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Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence
1. | Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada |
References:
[1] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347. |
[2] |
E. Beretta, Hara T., Ma W. and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115.
doi: doi:10.1016/S0362-546X(01)00528-4. |
[3] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.
doi: doi:10.1016/0025-5564(78)90006-8. |
[4] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math., 9 (1979), 31-42.
doi: doi:10.1216/RMJ-1979-9-1-31. |
[5] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: doi:10.1007/BF00178324. |
[6] |
Z. Feng and H. Thieme, Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: General theory, SIAM J. Appl. Math., 61 (2000), 803-833. |
[7] |
B.-S. Goh, Stability of some multispecies population models, in "Modeling and Differential Equations in Biology," Dekker, New York, 1980. |
[8] |
J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, 1993. |
[9] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. |
[10] |
H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.
doi: doi:10.1016/0025-5564(76)90132-2. |
[11] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: doi:10.1137/S0036144500371907. |
[12] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Springer-Verlag, 1993. |
[13] |
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stabilty for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2009), 1192-1207.
doi: doi:10.1007/s11538-009-9487-6. |
[14] |
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.
doi: doi:10.1007/s11538-007-9196-y. |
[15] |
A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. and Biol., 22 (2005), 113-128.
doi: doi:10.1093/imammb/dqi001. |
[16] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, vol 191, Academic Press, Cambridge, 1993. |
[17] |
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), 2434-2448.
doi: doi:10.1137/090779322. |
[18] |
W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54 (2002), 581-591.
doi: doi:10.2748/tmj/1113247650. |
[19] |
P. Magal, C. C. McCluskey and G. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140.
doi: doi:10.1080/00036810903208122. |
[20] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 6 (2009), 603-610.
doi: doi:10.3934/mbe.2009.6.603. |
[21] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59.
doi: doi:10.1016/j.nonrwa.2008.10.014. |
[22] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109.
doi: doi:10.1016/j.nonrwa.2009.11.005. |
[23] |
Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947.
doi: doi:10.1016/S0362-546X(99)00138-8. |
[24] |
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: doi:10.1016/j.nonrwa.2008.10.013. |
show all references
References:
[1] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347. |
[2] |
E. Beretta, Hara T., Ma W. and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115.
doi: doi:10.1016/S0362-546X(01)00528-4. |
[3] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.
doi: doi:10.1016/0025-5564(78)90006-8. |
[4] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math., 9 (1979), 31-42.
doi: doi:10.1216/RMJ-1979-9-1-31. |
[5] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: doi:10.1007/BF00178324. |
[6] |
Z. Feng and H. Thieme, Endemic models for the spread of infectious diseases with arbitrarily distributed disease stages I: General theory, SIAM J. Appl. Math., 61 (2000), 803-833. |
[7] |
B.-S. Goh, Stability of some multispecies population models, in "Modeling and Differential Equations in Biology," Dekker, New York, 1980. |
[8] |
J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, 1993. |
[9] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. |
[10] |
H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.
doi: doi:10.1016/0025-5564(76)90132-2. |
[11] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: doi:10.1137/S0036144500371907. |
[12] |
Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Springer-Verlag, 1993. |
[13] |
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stabilty for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2009), 1192-1207.
doi: doi:10.1007/s11538-009-9487-6. |
[14] |
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.
doi: doi:10.1007/s11538-007-9196-y. |
[15] |
A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. and Biol., 22 (2005), 113-128.
doi: doi:10.1093/imammb/dqi001. |
[16] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, vol 191, Academic Press, Cambridge, 1993. |
[17] |
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), 2434-2448.
doi: doi:10.1137/090779322. |
[18] |
W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54 (2002), 581-591.
doi: doi:10.2748/tmj/1113247650. |
[19] |
P. Magal, C. C. McCluskey and G. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Applicable Analysis, 89 (2010), 1109-1140.
doi: doi:10.1080/00036810903208122. |
[20] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 6 (2009), 603-610.
doi: doi:10.3934/mbe.2009.6.603. |
[21] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59.
doi: doi:10.1016/j.nonrwa.2008.10.014. |
[22] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109.
doi: doi:10.1016/j.nonrwa.2009.11.005. |
[23] |
Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947.
doi: doi:10.1016/S0362-546X(99)00138-8. |
[24] |
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: doi:10.1016/j.nonrwa.2008.10.013. |
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