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October  2013, 18(8): 1999-2017. doi: 10.3934/dcdsb.2013.18.1999

Global analysis of age-structured within-host virus model

Cameron J. Browne 1, and  Sergei S. Pilyugin 1,

1. 

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, United States

Received  March 2012 Revised  December 2012 Published  July 2013

A mathematical model of a within-host viral infection with explicit age-since-infection structure for infected cells is presented. A global analysis of the model is conducted. It is shown that when the basic reproductive number falls below unity, the infection dies out. On the contrary, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium that attracts all positive solutions of the model. The global stability analysis combines the existence of a compact global attractor and a Lyapunov function.
Keywords: Lyapunov functional., viral infection, Mathematical model, global stability analysis, age-structure.
Mathematics Subject Classification: Primary: 37N25, 92B05; Secondary: 37B2.
Citation: Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999
References:
[1]

R. Adams and J. Fournier, "Sobolev Spaces,'' Second edition, Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003.

[2]

C. L. Althaus, A. S. De Vos and R. J. De Boer, Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, $R_0$, J. Virol., 83 (2009), 7659-7667.

[3]

P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Math. Med. Biol., 25 (2008), 285-322.

[4]

M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, J. Theor. Biol., 229 (2004), 281-288. doi: 10.1016/j.jtbi.2004.04.015.

[5]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Math. Surv. Monogr., 25, Am. Math. Soc., Providence, RI, 1988.

[6]

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

[7]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588.

[8]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.

[9]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[10]

P. W. Nelson, M. A. Gilchrist, D. Coombs, J. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267.

[11]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8.

[12]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[13]

L. B. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945.

[14]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.

[15]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'' Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.

show all references

References:
[1]

R. Adams and J. Fournier, "Sobolev Spaces,'' Second edition, Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003.

[2]

C. L. Althaus, A. S. De Vos and R. J. De Boer, Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, $R_0$, J. Virol., 83 (2009), 7659-7667.

[3]

P. De Leenheer and S. S. Pilyugin, Multistrain virus dynamics with mutations: A global analysis, Math. Med. Biol., 25 (2008), 285-322.

[4]

M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, J. Theor. Biol., 229 (2004), 281-288. doi: 10.1016/j.jtbi.2004.04.015.

[5]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,'' Math. Surv. Monogr., 25, Am. Math. Soc., Providence, RI, 1988.

[6]

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

[7]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588.

[8]

P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.

[9]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173.

[10]

P. W. Nelson, M. A. Gilchrist, D. Coombs, J. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288. doi: 10.3934/mbe.2004.1.267.

[11]

P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94. doi: 10.1016/S0025-5564(02)00099-8.

[12]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[13]

L. B. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy, SIAM J. Appl. Math., 67 (2007), 731-756. doi: 10.1137/060663945.

[14]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.

[15]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics,'' Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.

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