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Optimal control of chikungunya disease: Larvae reduction, treatment and prevention
Impact of heterogeneity on the dynamics of an SEIR epidemic model
1. | Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., V8W 3R4, Canada, Canada |
References:
[1] |
H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection,, J. App. Prob., 35 (1998), 651.
|
[2] |
R. M. Anderson and R. M. May, Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes,, IMA J. Math. Appl. Med. Biol., 1 (1984), 233.
doi: 10.1093/imammb/1.3.233. |
[3] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992). Google Scholar |
[4] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331.
|
[5] |
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences,", Computer Science and Applied Mathematics, (1979).
|
[6] |
K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays,, J. Math. Biol., 35 (1996), 240.
doi: 10.1007/s002850050051. |
[7] |
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.
doi: 10.1007/BF00178324. |
[8] |
Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bull. Math. Biol., 69 (2007), 1511.
doi: 10.1007/s11538-006-9174-9. |
[9] |
H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases,, J. Biol. Dyn., 2 (2008), 154.
doi: 10.1080/17513750802120877. |
[10] |
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.
|
[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.
doi: 10.1090/S0002-9939-08-09341-6. |
[12] |
T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases,, J. Theor. Biol., 229 (2004), 349.
doi: 10.1016/j.jtbi.2004.04.002. |
[13] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations,", Appl. Math. Sci., (1993).
|
[14] |
H. W. Hethcote, H. W. Stech, and P. van den Driessche, Stability analysis for models of diseases without immunity,, J. Math. Biol., 13 (1981), 185.
doi: 10.1007/BF00275213. |
[15] |
H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases,, J. Math. Biol., 9 (1980), 37.
doi: 10.1007/BF00276034. |
[16] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lectures Notes in Math., (1473).
|
[17] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.
doi: 10.1007/s00285-010-0368-2. |
[18] |
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. App. Math., 70 (2010), 2693.
doi: 10.1137/090780821. |
[19] |
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.
doi: 10.1016/S0025-5564(98)10057-3. |
[20] |
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75.
doi: 10.1093/imammb/21.2.75. |
[21] |
A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57.
doi: 10.3934/mbe.2004.1.57. |
[22] |
A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.
doi: 10.1016/0025-5564(76)90125-5. |
[23] |
J. P. LaSalle, "The Stability of Dynamical Systems,", With an appendix: \emph{Limiting equations and stability of nonautonomous ordinary differential equations}, (1976).
|
[24] |
M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155.
doi: 10.1016/0025-5564(95)92756-5. |
[25] |
M. Y. Li and J. S. Muldowney, A geometric approach to global stability,, SIAM J. Math. Anal., 27 (1996), 1070.
|
[26] |
M. Y. Li, J. S. Muldowney and P. van den Driessche, Global stability of SEIRS models in epidemiology,, Can. Appl. Math. Q., 7 (1999), 409.
|
[27] |
M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.
doi: 10.1137/090779322. |
[28] |
M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Bio., 72 (2010), 1429.
doi: 10.1007/s11538-010-9503-x. |
[29] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1.
doi: 10.1016/j.jde.2009.09.003. |
[30] |
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.
doi: 10.1016/j.jmaa.2009.09.017. |
[31] |
S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal. Real World Appl., 12 (2011), 119.
doi: 10.1016/j.nonrwa.2010.06.001. |
[32] |
W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates,, J. Math. Biol., 25 (1987), 359.
doi: 10.1007/BF00277162. |
[33] |
A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models,, J. Theor. Biol., 179 (1996), 1.
doi: 10.1006/jtbi.1996.0042. |
[34] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603.
doi: 10.3934/mbe.2009.6.603. |
[35] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay--distributed or discrete,, Nonlinear Anal. Real World Appl., 11 (2010), 55.
doi: 10.1016/j.nonrwa.2008.10.014. |
[36] |
R. K. Miller, "Nonlinear Volterra Integral Equations,", Mathematics Lecture Note Series, (1971).
|
[37] |
M. E. J. Newman, The spread of epidemic disease on networks,, Phys. Rev. E, 66 (2002).
doi: 10.1103/PhysRevE.66.016128. |
[38] |
C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541.
doi: 10.1137/0152030. |
[39] |
P. van den Driessche, Some epidemiological models with delays,, in, (1996), 507.
|
[40] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205.
|
[41] |
P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics,, Math. Biosci., 228 (2010), 71.
doi: 10.1016/j.mbs.2010.08.008. |
[42] |
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.
doi: 10.1016/S0025-5564(02)00108-6. |
[43] |
P. van den Driessche and X. Zou, Modeling relapse in infectious diseases,, Math. Biosci., 207 (2007), 89.
doi: 10.1016/j.mbs.2006.09.017. |
[44] |
Z. Yuan and X. Zou, Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population,, Nonlinear Anal. Real World Appl., 11 (2010), 3479.
