# American Institute of Mathematical Sciences

June  2017, 14(3): 709-733. doi: 10.3934/mbe.2017040

## 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

* Corresponding author: pankavic@mines.edu

Received  November 04, 2015 Accepted  October 23, 2016 Published  December 2016

Fund Project: The second author is supported by NSF grants DMS-1211667 and DMS-1614586

Recently, a long-term model of HIV infection dynamics [8] was developed to describe the entire time course of the disease. It consists of a large system of ODEs with many parameters, and is expensive to simulate. In the current paper, this model is analyzed by determining all infection-free steady states and studying the local stability properties of the unique biologically-relevant equilibrium. Active subspace methods are then used to perform a global sensitivity analysis and study the dependence of an infected individual's T-cell count on the parameter space. Building on these results, a global-in-time approximation of the T-cell count is created by constructing dynamic active subspaces and reduced order models are generated, thereby allowing for inexpensive computation.

Citation: Tyson Loudon, Stephen Pankavich. Mathematical analysis and dynamic active subspaces for a long term model of HIV. Mathematical Biosciences & Engineering, 2017, 14 (3) : 709-733. doi: 10.3934/mbe.2017040
##### 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.
Ten simulations of (1) with representative parameter values.
Approximation of eigenvalues of C using 1000 random samples.
Measure of separation for the eigenvalues of C
Approximation of the 1st eigenvector of C using 1000 random samples. This is referred to as the first active variable vector and denoted by w.
Sufficient summary plot after 1700 days (left). Approximation to the T-cell count after 1700 days (right).
Relative errors in the approximation of T (1700)
Eigenvalues of the matrix C after 2000 days (left). Dimension of the active subspace for each time (right).
Sufficient summary plots after 2000 days, displaying the one-dimensional (left) and two-dimensional (right) active subspace representations
Sufficient summary plot after 2600 days using 1000 trials (left). Same plot with function approximation (right).
Sufficient summary plots representing the three stages of infection -Acute (left), Chronic (center), AIDS (right)
Slope (left) and T-intercept (right) functions, m(t) and b(t), respectively for t ∈ [55,1300].
Global-in-time approximation of the T-cell count
Relative error in the global approximation of the T-Student Version of MATLAB cell count.
Full HIV model versus reduced HIV model for the first 100 days. Parameter values within the reduced model are s1 = 10, p1 = 0.2, C1 = 55.6, δ1 = 0.01, K1 = 4.72 × 10−3, δ2 = 0.69, K9 = 5.37 × 10−1, and δ7 = 2.39
