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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

 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.
Citation: Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for virus-immune models with infinite delay. Discrete & 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.   Google Scholar [2] T. Burton and V. Hutson, Repellers in systems with infinite delay,, J. Math. Anal. Appl., 137 (1989), 240.  doi: 10.1016/0022-247X(89)90287-4.  Google Scholar [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.  doi: 10.1007/s11538-009-9465-z.  Google Scholar [4] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar [5] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388.  doi: 10.1137/0520025.  Google Scholar [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.  doi: 10.1137/050643271.  Google Scholar [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.  doi: 10.1080/17513750903180275.  Google Scholar [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.  doi: 10.1016/j.nonrwa.2011.12.011.  Google Scholar [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.   Google Scholar [10] A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar [11] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603.  doi: 10.3934/mbe.2009.6.603.  Google Scholar [12] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics,, J. Math. Biol., 51 (2005), 247.  doi: 10.1007/s00285-005-0321-y.  Google Scholar [13] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses,, Science, 272 (1996), 74.  doi: 10.1126/science.272.5258.74.  Google Scholar [14] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389.  doi: 10.3934/mbe.2008.5.389.  Google Scholar [15] H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proceedings of the American Mathematical Society, 127 (1999), 2395.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar [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.  doi: 10.1093/imammb/dqr009.  Google Scholar

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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.   Google Scholar [2] T. Burton and V. Hutson, Repellers in systems with infinite delay,, J. Math. Anal. Appl., 137 (1989), 240.  doi: 10.1016/0022-247X(89)90287-4.  Google Scholar [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.  doi: 10.1007/s11538-009-9465-z.  Google Scholar [4] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar [5] J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388.  doi: 10.1137/0520025.  Google Scholar [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.  doi: 10.1137/050643271.  Google Scholar [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.  doi: 10.1080/17513750903180275.  Google Scholar [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.  doi: 10.1016/j.nonrwa.2011.12.011.  Google Scholar [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.   Google Scholar [10] A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar [11] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603.  doi: 10.3934/mbe.2009.6.603.  Google Scholar [12] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics,, J. Math. Biol., 51 (2005), 247.  doi: 10.1007/s00285-005-0321-y.  Google Scholar [13] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses,, Science, 272 (1996), 74.  doi: 10.1126/science.272.5258.74.  Google Scholar [14] G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 5 (2008), 389.  doi: 10.3934/mbe.2008.5.389.  Google Scholar [15] H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proceedings of the American Mathematical Society, 127 (1999), 2395.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar [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.  doi: 10.1093/imammb/dqr009.  Google Scholar
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