doi: 10.1016/j.nonrwa.2009.12.008. |
show all references
References:
[1] |
H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection,, J. App. Prob., 35 (1998), 651.
|
[2] |
R. M. Anderson and R. M. May, Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes,, IMA J. Math. Appl. Med. Biol., 1 (1984), 233.
doi: 10.1093/imammb/1.3.233. |
[3] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992). Google Scholar |
[4] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331.
|
[5] |
A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences,", Computer Science and Applied Mathematics, (1979).
|
[6] |
K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays,, J. Math. Biol., 35 (1996), 240.
doi: 10.1007/s002850050051. |
[7] |
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.
doi: 10.1007/BF00178324. |
[8] |
Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bull. Math. Biol., 69 (2007), 1511.
doi: 10.1007/s11538-006-9174-9. |
[9] |
H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases,, J. Biol. Dyn., 2 (2008), 154.
doi: 10.1080/17513750802120877. |
[10] |
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.
|
[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.
doi: 10.1090/S0002-9939-08-09341-6. |
[12] |
T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases,, J. Theor. Biol., 229 (2004), 349.
doi: 10.1016/j.jtbi.2004.04.002. |
[13] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations,", Appl. Math. Sci., (1993).
|
[14] |
H. W. Hethcote, H. W. Stech, and P. van den Driessche, Stability analysis for models of diseases without immunity,, J. Math. Biol., 13 (1981), 185.
doi: 10.1007/BF00275213. |
[15] |
H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases,, J. Math. Biol., 9 (1980), 37.
doi: 10.1007/BF00276034. |
[16] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lectures Notes in Math., (1473).
|
[17] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125.
doi: 10.1007/s00285-010-0368-2. |
[18] |
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. App. Math., 70 (2010), 2693.
doi: 10.1137/090780821. |
[19] |
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.
doi: 10.1016/S0025-5564(98)10057-3. |
[20] |
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75.
doi: 10.1093/imammb/21.2.75. |
[21] |
A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57.
doi: 10.3934/mbe.2004.1.57. |
[22] |
A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.
doi: 10.1016/0025-5564(76)90125-5. |
[23] |
J. P. LaSalle, "The Stability of Dynamical Systems,", With an appendix: \emph{Limiting equations and stability of nonautonomous ordinary differential equations}, (1976).
|
[24] |
M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155.
doi: 10.1016/0025-5564(95)92756-5. |
[25] |
M. Y. Li and J. S. Muldowney, A geometric approach to global stability,, SIAM J. Math. Anal., 27 (1996), 1070.
|
[26] |
M. Y. Li, J. S. Muldowney and P. van den Driessche, Global stability of SEIRS models in epidemiology,, Can. Appl. Math. Q., 7 (1999), 409.
|
[27] |
M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.
doi: 10.1137/090779322. |
[28] |
M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Bio., 72 (2010), 1429.
doi: 10.1007/s11538-010-9503-x. |
[29] |
M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1.
doi: 10.1016/j.jde.2009.09.003. |
[30] |
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.
doi: 10.1016/j.jmaa.2009.09.017. |
[31] |
S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal. Real World Appl., 12 (2011), 119.
doi: 10.1016/j.nonrwa.2010.06.001. |
[32] |
W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates,, J. Math. Biol., 25 (1987), 359.
doi: 10.1007/BF00277162. |
[33] |
A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models,, J. Theor. Biol., 179 (1996), 1.
doi: 10.1006/jtbi.1996.0042. |
[34] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603.
doi: 10.3934/mbe.2009.6.603. |
[35] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay--distributed or discrete,, Nonlinear Anal. Real World Appl., 11 (2010), 55.
doi: 10.1016/j.nonrwa.2008.10.014. |
[36] |
R. K. Miller, "Nonlinear Volterra Integral Equations,", Mathematics Lecture Note Series, (1971).
|
[37] |
M. E. J. Newman, The spread of epidemic disease on networks,, Phys. Rev. E, 66 (2002).
doi: 10.1103/PhysRevE.66.016128. |
[38] |
C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541.
doi: 10.1137/0152030. |
[39] |
P. van den Driessche, Some epidemiological models with delays,, in, (1996), 507.
|
[40] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205.
|
[41] |
P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics,, Math. Biosci., 228 (2010), 71.
doi: 10.1016/j.mbs.2010.08.008. |
[42] |
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.
doi: 10.1016/S0025-5564(02)00108-6. |
[43] |
P. van den Driessche and X. Zou, Modeling relapse in infectious diseases,, Math. Biosci., 207 (2007), 89.
doi: 10.1016/j.mbs.2006.09.017. |
[44] |
Z. Yuan and X. Zou, Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population,, Nonlinear Anal. Real World Appl., 11 (2010), 3479.
doi: 10.1016/j.nonrwa.2009.12.008. |
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