Sufficient summary plots throughout the Acute stage
Sufficient summary plots throughout the Chronic stage
Sufficient summary plots during the progression to AIDS
Parameter values and ranges
 Parameter Value Range Value taken from: Units $s_1$ 10 5 -36 [13] mm$^{-3}$d$^{-1}$ $s_2$ 0.15 0.03 -0.15 [13] mm$^{-3}$d$^{-1}$ $s_3$ 5 - [8] mm$^{-3}$d$^{-1}$ $p_1$ 0.2 0.01 -0.5 [8] d$^{-1}$ $C_1$ 55.6 1 -188 [8] mm$^{-3}$ $K_1$ 3.87 x $10^{-3}$ 10$^{-8}$ -10$^{-2}$ [8] mm$^{3}$d$^{-1}$ $K_2$ $10^{-6}$ $10^{-6}$ [13] mm$^{3}$d$^{-1}$ $K_3$ 4.5 x 10$^{-4}$ 10$^{-4}$ -1 [8] mm$^{3}$d$^{-1}$ $K_4$ 7.45 x 10$^{-4}$ - [8] mm$^{3}$d$^{-1}$ $K_5$ 5.22 x 10$^{-4}$ 4.7 x 10$^{-9}$ -10$^{-3}$ [8] mm$^{3}$d$^{-1}$ $K_6$ 3 x 10$^{-6}$ - [8] mm$^{3}$d$^{-1}$ $K_7$ 3.3 x 10$^{-4}$ 10$^{-6}$ -10$^{-3}$ [8] mm$^{3}$d$^{-1}$ $K_8$ 6 x 10$^{-9}$ - [8] mm$^{3}$d$^{-1}$ $K_9$ 0.537 0.24 -500 [8] d$^{-1}$ $K_{10}$ 0.285 0.005 -300 [8] d$^{-1}$ $K_{11}$ 7.79 x 10$^{-6}$ - [8] mm$^{3}$d$^{-1}$ $K_{12}$ 10$^{-6}$ - [8] mm$^{3}$d$^{-1}$ $K_{13}$ 4 x 10$^{-5}$ - [8] mm$^{3}$d$^{-1}$ $\delta_1$ 0.01 0.01 -0.02 [8] d$^{-1}$ $\delta_2$ 0.28 0.24 -0.7 [8] d$^{-1}$ $\delta_3$ 0.05 0.02 -0.069 [8] d$^{-1}$ $\delta_4$ 0.005 0.005 [13] d$^{-1}$ $\delta_5$ 0.005 0.005 [13] d$^{-1}$ $\delta_6$ 0.015 0.015 -0.05 [27] d$^{-1}$ $\delta_7$ 2.39 2.39 -13 [13] d$^{-1}$ $\alpha_1$ 3 x 10$^{-4}$ - [8] d$^{-1}$ $\psi$ 0.97 0.93 -0.98 [8] -
 Parameter Value Range Value taken from: Units $s_1$ 10 5 -36 [13] mm$^{-3}$d$^{-1}$ $s_2$ 0.15 0.03 -0.15 [13] mm$^{-3}$d$^{-1}$ $s_3$ 5 - [8] mm$^{-3}$d$^{-1}$ $p_1$ 0.2 0.01 -0.5 [8] d$^{-1}$ $C_1$ 55.6 1 -188 [8] mm$^{-3}$ $K_1$ 3.87 x $10^{-3}$ 10$^{-8}$ -10$^{-2}$ [8] mm$^{3}$d$^{-1}$ $K_2$ $10^{-6}$ $10^{-6}$ [13] mm$^{3}$d$^{-1}$ $K_3$ 4.5 x 10$^{-4}$ 10$^{-4}$ -1 [8] mm$^{3}$d$^{-1}$ $K_4$ 7.45 x 10$^{-4}$ - [8] mm$^{3}$d$^{-1}$ $K_5$ 5.22 x 10$^{-4}$ 4.7 x 10$^{-9}$ -10$^{-3}$ [8] mm$^{3}$d$^{-1}$ $K_6$ 3 x 10$^{-6}$ - [8] mm$^{3}$d$^{-1}$ $K_7$ 3.3 x 10$^{-4}$ 10$^{-6}$ -10$^{-3}$ [8] mm$^{3}$d$^{-1}$ $K_8$ 6 x 10$^{-9}$ - [8] mm$^{3}$d$^{-1}$ $K_9$ 0.537 0.24 -500 [8] d$^{-1}$ $K_{10}$ 0.285 0.005 -300 [8] d$^{-1}$ $K_{11}$ 7.79 x 10$^{-6}$ - [8] mm$^{3}$d$^{-1}$ $K_{12}$ 10$^{-6}$ - [8] mm$^{3}$d$^{-1}$ $K_{13}$ 4 x 10$^{-5}$ - [8] mm$^{3}$d$^{-1}$ $\delta_1$ 0.01 0.01 -0.02 [8] d$^{-1}$ $\delta_2$ 0.28 0.24 -0.7 [8] d$^{-1}$ $\delta_3$ 0.05 0.02 -0.069 [8] d$^{-1}$ $\delta_4$ 0.005 0.005 [13] d$^{-1}$ $\delta_5$ 0.005 0.005 [13] d$^{-1}$ $\delta_6$ 0.015 0.015 -0.05 [27] d$^{-1}$ $\delta_7$ 2.39 2.39 -13 [13] d$^{-1}$ $\alpha_1$ 3 x 10$^{-4}$ - [8] d$^{-1}$ $\psi$ 0.97 0.93 -0.98 [8] -
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