Lyapunov functionals for virus-immune models with infinite delay

Yoji  Otani; Tsuyoshi Kajiwara; Toru  Sasaki

doi:10.3934/dcdsb.2015.20.3093

Article Contents

2015, Volume 20, Issue 9: 3093-3114. Doi: 10.3934/dcdsb.2015.20.3093

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Lyapunov functionals for virus-immune models with infinite delay

1.
Graduate School of Environmental Science, Okayama University, Okayama, 700-8530
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

Received: October 2014

Revised: April 2015

Published online: September 2015

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Abstract

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:

Mathematics Subject Classification: Primary: 92D25; Secondary: 34D20.

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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.