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

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.
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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.  doi: 10.1016/j.mbs.2010.01.002.  Google Scholar

[1]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

[2]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116

[3]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39

[4]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

[5]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[6]

Zhilan Feng, Carlos Castillo-Chavez. The influence of infectious diseases on population genetics. Mathematical Biosciences & Engineering, 2006, 3 (3) : 467-483. doi: 10.3934/mbe.2006.3.467

[7]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87

[8]

Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353

[9]

Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031

[10]

Hongbin Guo, Michael Yi Li. Global dynamics of a staged progression model for infectious diseases. Mathematical Biosciences & Engineering, 2006, 3 (3) : 513-525. doi: 10.3934/mbe.2006.3.513

[11]

M. H. A. Biswas, L. T. Paiva, MdR de Pinho. A SEIR model for control of infectious diseases with constraints. Mathematical Biosciences & Engineering, 2014, 11 (4) : 761-784. doi: 10.3934/mbe.2014.11.761

[12]

Darja Kalajdzievska, Michael Yi Li. Modeling the effects of carriers on transmission dynamics of infectious diseases. Mathematical Biosciences & Engineering, 2011, 8 (3) : 711-722. doi: 10.3934/mbe.2011.8.711

[13]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209

[14]

Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165

[15]

Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040

[16]

Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control & Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517

[17]

Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283

[18]

Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91

[19]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

[20]

Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

[Back to Top]