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A minimal mathematical model for the initial molecular interactions of death receptor signalling
Global properties of a delayed SIR epidemic model with multiple parallel infectious stages
1. | Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street Harbin, 150080 and College of Mathematics and Information Science, Xinyang Normal University, Xinyang, 464000, China |
2. | Academy of Fundamental and Interdisciplinary Sciences, Harbin Institute of Technology, 3041#, 2 Yi-Kuang Street, Harbin, 150080 |
References:
[1] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math., 9 (1979), 31-42. |
[2] |
R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T cells, Math. Biosci., 165 (2000), 27-39.
doi: 10.1016/S0025-5564(00)00006-7. |
[3] |
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), 365-382.
doi: 10.1007/BF00178324. |
[4] |
S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dynam., 2 (2008), 140-153. |
[5] |
H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.
doi: 10.1016/0025-5564(76)90132-2. |
[6] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[7] |
V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251.
doi: 10.1073/pnas.93.14.7247. |
[8] |
J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. |
[9] |
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/s11538-008-9352-z. |
[10] |
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), 57-60. |
[11] |
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. |
[12] |
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. |
[13] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. |
[14] |
M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505.
doi: 10.1007/s11538-010-9503-x. |
[15] |
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), 119-127.
doi: 10.1016/j.nonrwa.2010.06.001. |
[16] |
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: 10.1137/090779322. |
[17] |
C. C. McCluskey, Global stability of an epidemic model with delay and general nonlinear incidence, Math. Biosci. and Eng., 7 (2010), 837-850. |
[18] |
P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.
doi: 10.1080/00036810903208122. |
[19] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 6 (2009), 603-610. |
[20] |
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. |
[21] |
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. |
[22] |
K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109.
doi: 10.1016/j.mbs.2011.11.002. |
[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: 10.1016/S0362-546X(99)00138-8. |
[24] |
J. Wang, G. Huang, Y. Takeuchi and S. Liu, Sveir epidemiological model with varying infectivity and distributed delays, Math. Biosci. and Eng., 8 (2011), 875-888. |
[25] |
X. Wang, Y. D. Tao and X. Y. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dyn., 62 (2010), 67-72.
doi: 10.1007/s11071-010-9699-1. |
show all references
References:
[1] |
K. L. Cooke, Stability analysis for a vector disease model, Rocky Mount. J. Math., 9 (1979), 31-42. |
[2] |
R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T cells, Math. Biosci., 165 (2000), 27-39.
doi: 10.1016/S0025-5564(00)00006-7. |
[3] |
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), 365-382.
doi: 10.1007/BF00178324. |
[4] |
S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biol. Dynam., 2 (2008), 140-153. |
[5] |
H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.
doi: 10.1016/0025-5564(76)90132-2. |
[6] |
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
doi: 10.1137/S0036144500371907. |
[7] |
V. Herz, S. Bonhoeffer, R. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimations on intracellular delay and virus delay, Proc. Natl. Acad. Sci. USA, 93 (1996), 7247-7251.
doi: 10.1073/pnas.93.14.7247. |
[8] |
J. Hale and S. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. |
[9] |
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/s11538-008-9352-z. |
[10] |
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), 57-60. |
[11] |
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. |
[12] |
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. |
[13] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. |
[14] |
M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505.
doi: 10.1007/s11538-010-9503-x. |
[15] |
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), 119-127.
doi: 10.1016/j.nonrwa.2010.06.001. |
[16] |
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: 10.1137/090779322. |
[17] |
C. C. McCluskey, Global stability of an epidemic model with delay and general nonlinear incidence, Math. Biosci. and Eng., 7 (2010), 837-850. |
[18] |
P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.
doi: 10.1080/00036810903208122. |
[19] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. and Eng., 6 (2009), 603-610. |
[20] |
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. |
[21] |
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. |
[22] |
K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109.
doi: 10.1016/j.mbs.2011.11.002. |
[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: 10.1016/S0362-546X(99)00138-8. |
[24] |
J. Wang, G. Huang, Y. Takeuchi and S. Liu, Sveir epidemiological model with varying infectivity and distributed delays, Math. Biosci. and Eng., 8 (2011), 875-888. |
[25] |
X. Wang, Y. D. Tao and X. Y. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dyn., 62 (2010), 67-72.
doi: 10.1007/s11071-010-9699-1. |
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