American Institute of Mathematical Sciences

Advanced
2015, 12(4): 859-877. doi: 10.3934/mbe.2015.12.859

Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function

Yu Yang 1, , Shigui Ruan 2, and  Dongmei Xiao 3,

1. 

School of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, China
2. 

Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250
3. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received  April 2014 Revised  December 2014 Published  April 2015

In this paper, we study an age-structured virus dynamics model with Beddington-DeAngelis infection function. An explicit formula for the basic reproductive number $\mathcal{R}_{0}$ of the model is obtained. We investigate the global behavior of the model in terms of $\mathcal{R}_{0}$: if $\mathcal{R}_{0}\leq1$, then the infection-free equilibrium is globally asymptotically stable, whereas if $\mathcal{R}_{0}>1$, then the infection equilibrium is globally asymptotically stable. Finally, some special cases, which reduce to some known HIV infection models studied by other researchers, are considered.
Keywords: Age structure, virus dynamics model, Liapunov function, infection equilibrium, global stability..
Mathematics Subject Classification: Primary: 35L60, 92C37; Secondary: 35B35, 34K2.
Citation: Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
References:
[1]

C. L. Althaus and R. J. De Boer, Dynamics of immune escape during HIV/SIV infection, PLoS Comput. Biol., 4 (2008), e1000103, 9pp. doi: 10.1371/journal.pcbi.1000103.  Google Scholar

[2]

F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10 (2013), 1335-1349. doi: 10.3934/mbe.2013.10.1335.  Google Scholar

[3]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[4]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[5]

C. Cosner, D.L. DeAngelis, J. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Pop. Biol., 56 (1999), 65-75. doi: 10.1006/tpbi.1999.1414.  Google Scholar

[6]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol., 190 (1998), 201-214. Google Scholar

[7]

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

[8]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.  Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Vol 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[10]

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

[11]

G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004.  Google Scholar

[12]

G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

G. Huisman and R. J. De Boer, A formal derivation of the "Beddington" functional response, J. Theoret. Biol., 185 (1997), 389-400. doi: 10.1006/jtbi.1996.0318.  Google Scholar

[16]

D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390. doi: 10.1016/0092-8240(95)00345-2.  Google Scholar

[17]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.  Google Scholar

[18]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322.  Google Scholar

[19]

P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.  Google Scholar

[20]

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.  Google Scholar

[21]

P. Magal and C. C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056.  Google Scholar

[22]

P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695.  Google Scholar

[23]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819.  Google Scholar

[24]

P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. 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.  Google Scholar

[25]

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

[26]

M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.  Google Scholar

[27]

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.  Google Scholar

[28]

L. 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.  Google Scholar

[29]

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

show all references

References:
[1]

C. L. Althaus and R. J. De Boer, Dynamics of immune escape during HIV/SIV infection, PLoS Comput. Biol., 4 (2008), e1000103, 9pp. doi: 10.1371/journal.pcbi.1000103.  Google Scholar

[2]

F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 10 (2013), 1335-1349. doi: 10.3934/mbe.2013.10.1335.  Google Scholar

[3]

C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017. doi: 10.3934/dcdsb.2013.18.1999.  Google Scholar

[4]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T-cells, Math. Biosci., 165 (2000), 27-39. doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[5]

C. Cosner, D.L. DeAngelis, J. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theoret. Pop. Biol., 56 (1999), 65-75. doi: 10.1006/tpbi.1999.1414.  Google Scholar

[6]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, J. Theoret. Biol., 190 (1998), 201-214. Google Scholar

[7]

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

[8]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327. doi: 10.1137/S0036139902406905.  Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Vol 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[10]

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

[11]

G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 22 (2009), 1690-1693. doi: 10.1016/j.aml.2009.06.004.  Google Scholar

[12]

G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Appl. Math. Lett., 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007.  Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

G. Huisman and R. J. De Boer, A formal derivation of the "Beddington" functional response, J. Theoret. Biol., 185 (1997), 389-400. doi: 10.1006/jtbi.1996.0318.  Google Scholar

[16]

D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390. doi: 10.1016/0092-8240(95)00345-2.  Google Scholar

[17]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.  Google Scholar

[18]

M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM J. Appl. Math., 70 (2010), 2434-2448. doi: 10.1137/090779322.  Google Scholar

[19]

P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 65 (2001), 1-35.  Google Scholar

[20]

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.  Google Scholar

[21]

P. Magal and C. C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095. doi: 10.1137/120882056.  Google Scholar

[22]

P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727. doi: 10.3934/cpaa.2004.3.695.  Google Scholar

[23]

C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes, Math. Biosci. Eng., 9 (2012), 819-841. doi: 10.3934/mbe.2012.9.819.  Google Scholar

[24]

P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. 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.  Google Scholar

[25]

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

[26]

M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.  Google Scholar

[27]

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.  Google Scholar

[28]

L. 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.  Google Scholar

[29]

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

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