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Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method
Hopf bifurcation of an age-structured virus infection model
1. | Faculty of Mathematics, K. N. Toosi University of Technology, P. O. Box:16315-1618, Tehran, Iran |
2. | Department of Mathematics, University of Louisiana, Lafayette, LA, USA |
3. | Department of Mathematical Sciences, Sharif University of Technology, P. O. Box: 11155-9415, Tehran, Iran |
In this paper, we introduce and analyze a mathematical model of a viral infection with explicit age-since infection structure for infected cells. We extend previous age-structured within-host virus models by including logistic growth of target cells and allowing for absorption of multiple virus particles by infected cells. The persistence of the virus is shown to depend on the basic reproduction number $R_{0}$. In particular, when $R_{0}≤1$, the infection free equilibrium is globally asymptotically stable, and conversely if $R_{0}> 1$, then the infection free equilibrium is unstable, the system is uniformly persistent and there exists a unique positive equilibrium. We show that our system undergoes a Hopf bifurcation through which the infection equilibrium loses the stability and periodic solutions appear.
References:
[1] |
R. Adams and J. Fournier, Sobolev spaces, "Second edition", Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar |
[2] |
I. Rob De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, Journal of Theoretical Biology, 190 (1998), 201-214. Google Scholar |
[3] |
B. Brandenburg, L. Y. Lee, M. Lakadamyali, M. J. Rust, X. Zhuang and J. M. Hogle, Imaging poliovirus entry in live cells, PLoS Biology, 5 (2007), e183, http://doi.org/10.1371/journal.pbio.0050183. Google Scholar |
[4] |
C. J. Browne,
A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Analysis: Real World Applications, 22 (2015), 354-372.
doi: 10.1016/j.nonrwa.2014.10.004. |
[5] |
C. J. Browne,
Immune response in virus model structured by cell infection-age, Mathematical Biosciences and Engineering, 13 (2016), 887-909.
doi: 10.3934/mbe.2016022. |
[6] |
C. J. Browne and S. S. Pilyugin,
Global analysis of age-structured within-host virus model, DCDS-B, 18 (2013), 1999-2017.
doi: 10.3934/dcdsb.2013.18.1999. |
[7] |
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. |
[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. |
[9] |
M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945. Google Scholar |
[10] |
P. Dustin and D. Wodarz,
Modeling multiple infection of cells by viruses: Challenges and insights, Mathematical biosciences, 264 (2015), 21-28.
doi: 10.1016/j.mbs.2015.03.001. |
[11] |
J. K. Hale, J. P. Lasalle and M. Slemrod,
Theory of a general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191.
doi: 10.1016/0022-247X(72)90233-8. |
[12] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. |
[13] |
J. K. Hale and P. Waltman,
Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[14] |
S. Hongying, L. Wang and J. Watmough,
Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM Journal on Applied Mathematics, 73 (2013), 1280-1302.
doi: 10.1137/120896463. |
[15] |
G. Huang, X. Liu and Y. Takeuchi,
Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-58.
doi: 10.1137/110826588. |
[16] |
L. Josefsson, M. S. King, B. Makitalo, J. Brännström, W. Shao, F. Maldarelli and M. F. Kearney et al. Majority of CD4+ T cells from peripheral blood of HIV-1 infected individuals contain only one HIV DNA molecule, Proceedings of the National Academy of Sciences, 108 (2011), 11199-11204. Google Scholar |
[17] |
D. Kirschner and G. F. Webb,
A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390.
doi: 10.1007/BF02458312. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000. |
[22] |
A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho,
HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span and viral generation time, Science, 271 (1996), 1582-1586.
doi: 10.1126/science.271.5255.1582. |
[23] |
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. |
[24] |
H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. C. McCluskey,
Global analysis of within host virus models with cell-to-cell viral transmission, DCDS-B, 19 (2014), 3341-3357.
