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A note on global stability for malaria infections model with latencies
1. | School of Mathematical Science, Heilongjiang University, Harbin 150080, China, China |
2. | Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914 |
References:
[1] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347. |
[2] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. |
[3] |
J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol.99. Applied Mathematical Science, New York, 1993. |
[4] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. |
[5] |
W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.
doi: 10.1002/cpa.3160380607. |
[6] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.
doi: 10.1007/s00285-010-0368-2. |
[7] |
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708.
doi: 10.1137/090780821. |
[8] |
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.
doi: 10.3934/mbe.2004.1.57. |
[9] |
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. |
[10] |
J. P. Lasalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. |
[11] |
R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., New York, 1971. |
[12] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delaydistributed or discrete, Nonlinear Anal. RWA., 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[13] |
C. C. McCluskey, Global stability for an SIER epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[14] |
H. R. Thieme, Mathematics In Population Biology, Princeton University Press, Princeton, NJ, 2003. |
[15] |
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300.
doi: 10.1093/imammb/dqr009. |
[16] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219.
doi: 10.3934/mbe.2007.4.205. |
[17] |
Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10 (2013), 463-481.
doi: 10.3934/mbe.2013.10.463. |
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] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. |
[3] |
J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol.99. Applied Mathematical Science, New York, 1993. |
[4] |
J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41. |
[5] |
W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.
doi: 10.1002/cpa.3160380607. |
[6] |
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.
doi: 10.1007/s00285-010-0368-2. |
[7] |
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708.
doi: 10.1137/090780821. |
[8] |
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.
doi: 10.3934/mbe.2004.1.57. |
[9] |
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. |
[10] |
J. P. Lasalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. |
[11] |
R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., New York, 1971. |
[12] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delaydistributed or discrete, Nonlinear Anal. RWA., 11 (2010), 55-59.
doi: 10.1016/j.nonrwa.2008.10.014. |
[13] |
C. C. McCluskey, Global stability for an SIER epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[14] |
H. R. Thieme, Mathematics In Population Biology, Princeton University Press, Princeton, NJ, 2003. |
[15] |
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300.
doi: 10.1093/imammb/dqr009. |
[16] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219.
doi: 10.3934/mbe.2007.4.205. |
[17] |
Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10 (2013), 463-481.
doi: 10.3934/mbe.2013.10.463. |
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