# 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:

show all references

##### 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] -
 [1] Hamed Azizollahi, Marion Darbas, Mohamadou M. Diallo, Abdellatif El Badia, Stephanie Lohrengel. EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity. Mathematical Biosciences & Engineering, 2018, 15 (4) : 905-932. doi: 10.3934/mbe.2018041 [2] Brandy Rapatski, Juan Tolosa. Modeling and analysis of the San Francisco City Clinic Cohort (SFCCC) HIV-epidemic including treatment. Mathematical Biosciences & Engineering, 2014, 11 (3) : 599-619. doi: 10.3934/mbe.2014.11.599 [3] Arni S. R. Srinivasa Rao, Kurien Thomas, Kurapati Sudhakar, Philip K. Maini. HIV/AIDS epidemic in India and predicting the impact of the national response: Mathematical modeling and analysis. Mathematical Biosciences & Engineering, 2009, 6 (4) : 779-813. doi: 10.3934/mbe.2009.6.779 [4] Claude-Michel Brauner, Xinyue Fan, Luca Lorenzi. Two-dimensional stability analysis in a HIV model with quadratic logistic growth term. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1813-1844. doi: 10.3934/cpaa.2013.12.1813 [5] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157 [6] Jaroslaw Smieja, Malgorzata Kardynska, Arkadiusz Jamroz. The meaning of sensitivity functions in signaling pathways analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2697-2707. doi: 10.3934/dcdsb.2014.19.2697 [7] Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241 [8] Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019041 [9] Graziano Guerra, Michael Herty, Francesca Marcellini. Modeling and analysis of pooled stepped chutes. Networks & Heterogeneous Media, 2011, 6 (4) : 665-679. doi: 10.3934/nhm.2011.6.665 [10] Sarbaz H. A. Khoshnaw. Reduction of a kinetic model of active export of importins. Conference Publications, 2015, 2015 (special) : 705-722. doi: 10.3934/proc.2015.0705 [11] Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034 [12] Lorena Bociu, Jean-Paul Zolésio. Sensitivity analysis for a free boundary fluid-elasticity interaction. Evolution Equations & Control Theory, 2013, 2 (1) : 55-79. doi: 10.3934/eect.2013.2.55 [13] Qilin Wang, S. J. Li. Higher-order sensitivity analysis in nonconvex vector optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 381-392. doi: 10.3934/jimo.2010.6.381 [14] Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1 [15] Krzysztof Fujarewicz, Krzysztof Łakomiec. Parameter estimation of systems with delays via structural sensitivity analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2521-2533. doi: 10.3934/dcdsb.2014.19.2521 [16] Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022 [17] Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial & Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611 [18] Behrouz Kheirfam, Kamal mirnia. Multi-parametric sensitivity analysis in piecewise linear fractional programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 343-351. doi: 10.3934/jimo.2008.4.343 [19] Seung-Yeal Ha, Shi Jin, Jinwook Jung. A local sensitivity analysis for the kinetic Kuramoto equation with random inputs. Networks & Heterogeneous Media, 2019, 14 (2) : 317-340. doi: 10.3934/nhm.2019013 [20] Zindoga Mukandavire, Abba B. Gumel, Winston Garira, Jean Michel Tchuenche. Mathematical analysis of a model for HIV-malaria co-infection. Mathematical Biosciences & Engineering, 2009, 6 (2) : 333-362. doi: 10.3934/mbe.2009.6.333

2018 Impact Factor: 1.313