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Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey
Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility
1. | OCCAM, Mathematical Institute, 24 - 29 St Giles', Oxford, OX1 3LB, United Kingdom |
2. | Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain |
References:
[1] |
P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis, Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1-17. |
[2] |
A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 62-83.
doi: 10.1051/mmnp:2008011. |
[3] |
P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 2 (2006), 337-353.
doi: 10.1137/060654876. |
[4] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525.
doi: 10.3934/mbe.2006.3.513. |
[5] |
H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154-168.
doi: 10.1080/17513750802120877. |
[6] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[7] |
G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693.
doi: 10.1016/j.aml.2009.06.004. |
[8] |
G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203.
doi: 10.1016/j.aml.2011.02.007. |
[9] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 129-139.
doi: 10.1007/s00285-010-0368-2. |
[10] |
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differentical equations models of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708.
doi: 10.1137/090780821. |
[11] |
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: 10.1007/s11538-009-9487-6. |
[12] |
A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays, Math. Biosci., 209 (2007), 51-75.
doi: 10.1016/j.mbs.2007.01.008. |
[13] |
A. Korobeinikov, P. K. Maini and W. J. Walker, Estimation of effective vaccination rate: Pertussis in New Zealand as a case study, J. Theor. Biol., 224 (2003), 269-275.
doi: 10.1016/S0022-5193(03)00163-2. |
[14] |
A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83.
doi: 10.1007/s11538-007-9196-y. |
[15] |
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. |
[16] |
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239.
doi: 10.1093/imammb/dqp009. |
[17] |
A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321.
doi: 10.1093/imammb/dqp009. |
[18] |
A. Korobeinikov, Global properties of a general predator-prey model with non-symmetric attack and consumption rate, Discrete Cont. Dyn. Syst. Ser. B, 14 (2010), 1095-1103.
doi: 10.3934/dcdsb.2010.14.1095. |
[19] |
J. P. LaSalle, "The Stability of Dynamical Systems," SIAM, Philadelphia, 1976. |
[20] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.
doi: 10.1016/j.jmaa.2009.09.017. |
[21] |
S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosci. Eng., 7 (2010), 675-685.
doi: 10.3934/mbe.2010.7.675. |
[22] |
A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992. |
[23] |
P. Magal, C. C. McCluskey and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.
doi: 10.1080/00036810903208122. |
[24] |
V. G. Matsenko and V. N. Rubanovskii, The Lyapunov direct method for analyzing the dynamics of the age structure of biological populations, USSR Comput Maths Math. Phys., 23 (1983), 45-49. |
[25] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[26] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[27] |
C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.
doi: 10.1016/j.amc.2010.08.037. |
[28] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl., 11 (2010), 3106-3109.
doi: 10.1016/j.nonrwa.2009.11.005. |
[29] |
A. V. Melnik and A. Korobeinikov, Global asymptotic properties of staged models with multiple progression pathways for infectious diseases, Math. Biosci. Eng., 8 (2011), 1019-1034.
doi: 10.3934/mbe.2011.8.1019. |
[30] |
H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differ. Equations, 250 (2011), 3772-3801.
doi: 10.1016/j.jde.2011.01.007. |
[31] |
H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003. |
[32] |
H. R. Thieme, Disease extinction and disease persistence in age structured epidemic models, Nonlinear Anal., 47 (2001), 6181-6194.
doi: 10.1016/S0362-546X(01)00677-0. |
[33] |
V. Volterra, "Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie," Gauthier-Villars, Paris, 1931. |
show all references
References:
[1] |
P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis, Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1-17. |
[2] |
A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions, Math. Model. Nat. Phenom., 2 (2007), 62-83.
doi: 10.1051/mmnp:2008011. |
[3] |
P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM J. Appl. Math., 2 (2006), 337-353.
doi: 10.1137/060654876. |
[4] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525.
doi: 10.3934/mbe.2006.3.513. |
[5] |
H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154-168.
doi: 10.1080/17513750802120877. |
[6] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[7] |
G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693.
doi: 10.1016/j.aml.2009.06.004. |
[8] |
G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203.
doi: 10.1016/j.aml.2011.02.007. |
[9] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 129-139.
doi: 10.1007/s00285-010-0368-2. |
[10] |
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differentical equations models of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708.
doi: 10.1137/090780821. |
[11] |
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: 10.1007/s11538-009-9487-6. |
[12] |
A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays, Math. Biosci., 209 (2007), 51-75.
doi: 10.1016/j.mbs.2007.01.008. |
[13] |
A. Korobeinikov, P. K. Maini and W. J. Walker, Estimation of effective vaccination rate: Pertussis in New Zealand as a case study, J. Theor. Biol., 224 (2003), 269-275.
doi: 10.1016/S0022-5193(03)00163-2. |
[14] |
A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83.
doi: 10.1007/s11538-007-9196-y. |
[15] |
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. |
[16] |
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225-239.
doi: 10.1093/imammb/dqp009. |
[17] |
A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321.
doi: 10.1093/imammb/dqp009. |
[18] |
A. Korobeinikov, Global properties of a general predator-prey model with non-symmetric attack and consumption rate, Discrete Cont. Dyn. Syst. Ser. B, 14 (2010), 1095-1103.
doi: 10.3934/dcdsb.2010.14.1095. |
[19] |
J. P. LaSalle, "The Stability of Dynamical Systems," SIAM, Philadelphia, 1976. |
[20] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.
doi: 10.1016/j.jmaa.2009.09.017. |
[21] |
S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. Biosci. Eng., 7 (2010), 675-685.
doi: 10.3934/mbe.2010.7.675. |
[22] |
A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992. |
[23] |
P. Magal, C. C. McCluskey and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.
doi: 10.1080/00036810903208122. |
[24] |
V. G. Matsenko and V. N. Rubanovskii, The Lyapunov direct method for analyzing the dynamics of the age structure of biological populations, USSR Comput Maths Math. Phys., 23 (1983), 45-49. |
[25] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[26] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[27] |
C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.
doi: 10.1016/j.amc.2010.08.037. |
[28] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl., 11 (2010), 3106-3109.
doi: 10.1016/j.nonrwa.2009.11.005. |
[29] |
A. V. Melnik and A. Korobeinikov, Global asymptotic properties of staged models with multiple progression pathways for infectious diseases, Math. Biosci. Eng., 8 (2011), 1019-1034.
doi: 10.3934/mbe.2011.8.1019. |
[30] |
H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differ. Equations, 250 (2011), 3772-3801.
doi: 10.1016/j.jde.2011.01.007. |
[31] |
H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003. |
[32] |
H. R. Thieme, Disease extinction and disease persistence in age structured epidemic models, Nonlinear Anal., 47 (2001), 6181-6194.
doi: 10.1016/S0362-546X(01)00677-0. |
[33] |
V. Volterra, "Leçons Sur la Théorie Mathématique de la Lutte Pour la Vie," Gauthier-Villars, Paris, 1931. |
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