American Institute of Mathematical Sciences

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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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