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Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays
1. | Department of Pure and Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555 |
2. | Basque Center for Applied Mathematics, Bizkaia Technology Park, Building 500 E-48160 Derio, Spain |
3. | Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555 |
References:
[1] |
R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367.
doi: doi:10.1038/280361a0. |
[2] |
E. Beretta and Y. Takeuchi, Convergence results in SIR epidemic models with varying population size, Nonlinear Anal., 28 (1997), 1909-1921.
doi: doi:10.1016/S0362-546X(96)00035-1. |
[3] |
G. C. Brown and R. Hasibuan, Conidial discharge and transmission efficiency of "Neozygites floridana," an entomopathogenic fungus infeccting two-spotted spider mites under laboratory conditions, J. Invertebr. Pathol., 65 (1995), 10-16.
doi: doi:10.1006/jipa.1995.1002. |
[4] |
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. |
[5] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42.
doi: doi:10.1216/RMJ-1979-9-1-31. |
[6] |
J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, 1993. |
[7] |
H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci. 28 (1976), 335-356.
doi: doi:10.1016/0025-5564(76)90132-2. |
[8] |
H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev., 42 (2000), 599-653. |
[9] |
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.
doi: doi:10.1007/s11538-009-9487-6. |
[10] |
A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.
doi: doi:10.1093/imammb/dqi001. |
[11] |
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626.
doi: doi:10.1007/s11538-005-9037-9. |
[12] |
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. |
[13] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993. |
[14] |
J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications," Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961. |
[15] |
W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed delays, Tohoku. Math. J., 54 (2002), 581-591.
doi: doi:10.2748/tmj/1113247650. |
[16] |
W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141-1145.
doi: doi:10.1016/j.aml.2003.11.005. |
[17] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonl. Anal. RWA., 11 (2010), 55-59.
doi: doi:10.1016/j.nonrwa.2008.10.014. |
[18] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonl. Anal. RWA., 11 (2010), 3106-3109.
doi: doi:10.1016/j.nonrwa.2009.11.005. |
[19] |
M. Song, W. Ma and Y. Takeuchi, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. Comp. Appl. Math., 201 (2007), 389-394.
doi: doi:10.1016/j.cam.2005.12.039. |
[20] |
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. |
[21] |
W. Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428.
doi: doi:10.1016/S0893-9659(01)00153-7. |
[22] |
R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonl. Anal. RWA., 10 (2009), 3175-3189.
doi: doi:10.1016/j.nonrwa.2008.10.013. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I, Nature, 280 (1979), 361-367.
doi: doi:10.1038/280361a0. |
[2] |
E. Beretta and Y. Takeuchi, Convergence results in SIR epidemic models with varying population size, Nonlinear Anal., 28 (1997), 1909-1921.
doi: doi:10.1016/S0362-546X(96)00035-1. |
[3] |
G. C. Brown and R. Hasibuan, Conidial discharge and transmission efficiency of "Neozygites floridana," an entomopathogenic fungus infeccting two-spotted spider mites under laboratory conditions, J. Invertebr. Pathol., 65 (1995), 10-16.
doi: doi:10.1006/jipa.1995.1002. |
[4] |
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. |
[5] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42.
doi: doi:10.1216/RMJ-1979-9-1-31. |
[6] |
J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Springer-Verlag, 1993. |
[7] |
H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci. 28 (1976), 335-356.
doi: doi:10.1016/0025-5564(76)90132-2. |
[8] |
H. W. Hethcote, The Mathematics of Infectious Diseases, SIAM Rev., 42 (2000), 599-653. |
[9] |
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.
doi: doi:10.1007/s11538-009-9487-6. |
[10] |
A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.
doi: doi:10.1093/imammb/dqi001. |
[11] |
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626.
doi: doi:10.1007/s11538-005-9037-9. |
[12] |
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. |
[13] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press, San Diego, 1993. |
[14] |
J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications," Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961. |
[15] |
W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed delays, Tohoku. Math. J., 54 (2002), 581-591.
doi: doi:10.2748/tmj/1113247650. |
[16] |
W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141-1145.
doi: doi:10.1016/j.aml.2003.11.005. |
[17] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonl. Anal. RWA., 11 (2010), 55-59.
doi: doi:10.1016/j.nonrwa.2008.10.014. |
[18] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonl. Anal. RWA., 11 (2010), 3106-3109.
doi: doi:10.1016/j.nonrwa.2009.11.005. |
[19] |
M. Song, W. Ma and Y. Takeuchi, Permanence of a delayed SIR epidemic model with density dependent birth rate, J. Comp. Appl. Math., 201 (2007), 389-394.
doi: doi:10.1016/j.cam.2005.12.039. |
[20] |
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. |
[21] |
W. Wang, Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15 (2002), 423-428.
doi: doi:10.1016/S0893-9659(01)00153-7. |
[22] |
R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonl. Anal. RWA., 10 (2009), 3175-3189.
doi: doi:10.1016/j.nonrwa.2008.10.013. |
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