016/0040-5809(78)90011-4. |
[10] |
W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854.
doi: 10.1137/0152047. |
[11] |
J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109.
doi: 10.1016/S0025-5564(98)10057-3. |
[12] |
A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations, Proceedings of the 6$^th$ AIMS International Conference, suppl., 506-519. |
[13] |
J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analysis of HIV transmission: The effect of contact patterns, Math. Biosci., 92 (1988), 119-199.
doi: 10.1016/0025-5564(88)90031-4. |
[14] |
A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models withnonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. |
[15] |
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. |
[16] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[17] |
G. Li, W. Wang and Z. Jin, Global stability of an SEIR epidemicmodel with constant immigration, Chaos, Solitons & Fractals, 30 (2006), 1012-1019. |
[18] |
A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11.
doi: 10.1006/jtbi.1996.0042. |
[19] |
C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.
doi: 10.1016/S0025-5564(02)00149-9. |
[20] |
C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Differential Equations, 16 (2004), 139-166.
doi: 10.1023/B:JODY.0000041283.66784.3e. |
[21] |
S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan and A. V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett., 23 (2010), 446-448.
doi: 10.1016/j.aml.2009.11.014. |
[22] |
H. R. Thieme, Local stability in epidemic models for heterogeneous populations, "Mathematics in Biology and Medicine" (Bari, 1983) (eds. V. Capasso, E. Grosso and S. L. Paveri-Fontana), Lecture Notes in Biomath., 57, Springer, Berlin, (1985), 185-189. |
[23] |
H. R. Thieme, "Mathematics in Population Biology," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1992. |
[2] |
N. Bame, S. Bowong, J. Mbang, G. Sallet and J. Tewa, Global stability analysis for SEIS models with n latent classes, Math. Biosci. Eng., 5 (2008), 20-33. |
[3] |
F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154.
doi: 10.1016/S0025-5564(01)00057-8. |
[4] |
A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 55-73. |
[5] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525. |
[6] |
H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284. |
[7] |
H. Guo, M. Y. Li and Z. Shuai, A graph-theoretical approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802.
doi: 10.1090/S0002-9939-08-09341-6. |
[8] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model with amelioration for infectious diseases, J. Biol. Dyn., 2 (2008), 154-168. |
[9] |
H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol., 14 (1978), 338-349.
doi: 10.1016/0040-5809(78)90011-4. |
[10] |
W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854.
doi: 10.1137/0152047. |
[11] |
J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77-109.
doi: 10.1016/S0025-5564(98)10057-3. |
[12] |
A. Iggidr, J. Mbang, G. Sallet and J.-J. Tewa, Multi-compartment models, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations, Proceedings of the 6$^th$ AIMS International Conference, suppl., 506-519. |
[13] |
J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analysis of HIV transmission: The effect of contact patterns, Math. Biosci., 92 (1988), 119-199.
doi: 10.1016/0025-5564(88)90031-4. |
[14] |
A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models withnonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. |
[15] |
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. |
[16] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248 (2010), 1-20.
doi: 10.1016/j.jde.2009.09.003. |
[17] |
G. Li, W. Wang and Z. Jin, Global stability of an SEIR epidemicmodel with constant immigration, Chaos, Solitons & Fractals, 30 (2006), 1012-1019. |
[18] |
A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11.
doi: 10.1006/jtbi.1996.0042. |
[19] |
C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.
doi: 10.1016/S0025-5564(02)00149-9. |
[20] |
C. C. McCluskey and P. van den Driessche, Global analysis of two tuberculosis models, J. Dyn. Differential Equations, 16 (2004), 139-166.
doi: 10.1023/B:JODY.0000041283.66784.3e. |
[21] |
S. M. O'Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O'Callaghan and A. V. Pokrovskii, Lyapunov functions for SIR and SIRS epidemic models, Appl. Math. Lett., 23 (2010), 446-448.
doi: 10.1016/j.aml.2009.11.014. |
[22] |
H. R. Thieme, Local stability in epidemic models for heterogeneous populations, "Mathematics in Biology and Medicine" (Bari, 1983) (eds. V. Capasso, E. Grosso and S. L. Paveri-Fontana), Lecture Notes in Biomath., 57, Springer, Berlin, (1985), 185-189. |
[23] |
H. R. Thieme, "Mathematics in Population Biology," Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003. |
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