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Mathematical analysis and dynamic active subspaces for a long term model of HIV
1. | School of Mathematics, University of Minnesota-Twin Cities, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, USA |
2. | Department of Applied Mathematics and Statistics, Colorado School of Mines, 1500 Illinois St, Golden, CO 80401, USA |
Recently, a long-term model of HIV infection dynamics [
References:
[1] |
D. Callaway and A. Perelson,
HIV-1 infection and low steady state viral loads, Bull.Math.Biol., 64 (2002), 29-64.
doi: 10.1006/bulm.2001.0266. |
[2] |
P. Constantine, Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies SIAM, 2015. |
[3] |
P. Constantine and D. Gleich, Computing active subspaces with monte carlo, arXiv: 1408.0545 |
[4] |
P. Constantine, B. Zaharatos and M. Campanelli,
Discovering an active subspace in a single-diode solar cell model, Statistical Analysis and Data Mining: The ASA Data Science Journal, 8 (2015), 264-273.
doi: 10.1002/sam.11281. |
[5] |
A. S. Fauci, G. Pantaleo and S. Stanley,
Immunopathogenic mechanisms of HIV infection, Annals of Internal Medicine, 124 (1996), 654-663.
|
[6] |
T. C. Greenough, D. B. Brettler and F. Kirchhoff,
Long-term non-progressive infection with Human Immunodeficiency Virus in a Hemophilia cohort, J Infect Dis, 180 (1999), 1790-1802.
|
[7] |
A. B. Gumel, P. N. Shivakumar and B. M. Sahai,
A mathematical model for the dynamics of HIV-1 during the typical course of infection, Nonlinear Analysis, 47 (2001), 1773-1783.
doi: 10.1016/S0362-546X(01)00309-1. |
[8] |
M. Hadjiandreou, R. Conejeros and V. S. Vassiliadis,
Towards a long-term model construction for the dynamic simulation of HIV infection, Mathematical Biosciences and Engineering, 4 (2007), 489-504.
|
[9] |
E. Hernandez-Vargas and R. Middleton,
Modeling the three stages in HIV infection, J TheorBiol., 320 (2013), 33-40.
doi: 10.1016/j.jtbi.2012.11.028. |
[10] |
T. Igarashi, C. R. Brown and Y. Endo,
Macrophages are the principal reservoir and sustain high virus loads in Rhesus Macaques following the depletion of CD4+ T-cells by a highly pathogenic SIV: Implications for HIV-1 infections of man, Proc Natl Acad Sci., 98 (2001), 658-663.
|
[11] |
E. Jones and P. Roemer (sponsors: S. Pankavich and M. Raghupathi), Analysis and simulation of the three-component model of HIV dynamics, SIAM Undergraduate Research Online, 7 (2014), 89–106
doi: 10.1137/13S012698. |
[12] |
D. Kirschner,
Using mathematics to understand HIV immunodynamics, Am. Math. Soc., 43 (1996), 191-202.
|
[13] |
D. E. Kirschner and A. S. Perelson, A model for the immune response to HIV: AZT treatment studies, Mathematical Population Dynamics: Analysis of Heterogeneity, Volume One: Theory of Epidemics Eds. O. Arino, D. Axelrod, M. Kimmel, and M. Langlais, Wuerz Publishing Ltd., Winnipeg, Canada, (1993), 295–310. |
[14] |
D. Kirschner and G. F. Webb,
Immunotherapy of HIV-1 infection, J Biological Systems, 6 (1998), 71-83.
doi: 10.1142/S0218339098000091. |
[15] |
D. Kirschner, G. F. Webb and M. Cloyd,
A model of HIV-1 disease progression based on virus-induced lymph node homing-induced apoptosis of CD4+ lymphocytes, J Acquir Immune Dec Syndr, 24 (2000), 352-362.
|
[16] |
J. M. Murray, G. Kaufmann and A. D. Kelleher,
A model of primary HIV-1 infection, Math Biosci, 154 (1998), 57-85.
doi: 10.1016/S0025-5564(98)10046-9. |
[17] |
M. Nowak and R. May, Virus Dynamics: Mathematical Principles of Immunology and Virology Oxford University Press, NewYork, 2000. |
[18] |
S. Pankavich,
The effects of latent infection on the dynamics of HIV, Differential Equations and Dynamical Systems, 24 (2016), 281-303.
doi: 10.1007/s12591-014-0234-6. |
[19] |
S. Pankavich and D. Shutt,
An in-host model of HIV incorporating latent infection and viral mutation, Dynamical Systems, Differential Equations, and Applications, AIMS Proceedings, (2015), 913-922.
doi: 10.3934/proc.2015.0913. |
[20] |
S. Pankavich, N. Neri and D. Shutt, Bistable dynamics and Hopf bifurcation in a refined model of the acute stage of HIV infection, submitted, (2015). |
[21] |
S. Pankavich and C. Parkinson,
Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity, Discrete and Continuous Dynamical Systems B, 21 (2016), 1237-1257.