doi: 10.3934/dcdsb.2014.19.3341. |
[25] |
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. |
[26] |
Libin Rong and A. S. Perelson,
Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331.
doi: 10.1016/j.jtbi.2009.06.011. |
[27] |
M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson,
Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 23 (2000), 285-301.
doi: 10.1006/jtbi.2000.1076. |
[28] |
R. V. Ursache, Y. E. Thomassen, G. Van Eikenhorst, P. J. T. Verheijen and W. A. M. Bakker,
Mathematical model of adherent Vero cell growth and poliovirus production in animal component free medium, Bioprocess Biosyst Eng., 38 (2015), 543-555.
doi: 10.1007/s00449-014-1294-2. |
[29] |
Y. Wang,
Viral dynamics model with CTL immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67 (2013), 901-934.
doi: 10.1007/s00285-012-0580-3. |
[30] |
G. F. Webb and C. J. Browne. A model of the Ebola epidemics in West Africa incorporating age of infection,
A model of the Ebola epidemics in West Africa incorporating age of infection, Journal of Biological Dynamics, 10 (2016), 18-30.
doi: 10.1080/17513758.2015.1090632. |
[31] |
Y. Yang, S. Ruan and D. Xiao,
Global stability of an age-structured virus dynamics model with Bedington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877.
doi: 10.3934/mbe.2015.12.859. |
[32] |
J. A. Zack, S. J. Arrigo, S. R. Weitsman, A. S. Go, A. Haislip and I. S. Chen,
HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viralstructure, Cell, 61 (1990), 213-222.
doi: 10.1016/0092-8674(90)90802-L. |
show all references
References:
[1] |
R. Adams and J. Fournier, Sobolev spaces, "Second edition", Pure Appl. Math., 140, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar |
[2] |
I. Rob De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison, Journal of Theoretical Biology, 190 (1998), 201-214. Google Scholar |
[3] |
B. Brandenburg, L. Y. Lee, M. Lakadamyali, M. J. Rust, X. Zhuang and J. M. Hogle, Imaging poliovirus entry in live cells, PLoS Biology, 5 (2007), e183, http://doi.org/10.1371/journal.pbio.0050183. Google Scholar |
[4] |
C. J. Browne,
A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Analysis: Real World Applications, 22 (2015), 354-372.
doi: 10.1016/j.nonrwa.2014.10.004. |
[5] |
C. J. Browne,
Immune response in virus model structured by cell infection-age, Mathematical Biosciences and Engineering, 13 (2016), 887-909.
doi: 10.3934/mbe.2016022. |
[6] |
C. J. Browne and S. S. Pilyugin,
Global analysis of age-structured within-host virus model, DCDS-B, 18 (2013), 1999-2017.
doi: 10.3934/dcdsb.2013.18.1999. |
[7] |
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. |
[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. |
[9] |
M. N. Dixit and A. S. Perelson, Multiplicity of human immunodeficiency virus infections in lymphoid tissue, Journal of Virology, 78 (2004), 8942-8945. Google Scholar |
[10] |
P. Dustin and D. Wodarz,
Modeling multiple infection of cells by viruses: Challenges and insights, Mathematical biosciences, 264 (2015), 21-28.
doi: 10.1016/j.mbs.2015.03.001. |
[11] |
J. K. Hale, J. P. Lasalle and M. Slemrod,
Theory of a general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191.
doi: 10.1016/0022-247X(72)90233-8. |
[12] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematics Surveys and Monographs, American Mathematical Society, Providence, RI, 1988. |
[13] |
J. K. Hale and P. Waltman,
Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395.
doi: 10.1137/0520025. |
[14] |
S. Hongying, L. Wang and J. Watmough,
Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM Journal on Applied Mathematics, 73 (2013), 1280-1302.
doi: 10.1137/120896463. |
[15] |
G. Huang, X. Liu and Y. Takeuchi,
Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-58.
