Mathematical analysis and dynamic active subspaces for a long term model of HIV

Tyson Loudon; Stephen Pankavich

doi:10.3934/mbe.2017040

Article Contents

2017, Volume 14, Issue 3: 709-733. Doi: 10.3934/mbe.2017040

This issue Previous Article Mixed vaccination strategy for the control of tuberculosis: A case study in China Next Article Effect of the epidemiological heterogeneity on the outbreak outcomes

Mathematical analysis and dynamic active subspaces for a long term model of HIV

Tyson Loudon^1, and
Stephen Pankavich^2, ,

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
* Corresponding author: pankavic@mines.edu

Received: November 03, 2015

Accepted: October 22, 2016

Published: September 2017

The second author is supported by NSF grants DMS-1211667 and DMS-1614586.

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Abstract

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.

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Mathematics Subject Classification: Primary: 92B99; Secondary: 34A34.

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Figure 1. Ten simulations of (1) with representative parameter values.

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Figure 2. Approximation of eigenvalues of C using 1000 random samples.

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Figure 4. Measure of separation for the eigenvalues of C

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Figure 3. 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.

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Figure 5. Sufficient summary plot after 1700 days (left). Approximation to the T-cell count after 1700 days (right).

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Figure 6. Relative errors in the approximation of T (1700)

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Figure 7. Eigenvalues of the matrix C after 2000 days (left). Dimension of the active subspace for each time (right).

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Figure 8. Sufficient summary plots after 2000 days, displaying the one-dimensional (left) and two-dimensional (right) active subspace representations

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Figure 9. Sufficient summary plot after 2600 days using 1000 trials (left). Same plot with function approximation (right).

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Figure 10. Sufficient summary plots representing the three stages of infection -Acute (left), Chronic (center), AIDS (right)

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Figure 11. Slope (left) and T-intercept (right) functions, m(t) and b(t), respectively for t ∈ [55,1300].

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Figure 12. Global-in-time approximation of the T-cell count

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Figure 13. Relative error in the global approximation of the T-Student Version of MATLAB cell count.

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Figure 14. 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

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Figure 15. Sufficient summary plots throughout the Acute stage

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Figure 16. Sufficient summary plots throughout the Chronic stage

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Figure 17. Sufficient summary plots during the progression to AIDS

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Table 1. 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]	-

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