doi: 10.3934/dcdsb.2016.21.1237. |
[22] |
E. Pennisi and J. Cohen, Eradicating HIV from a patient: Not just a dream?, Science, 272 (1996), 1884. |
[23] |
A. S. Perelson, Modeling the Interaction of the Immune System with HIV, Lecture Notes in Biomath. Berlin: Springer, 1989.
doi: 10.1007/978-3-642-93454-4_17. |
[24] |
A. Perelson and P. Nelson,
Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.
doi: 10.1137/S0036144598335107. |
[25] |
T. M. Russi, Uncertainty Quantification with Experimental data and Complex System Models, Ph. D. thesis, UC Berkeley, 2010. |
[26] |
W. Y. Tan and H. Wu,
Stochastic modeing of the dynamics of CD4+ T-cell infection by HIV and some monte carlo studies, Math Biosci, 147 (1997), 173-205.
doi: 10.1016/S0025-5564(97)00094-1. |
[27] |
E. Vergu, A. Mallet and J. Golmard,
A modeling approach to the impact of HIV mutations on the immune system, Comput Biol Med., 35 (2005), 1-24.
doi: 10.1016/j.compbiomed.2004.01.001. |
show all references
References:
[1] |
D. Callaway and A. Perelson,
HIV-1 infection and low steady state viral loads, Bull.Math.Biol., 64 (2002), 29-64.
doi: 10.1006/bulm.2001.0266. |
[2] |
P. Constantine, Active Subspaces: Emerging Ideas for Dimension Reduction in Parameter Studies SIAM, 2015. |
[3] |
P. Constantine and D. Gleich, Computing active subspaces with monte carlo, arXiv: 1408.0545 |
[4] |
P. Constantine, B. Zaharatos and M. Campanelli,
Discovering an active subspace in a single-diode solar cell model, Statistical Analysis and Data Mining: The ASA Data Science Journal, 8 (2015), 264-273.
doi: 10.1002/sam.11281. |
[5] |
A. S. Fauci, G. Pantaleo and S. Stanley,
Immunopathogenic mechanisms of HIV infection, Annals of Internal Medicine, 124 (1996), 654-663.
|
[6] |
T. C. Greenough, D. B. Brettler and F. Kirchhoff,
Long-term non-progressive infection with Human Immunodeficiency Virus in a Hemophilia cohort, J Infect Dis, 180 (1999), 1790-1802.
|
[7] |
A. B. Gumel, P. N. Shivakumar and B. M. Sahai,
A mathematical model for the dynamics of HIV-1 during the typical course of infection, Nonlinear Analysis, 47 (2001), 1773-1783.
doi: 10.1016/S0362-546X(01)00309-1. |
[8] |
M. Hadjiandreou, R. Conejeros and V. S. Vassiliadis,
Towards a long-term model construction for the dynamic simulation of HIV infection, Mathematical Biosciences and Engineering, 4 (2007), 489-504.
|
[9] |
E. Hernandez-Vargas and R. Middleton,
Modeling the three stages in HIV infection, J TheorBiol., 320 (2013), 33-40.
doi: 10.1016/j.jtbi.2012.11.028. |
[10] |
T. Igarashi, C. R. Brown and Y. Endo,
Macrophages are the principal reservoir and sustain high virus loads in Rhesus Macaques following the depletion of CD4+ T-cells by a highly pathogenic SIV: Implications for HIV-1 infections of man, Proc Natl Acad Sci., 98 (2001), 658-663.
|
[11] |
E. Jones and P. Roemer (sponsors: S. Pankavich and M. Raghupathi), Analysis and simulation of the three-component model of HIV dynamics, SIAM Undergraduate Research Online, 7 (2014), 89–106
doi: 10.1137/13S012698. |
[12] |
D. Kirschner,
Using mathematics to understand HIV immunodynamics, Am. Math. Soc., 43 (1996), 191-202.
|
[13] |
D. E. Kirschner and A. S. Perelson, A model for the immune response to HIV: AZT treatment studies, Mathematical Population Dynamics: Analysis of Heterogeneity, Volume One: Theory of Epidemics Eds. O. Arino, D. Axelrod, M. Kimmel, and M. Langlais, Wuerz Publishing Ltd., Winnipeg, Canada, (1993), 295–310. |
[14] |
D. Kirschner and G. F. Webb,
Immunotherapy of HIV-1 infection, J Biological Systems, 6 (1998), 71-83.
doi: 10.1142/S0218339098000091. |
[15] |
D. Kirschner, G. F. Webb and M. Cloyd,
A model of HIV-1 disease progression based on virus-induced lymph node homing-induced apoptosis of CD4+ lymphocytes, J Acquir Immune Dec Syndr, 24 (2000), 352-362.
|
[16] |
J. M. Murray, G. Kaufmann and A. D. Kelleher,
A model of primary HIV-1 infection, Math Biosci, 154 (1998), 57-85.
doi: 10.1016/S0025-5564(98)10046-9. |
[17] |
M. Nowak and R. May, Virus Dynamics: Mathematical Principles of Immunology and Virology Oxford University Press, NewYork, 2000. |
[18] |
S. Pankavich,
The effects of latent infection on the dynamics of HIV, Differential Equations and Dynamical Systems, 24 (2016), 281-303.