doi: 10.1137/110826588. |
[16] |
L. Josefsson, M. S. King, B. Makitalo, J. Brännström, W. Shao, F. Maldarelli and M. F. Kearney et al. Majority of CD4+ T cells from peripheral blood of HIV-1 infected individuals contain only one HIV DNA molecule, Proceedings of the National Academy of Sciences, 108 (2011), 11199-11204. Google Scholar |
[17] |
D. Kirschner and G. F. Webb,
A model for treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58 (1996), 367-390.
doi: 10.1007/BF02458312. |
[18] |
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. |
[19] |
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. |
[20] |
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. |
[21] |
M. A. Nowak and R. M. May, Virus dynamics: Mathematical Principles of Immunology and Virology, Oxford University Press, 2000. |
[22] |
A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho,
HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span and viral generation time, Science, 271 (1996), 1582-1586.
doi: 10.1126/science.271.5255.1582. |
[23] |
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. |
[24] |
H. Pourbashash, S. S. Pilyugin, P. De Leenheer and C. C. McCluskey,
Global analysis of within host virus models with cell-to-cell viral transmission, DCDS-B, 19 (2014), 3341-3357.
doi: 10.3934/dcdsb.2014.19.3341. |
[25] |
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. |
[26] |
Libin Rong and A. S. Perelson,
Modeling HIV persistence, the latent reservoir, and viral blips, Journal of Theoretical Biology, 260 (2009), 308-331.
doi: 10.1016/j.jtbi.2009.06.011. |
[27] |
M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson,
Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 23 (2000), 285-301.
doi: 10.1006/jtbi.2000.1076. |
[28] |
R. V. Ursache, Y. E. Thomassen, G. Van Eikenhorst, P. J. T. Verheijen and W. A. M. Bakker,
Mathematical model of adherent Vero cell growth and poliovirus production in animal component free medium, Bioprocess Biosyst Eng., 38 (2015), 543-555.
doi: 10.1007/s00449-014-1294-2. |
[29] |
Y. Wang,
Viral dynamics model with CTL immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67 (2013), 901-934.
doi: 10.1007/s00285-012-0580-3. |
[30] |
G. F. Webb and C. J. Browne. A model of the Ebola epidemics in West Africa incorporating age of infection,
A model of the Ebola epidemics in West Africa incorporating age of infection, Journal of Biological Dynamics, 10 (2016), 18-30.
doi: 10.1080/17513758.2015.1090632. |
[31] |
Y. Yang, S. Ruan and D. Xiao,
Global stability of an age-structured virus dynamics model with Bedington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877.
doi: 10.3934/mbe.2015.12.859. |
[32] |
J. A. Zack, S. J. Arrigo, S. R. Weitsman, A. S. Go, A. Haislip and I. S. Chen,
HIV-1 entry into quiescent primary lymphocytes: Molecular analysis reveals a labile latent viralstructure, Cell, 61 (1990), 213-222.
doi: 10.1016/0092-8674(90)90802-L. |






Parameter | Value | Description | Reference |
day |
Maximum proliferation rate | See text | |
0.008 day |
Death rate of uninfected cells | [27] | |
mm |
Density of |
See text | |
Infection rate of target cells by virus | [27] | ||
0.8 day |
Death rate of infected cells | [32] | |
p | Varied | Virion production rate of an infected cell | See text |
3 day |
clearance rate of free virus | [22] | |
day |
Reinfection rate of infected cells by virus | See text |
Parameter | Value | Description | Reference |
day |
Maximum proliferation rate | See text | |
0.008 day |
Death rate of uninfected cells | [27] | |
mm |
Density of |
See text | |
Infection rate of target cells by virus | [27] | ||
0.8 day |
Death rate of infected cells | [32] | |
p | Varied | Virion production rate of an infected cell | See text |
3 day |
clearance rate of free virus | [22] | |
day |
Reinfection rate of infected cells by virus | See text |
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