Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models

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2015, 12(1): 117-133. doi: 10.3934/mbe.2015.12.117

Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models

Tsuyoshi Kajiwara ^1, , Toru Sasaki ^1, and Yasuhiro Takeuchi ^2,

1.	Graduate School of Environmental and Life Sciences, Okayama University, 3-1-1, Tsushima-Naka, Okayama, Japan, Japan
2.	College of Science and Engineering, Aoyama Gakuin University, 5-10-1, Fuchinobe, Sagamihara, Japan

Received March 2014 Revised September 2014 Published December 2014

Abstract
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We present a constructive method for Lyapunov functions for ordinary differential equation models of infectious diseases in vivo. We consider models derived from the Nowak-Bangham models. We construct Lyapunov functions for complex models using those of simpler models. Especially, we construct Lyapunov functions for models with an immune variable from those for models without an immune variable, a Lyapunov functions of a model with absorption effect from that for a model without absorption effect. We make the construction clear for Lyapunov functions proposed previously, and present new results with our method.

Keywords: stability, Lyapunov functions, immunity., ordinary differential equations, infectious diseases.

Mathematics Subject Classification: Primary: 92B05, 92D25; Secondary: 34D2.

Citation: Tsuyoshi Kajiwara, Toru Sasaki, Yasuhiro Takeuchi. Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models. Mathematical Biosciences & Engineering, 2015, 12 (1) : 117-133. doi: 10.3934/mbe.2015.12.117

References:

[1]	H. Gomez-Acevedo, M. Y. Li and J. Steven, 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.
[2]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[3]	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.
[4]	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.
[5]	T. Kajiwara and T. Sasaki, Global stability of pathogen-immune dynamics with absorption, J. Biol Dyn. , 4 (2010), 258-269. doi: 10.1080/17513750903051989.
[6]	T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonl. Anal. RWA, 13 (2012), 1802-1826. doi: 10.1016/j.nonrwa.2011.12.011.
[7]	A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.
[8]	A. Korobeinikov, Global properties of infectious disease model with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y.
[9]	M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Eq., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.
[10]	J. Lang and M. Y. Li, Stable and transient periodic model for CTL response to HTLV-I infection, J. Math. Biol., 65 (2012), 181-199. doi: 10.1007/s00285-011-0455-z.
[11]	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.
[12]	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.
[13]	H. Pang, W. Wang and K. Wang, Global properties of virus dynamics with CTL immune response, Southwest China Normal Univ., 30 (2005), 797-799.
[14]	R. Qesmi, J. Wu, J. Wu and J. M. Heffernan, Influence of backward bifurcation in a model of hepatitis B and C virus, Math. Biosc., 224 (2010), 118-125. doi: 10.1016/j.mbs.2010.01.002.

show all references

References:

[1]	H. Gomez-Acevedo, M. Y. Li and J. Steven, 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.
[2]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.
[3]	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.
[4]	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.
[5]	T. Kajiwara and T. Sasaki, Global stability of pathogen-immune dynamics with absorption, J. Biol Dyn. , 4 (2010), 258-269. doi: 10.1080/17513750903051989.
[6]	T. Kajiwara, T. Sasaki and Y. Takeuchi, Construction of Lyapunov functionals for delay differential equations in virology and epidemiology, Nonl. Anal. RWA, 13 (2012), 1802-1826. doi: 10.1016/j.nonrwa.2011.12.011.
[7]	A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001.
[8]	A. Korobeinikov, Global properties of infectious disease model with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y.
[9]	M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Eq., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.
[10]	J. Lang and M. Y. Li, Stable and transient periodic model for CTL response to HTLV-I infection, J. Math. Biol., 65 (2012), 181-199. doi: 10.1007/s00285-011-0455-z.
[11]	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.
[12]	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.
[13]	H. Pang, W. Wang and K. Wang, Global properties of virus dynamics with CTL immune response, Southwest China Normal Univ., 30 (2005), 797-799.
[14]	R. Qesmi, J. Wu, J. Wu and J. M. Heffernan, Influence of backward bifurcation in a model of hepatitis B and C virus, Math. Biosc., 224 (2010), 118-125. doi: 10.1016/j.mbs.2010.01.002.

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