doi: 10.1007/s12591-014-0234-6. |
[19] |
S. Pankavich and D. Shutt,
An in-host model of HIV incorporating latent infection and viral mutation, Dynamical Systems, Differential Equations, and Applications, AIMS Proceedings, (2015), 913-922.
doi: 10.3934/proc.2015.0913. |
[20] |
S. Pankavich, N. Neri and D. Shutt, Bistable dynamics and Hopf bifurcation in a refined model of the acute stage of HIV infection, submitted, (2015). |
[21] |
S. Pankavich and C. Parkinson,
Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity, Discrete and Continuous Dynamical Systems B, 21 (2016), 1237-1257.
doi: 10.3934/dcdsb.2016.21.1237. |
[22] |
E. Pennisi and J. Cohen, Eradicating HIV from a patient: Not just a dream?, Science, 272 (1996), 1884. |
[23] |
A. S. Perelson, Modeling the Interaction of the Immune System with HIV, Lecture Notes in Biomath. Berlin: Springer, 1989.
doi: 10.1007/978-3-642-93454-4_17. |
[24] |
A. Perelson and P. Nelson,
Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.
doi: 10.1137/S0036144598335107. |
[25] |
T. M. Russi, Uncertainty Quantification with Experimental data and Complex System Models, Ph. D. thesis, UC Berkeley, 2010. |
[26] |
W. Y. Tan and H. Wu,
Stochastic modeing of the dynamics of CD4+ T-cell infection by HIV and some monte carlo studies, Math Biosci, 147 (1997), 173-205.
doi: 10.1016/S0025-5564(97)00094-1. |
[27] |
E. Vergu, A. Mallet and J. Golmard,
A modeling approach to the impact of HIV mutations on the immune system, Comput Biol Med., 35 (2005), 1-24.
doi: 10.1016/j.compbiomed.2004.01.001. |

















Parameter | Value | Range | Value taken from: | Units |
10 | 5 -36 | [13] | mm |
|
0.15 | 0.03 -0.15 | [13] | mm |
|
5 | - | [8] | mm |
|
0.2 | 0.01 -0.5 | [8] | d |
|
55.6 | 1 -188 | [8] | mm |
|
3.87 x |
10 |
[8] | mm |
|
[13] | mm |
|||
4.5 x 10 |
10 |
[8] | mm |
|
7.45 x 10 |
- | [8] | mm |
|
5.22 x 10 |
4.7 x 10 |
[8] | mm |
|
3 x 10 |
- | [8] | mm |
|
3.3 x 10 |
10 |
[8] | mm |
|
6 x 10 |
- | [8] | mm |
|
0.537 | 0.24 -500 | [8] | d |
|
0.285 | 0.005 -300 | [8] | d |
|
7.79 x 10 |
- | [8] | mm |
|
10 |
- | [8] | mm |
|
4 x 10 |
- | [8] | mm |
|
0.01 | 0.01 -0.02 | [8] | d |
|
0.28 | 0.24 -0.7 | [8] | d |
|
0.05 | 0.02 -0.069 | [8] | d |
|
0.005 | 0.005 | [13] | d |
|
0.005 | 0.005 | [13] | d |
|
0.015 | 0.015 -0.05 | [27] | d |
|
2.39 | 2.39 -13 | [13] | d |
|
3 x 10 |
- | [8] | d |
|
0.97 | 0.93 -0.98 | [8] | - |
Parameter | Value | Range | Value taken from: | Units |
10 | 5 -36 | [13] | mm |
|
0.15 | 0.03 -0.15 | [13] | mm |
|
5 | - | [8] | mm |
|
0.2 | 0.01 -0.5 | [8] | d |
|
55.6 | 1 -188 | [8] | mm |
|
3.87 x |
10 |
[8] | mm |
|
[13] | mm |
|||
4.5 x 10 |
10 |
[8] | mm |
|
7.45 x 10 |
- | [8] | mm |
|
5.22 x 10 |
4.7 x 10 |
[8] | mm |
|
3 x 10 |
- | [8] | mm |
|
3.3 x 10 |
10 |
[8] | mm |
|
6 x 10 |
- | [8] | mm |
|
0.537 | 0.24 -500 | [8] | d |
|
0.285 | 0.005 -300 | [8] | d |
|
7.79 x 10 |
- | [8] | mm |
|
10 |
- | [8] | mm |
|
4 x 10 |
- | [8] | mm |
|
0.01 | 0.01 -0.02 | [8] | d |
|
0.28 | 0.24 -0.7 | [8] | d |
|
0.05 | 0.02 -0.069 | [8] | d |
|
0.005 | 0.005 | [13] | d |
|
0.005 | 0.005 | [13] | d |
|
0.015 | 0.015 -0.05 | [27] | d |
|
2.39 | 2.39 -13 | [13] | d |
|
3 x 10 |
- | [8] | d |
|
0.97 | 0.93 -0.98 | [8] | - |
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