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March  2017, 22(2): 507-536. doi: 10.3934/dcdsb.2017025

Lyapunov functionals for multistrain models with infinite delay

Graduate School of Environmental and Life Science, Okayama University, Okayama, 700-8530, Japan

Received  February 2016 Revised  October 2016 Published  December 2016

We construct Lyapunov functionals for delay differential equation models of infectious diseases in vivo to analyze the stability of the equilibria. The Lyapunov functionals contain the terms that integrate over all previous states. An appropriate evaluation of the logarithm functions in those terms guarantees the existence of the integrals. We apply the rigorous analysis for the one-strain models to multistrain models by using mathematical induction.

Citation: Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for multistrain models with infinite delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 507-536. doi: 10.3934/dcdsb.2017025
References:
[1]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347.   Google Scholar

[2]

C. J. Browne, A multi-strain virus model with infected cell age structure: application to HIV, Nonlinear Anal., 22 (2015), 354-372.  doi: 10.1016/j.nonrwa.2014.10.004.  Google Scholar

[3]

R. D. Demasse and A. Ducrot, An age-structured within-host model for multistrain malaria infections, SIAM J. Math. Anal., 73 (2013), 572-593.  doi: 10.1137/120890351.  Google Scholar

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[5]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[6]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473. Springer, Berlin, 1991. Google Scholar

[7]

A. IggidrJ.-C. KamgangG. 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

[8]

T. InoueT. 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

[9]

Y. IwasaF. Michor and M. Nowak, Some basic properties of immune selection, J. Theoret. Biol., 229 (2004), 179-188.  doi: 10.1016/j.jtbi.2004.03.013.  Google Scholar

[10]

T. KajiwaraT. 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.   Google Scholar

[11]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  doi: 10.3934/mbe.2009.6.603.  Google Scholar

[12]

Y. OtaniT. Kajiwara and T. Sasaki, Lyapunov functionals for virus-immune models with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3093-3114.  doi: 10.3934/dcdsb.2015.20.3093.  Google Scholar

[13] H. A. Priestley, Introduction to Integration, Oxford University Press, New York, 1997.   Google Scholar
[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.  Google Scholar

[15]

M. O. Souza and J. P. Zubelli, Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation, Bull. Math. Biol., 73 (2011), 609-625.  doi: 10.1007/s11538-010-9543-2.  Google Scholar

[16]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. Amer. Math. Soc., 127 (1999), 2395-2403.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar

[17]

H. R. Thieme, Pathogen competition and coexistence and the evolution of virulence, Mathematics for Life Sciences and Medicine, Springer, Berlin Heidelberg, (2007), 123-153. Google Scholar

[18]

J. WangG. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300.  doi: 10.1093/imammb/dqr009.  Google Scholar

[19]

D. Wodarz, Hepatitis C virus dynamics and pathology: The roles of CTL and antibody responces, J. Gen. Virol., 84 (2003), 1743-1750.   Google Scholar

show all references

References:
[1]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347.   Google Scholar

[2]

C. J. Browne, A multi-strain virus model with infected cell age structure: application to HIV, Nonlinear Anal., 22 (2015), 354-372.  doi: 10.1016/j.nonrwa.2014.10.004.  Google Scholar

[3]

R. D. Demasse and A. Ducrot, An age-structured within-host model for multistrain malaria infections, SIAM J. Math. Anal., 73 (2013), 572-593.  doi: 10.1137/120890351.  Google Scholar

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.   Google Scholar

[5]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.  doi: 10.1137/0520025.  Google Scholar

[6]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, vol. 1473. Springer, Berlin, 1991. Google Scholar

[7]

A. IggidrJ.-C. KamgangG. 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

[8]

T. InoueT. 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

[9]

Y. IwasaF. Michor and M. Nowak, Some basic properties of immune selection, J. Theoret. Biol., 229 (2004), 179-188.  doi: 10.1016/j.jtbi.2004.03.013.  Google Scholar

[10]

T. KajiwaraT. 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.   Google Scholar

[11]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  doi: 10.3934/mbe.2009.6.603.  Google Scholar

[12]

Y. OtaniT. Kajiwara and T. Sasaki, Lyapunov functionals for virus-immune models with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3093-3114.  doi: 10.3934/dcdsb.2015.20.3093.  Google Scholar

[13] H. A. Priestley, Introduction to Integration, Oxford University Press, New York, 1997.   Google Scholar
[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.  Google Scholar

[15]

M. O. Souza and J. P. Zubelli, Global stability for a class of virus models with cytotoxic T lymphocyte immune response and antigenic variation, Bull. Math. Biol., 73 (2011), 609-625.  doi: 10.1007/s11538-010-9543-2.  Google Scholar

[16]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. Amer. Math. Soc., 127 (1999), 2395-2403.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar

[17]

H. R. Thieme, Pathogen competition and coexistence and the evolution of virulence, Mathematics for Life Sciences and Medicine, Springer, Berlin Heidelberg, (2007), 123-153. Google Scholar

[18]

J. WangG. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300.  doi: 10.1093/imammb/dqr009.  Google Scholar

[19]

D. Wodarz, Hepatitis C virus dynamics and pathology: The roles of CTL and antibody responces, J. Gen. Virol., 84 (2003), 1743-1750.   Google Scholar

Figure 1.  $x^* \ne \hat{x}_i$ for all $i\in J$: in this case $K_J=\{1,2\}$
Figure 2.  $x^* = \hat{x}_i$ for some $i\in J$: in this case $K_J=\{1,2,3\}, x^* = \hat{x}_3$
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