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A SEIR model for control of infectious diseases with constraints
Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
1. | Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan |
2. | Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1, Hungary |
References:
[1] |
E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis, 47 (2001), 4107-4115.
doi: 10.1016/S0362-546X(01)00528-4. |
[2] |
E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952.
doi: 10.3934/mbe.2011.8.931. |
[3] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.
doi: 10.1016/0025-5564(78)90006-8. |
[4] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42.
doi: 10.1216/RMJ-1979-9-1-31. |
[5] |
O. Diekmann and K. Korvasova, A didactical note on the advantage of using two parameters in Hopf bifurcation studies, J. Biological Dynamics, 7 (2013), 21-30.
doi: 10.1080/17513758.2012.760758. |
[6] |
O. Diekmann, S. A. van Gils, S. M. V. Lunel and H. O. Walther, Delay Equations Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[7] |
Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal. RWA., 13 (2012), 2120-2133.
doi: 10.1016/j.nonrwa.2012.01.007. |
[8] |
Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 61-74.
doi: 10.3934/dcdsb.2011.15.61. |
[9] |
F. R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959 (Translated from Russian). |
[10] |
H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.
doi: 10.1007/BF00160539. |
[11] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[12] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamics models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.
doi: 10.1007/s00285-010-0368-2. |
[13] |
Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcation of an SIRS epidemic model with nonlinear incidence rate, Disc. Cont. Dynam. Sys. B, 15 (2011), 93-112.
doi: 10.3934/dcdsb.2011.15.93. |
[14] |
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.
doi: 10.1007/s11538-007-9196-y. |
[15] |
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: 10.1007/s11538-005-9037-9. |
[16] |
A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.
doi: 10.1093/imammb/dqi001. |
[17] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. |
[18] |
Y. N. Kyrychko and K. B. Blyuss, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. RWA., 6 (2005), 495-507.
doi: 10.1016/j.nonrwa.2004.10.001. |
[19] |
W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.
doi: 10.1007/BF00276956. |
[20] |
W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.
doi: 10.1007/BF00277162. |
[21] |
C. C. McCluskey, Global stability of an SIR epidemic model with delay and general incidence, Math. Biosci. Eng., 7 (2010), 837-850.
doi: 10.3934/mbe.2010.7.837. |
[22] |
Y. Muroya, Y. Enatsu and Y. Nakata, Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays, Nonlinear Anal. RWA., 12 (2011), 1897-1910.
doi: 10.1016/j.nonrwa.2010.12.002. |
[23] |
Y. Nakata, Y. Enatsu and Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays, Disc. Cont. Dynam. Sys. Supplement, II (2011), 1119-1128. |
[24] |
W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976. |
[25] |
S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163.
doi: 10.1016/S0022-0396(02)00089-X. |
[26] |
H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics Vol. 57, Springer, Berlin, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[27] |
Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delayed SIR epidemic model with finite incubation time, Nonlinear Anal. TMA., 42 (2000), 931-947.
doi: 10.1016/S0362-546X(99)00138-8. |
[28] |
W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279.
doi: 10.3934/mbe.2006.3.267. |
[29] |
D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
[30] |
R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos, Solitons & Fractals, 41 (2009), 2319-2325.
doi: 10.1016/j.chaos.2008.09.007. |
[31] |
Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B, 13 (2010), 195-211.
doi: 10.3934/dcdsb.2010.13.195. |
show all references
References:
[1] |
E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis, 47 (2001), 4107-4115.
doi: 10.1016/S0362-546X(01)00528-4. |
[2] |
E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952.
doi: 10.3934/mbe.2011.8.931. |
[3] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.
doi: 10.1016/0025-5564(78)90006-8. |
[4] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42.
doi: 10.1216/RMJ-1979-9-1-31. |
[5] |
O. Diekmann and K. Korvasova, A didactical note on the advantage of using two parameters in Hopf bifurcation studies, J. Biological Dynamics, 7 (2013), 21-30.
doi: 10.1080/17513758.2012.760758. |
[6] |
O. Diekmann, S. A. van Gils, S. M. V. Lunel and H. O. Walther, Delay Equations Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[7] |
Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal. RWA., 13 (2012), 2120-2133.
doi: 10.1016/j.nonrwa.2012.01.007. |
[8] |
Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 61-74.
doi: 10.3934/dcdsb.2011.15.61. |
[9] |
F. R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959 (Translated from Russian). |
[10] |
H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.
doi: 10.1007/BF00160539. |
[11] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[12] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamics models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.
doi: 10.1007/s00285-010-0368-2. |
[13] |
Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcation of an SIRS epidemic model with nonlinear incidence rate, Disc. Cont. Dynam. Sys. B, 15 (2011), 93-112.
doi: 10.3934/dcdsb.2011.15.93. |
[14] |
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.
doi: 10.1007/s11538-007-9196-y. |
[15] |
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: 10.1007/s11538-005-9037-9. |
[16] |
A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128.
doi: 10.1093/imammb/dqi001. |
[17] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. |
[18] |
Y. N. Kyrychko and K. B. Blyuss, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. RWA., 6 (2005), 495-507.
doi: 10.1016/j.nonrwa.2004.10.001. |
[19] |
W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.
doi: 10.1007/BF00276956. |
[20] |
W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.
doi: 10.1007/BF00277162. |
[21] |
C. C. McCluskey, Global stability of an SIR epidemic model with delay and general incidence, Math. Biosci. Eng., 7 (2010), 837-850.
doi: 10.3934/mbe.2010.7.837. |
[22] |
Y. Muroya, Y. Enatsu and Y. Nakata, Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays, Nonlinear Anal. RWA., 12 (2011), 1897-1910.
doi: 10.1016/j.nonrwa.2010.12.002. |
[23] |
Y. Nakata, Y. Enatsu and Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays, Disc. Cont. Dynam. Sys. Supplement, II (2011), 1119-1128. |
[24] |
W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976. |
[25] |
S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163.
doi: 10.1016/S0022-0396(02)00089-X. |
[26] |
H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics Vol. 57, Springer, Berlin, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[27] |
Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delayed SIR epidemic model with finite incubation time, Nonlinear Anal. TMA., 42 (2000), 931-947.
doi: 10.1016/S0362-546X(99)00138-8. |
[28] |
W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279.
doi: 10.3934/mbe.2006.3.267. |
[29] |
D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.
doi: 10.1016/j.mbs.2006.09.025. |
[30] |
R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos, Solitons & Fractals, 41 (2009), 2319-2325.
doi: 10.1016/j.chaos.2008.09.007. |
[31] |
Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B, 13 (2010), 195-211.
doi: 10.3934/dcdsb.2010.13.195. |
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