# 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:

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##### References:
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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