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November  2015, 20(9): 3093-3114. doi: 10.3934/dcdsb.2015.20.3093

Lyapunov functionals for virus-immune models with infinite delay

Yoji Otani 1, , Tsuyoshi Kajiwara 2, and  Toru Sasaki 3,

1. 

Graduate School of Environmental Science, Okayama University, Okayama, 700-8530, Japan
2. 

Graduate School of Environmental and Life Science, Okayama University, Tsushimanaka 3-1-1, Okayama, 700-8530
3. 

Graduate School of Environmental and Life Science, Okayama University, Okayama, 700-8530, Japan

Received  October 2014 Revised  April 2015 Published  September 2015

We present a systematic method to construct Lyapunov functionals of delay differential equation models of infectious diseases in vivo. For generality we construct Lyapunov functionals of models with infinitely distributed delay. We begin with simpler models without delay and construct Lyapunov functionals for the complex models progressively. We construct those functionals using our result obtained previously instead of constructing each functional independently. Additionally we discuss some problems that arise from the mathematical requirements caused by the infinitely distributed delay.
Keywords: global stability., Lyapunov functional, infinitely distributed delay, delay differential equation, virus-immune model.
Mathematics Subject Classification: Primary: 92D25; Secondary: 34D2.
Citation: Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for virus-immune models with infinite delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3093-3114. doi: 10.3934/dcdsb.2015.20.3093
References:
[1]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347.

[2]

T. Burton and V. Hutson, Repellers in systems with infinite delay, J. Math. Anal. Appl., 137 (1989), 240-263. doi: 10.1016/0022-247X(89)90287-4.

[3]

H. Gomez-Acevedo , M. Y. Li and S. Jacobson, Multistability in a model for CTL response to HTLV-I infection and its implications to HAM/TSP development and prevention, Bull. Math. Biol., 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z.

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.

[5]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.

[6]

A. Iggidr, J.-C. Kamgang, G. Sallet and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle, SIAM J. Appl. Math., 67 (2006), 260-278. doi: 10.1137/050643271.

[7]

T. Inoue, T. Kajiwara and T. Sasaki, Global stability of models of humoral immunity against multiple viral strains, J. Biol. Dyn., 4 (2010), 282-295. doi: 10.1080/17513750903180275.

[8]

T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Analysis RWA, 13 (2012), 1802-1826. doi: 10.1016/j.nonrwa.2011.12.011.

[9]

T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functions of the models for infectious diseases in vivo: from simple models to complex models, Math. Biosci. Eng., 12 (2015), 117-133.

[10]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.

[11]

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.

[12]

A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267. doi: 10.1007/s00285-005-0321-y.

[13]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[14]

G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402. doi: 10.3934/mbe.2008.5.389.

[15]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proceedings of the American Mathematical Society, 127 (1999), 2395-2403. doi: 10.1090/S0002-9939-99-05034-0.

[16]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Mathematical Medicine and Biology, 29 (2012), 283-300. doi: 10.1093/imammb/dqr009.

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]

T. Burton and V. Hutson, Repellers in systems with infinite delay, J. Math. Anal. Appl., 137 (1989), 240-263. doi: 10.1016/0022-247X(89)90287-4.

[3]

H. Gomez-Acevedo , M. Y. Li and S. Jacobson, Multistability in a model for CTL response to HTLV-I infection and its implications to HAM/TSP development and prevention, Bull. Math. Biol., 72 (2010), 681-696. doi: 10.1007/s11538-009-9465-z.

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.

[5]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.

[6]

A. Iggidr, J.-C. Kamgang, G. Sallet and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle, SIAM J. Appl. Math., 67 (2006), 260-278. doi: 10.1137/050643271.

[7]

T. Inoue, T. Kajiwara and T. Sasaki, Global stability of models of humoral immunity against multiple viral strains, J. Biol. Dyn., 4 (2010), 282-295. doi: 10.1080/17513750903180275.

[8]

T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonlinear Analysis RWA, 13 (2012), 1802-1826. doi: 10.1016/j.nonrwa.2011.12.011.

[9]

T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functions of the models for infectious diseases in vivo: from simple models to complex models, Math. Biosci. Eng., 12 (2015), 117-133.

[10]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.

[11]

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.

[12]

A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247-267. doi: 10.1007/s00285-005-0321-y.

[13]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[14]

G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402. doi: 10.3934/mbe.2008.5.389.

[15]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proceedings of the American Mathematical Society, 127 (1999), 2395-2403. doi: 10.1090/S0002-9939-99-05034-0.

[16]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Mathematical Medicine and Biology, 29 (2012), 283-300. doi: 10.1093/imammb/dqr009.

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