# American Institute of Mathematical Sciences

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

 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.
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). 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0036139902406905. Google Scholar [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs Vol 25, (1988). Google Scholar [10] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2000). Google Scholar [27] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. 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. 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. Google Scholar

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##### References:
 [1] C. L. Althaus and R. J. De Boer, Dynamics of immune escape during HIV/SIV infection,, PLoS Comput. Biol., 4 (2008). 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. 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. 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. 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. 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. 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. 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. doi: 10.1137/S0036139902406905. Google Scholar [9] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs Vol 25, (1988). Google Scholar [10] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2000). Google Scholar [27] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. 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. 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. Google Scholar
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