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. ConstantineB. 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. FauciG. Pantaleo and S. Stanley, Immunopathogenic mechanisms of HIV infection, Annals of Internal Medicine, 124 (1996), 654-663.

[6]

T. C. GreenoughD. 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. GumelP. 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. HadjiandreouR. 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. IgarashiC. 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. KirschnerG. 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. MurrayG. 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. VerguA. 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. ConstantineB. 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. FauciG. Pantaleo and S. Stanley, Immunopathogenic mechanisms of HIV infection, Annals of Internal Medicine, 124 (1996), 654-663.

[6]

T. C. GreenoughD. 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. GumelP. 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. HadjiandreouR. 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. IgarashiC. 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. KirschnerG. 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. MurrayG. 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. VerguA. 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.

Figure 1.  Ten simulations of (1) with representative parameter values.
Figure 2.  Approximation of eigenvalues of C using 1000 random samples.
Figure 4.  Measure of separation for the eigenvalues of C
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.
Figure 5.  Sufficient summary plot after 1700 days (left). Approximation to the T-cell count after 1700 days (right).
Figure 6.  Relative errors in the approximation of T (1700)
Figure 7.  Eigenvalues of the matrix C after 2000 days (left). Dimension of the active subspace for each time (right).
Figure 8.  Sufficient summary plots after 2000 days, displaying the one-dimensional (left) and two-dimensional (right) active subspace representations
Figure 9.  Sufficient summary plot after 2600 days using 1000 trials (left). Same plot with function approximation (right).
Figure 10.  Sufficient summary plots representing the three stages of infection -Acute (left), Chronic (center), AIDS (right)
Figure 11.  Slope (left) and T-intercept (right) functions, m(t) and b(t), respectively for t ∈ [55,1300].
Figure 12.  Global-in-time approximation of the T-cell count
Figure 13.  Relative error in the global approximation of the T-Student Version of MATLAB cell count.
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
Figure 15.  Sufficient summary plots throughout the Acute stage
Figure 16.  Sufficient summary plots throughout the Chronic stage
Figure 17.  Sufficient summary plots during the progression to AIDS
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] -
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

Metrics

  • PDF downloads (10)
  • HTML views (2)
  • Cited by (0)

Other articles
by authors

[Back to Top]