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August  2020, 25(8): 2867-2893. doi: 10.3934/dcdsb.2020044

Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection

Stephen Pankavich 1,, , Nathan Neri 1, and  Deborah Shutt 2,

1. 

Colorado School of Mines, Department of Applied Mathematics and Statistics, 1500 Illinois St., Golden, CO 80401, USA
2. 

Los Alamos National Laboratory, Information Systems and Modeling, Los Alamos, NM 87544 USA

* Corresponding author: Stephen Pankavich

Received  August 2018 Revised  April 2019 Published  August 2020 Early access  February 2020

Fund Project: The first author is supported by NSF grants DMS-1551229 and DMS-1614586

Recent clinical studies have shown that HIV disease pathogenesis can depend strongly on many factors at the time of transmission, including the strength of the initial viral load and the local availability of CD4+ T-cells. In this article, a new within-host model of HIV infection that incorporates the homeostatic proliferation of T-cells is formulated and analyzed. Due to the effects of this biological process, the influence of initial conditions on the proliferation of HIV infection is further elucidated. The identifiability of parameters within the model is investigated and a local stability analysis, which displays additional complexity in comparison to previous models, is conducted. The current study extends previous theoretical and computational work on the early stages of the disease and leads to interesting nonlinear dynamics, including a parameter region featuring bistability of infectious and viral clearance equilibria and the appearance of a Hopf bifurcation within biologically relevant parameter regimes.

Keywords: HIV, in-host model, acute phase, dynamics, bistability, Hopf bifurcation.
Mathematics Subject Classification: Primary: 37N25, 92B05; Secondary: 34D20, 34C23, 37G15.
Citation: Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044
References:
[1]

B. Adams, H. T. Banks, M. Davidian and E. Rosenberg, Estimation and prediction with HIV-treatment interruption data, Bull. Math. Biol., 69 (2007), 563-584.  doi: 10.1007/s11538-006-9140-6.  Google Scholar

[2]

H. T. Banks, R. Baraldi, K. Cross, K. Flores, C. McChesney, L. Poag and E. Thorpe, Uncertainty quantification in modeling HIV viral mechanics, Math. Biosci. Eng., 12 (2015), 937-964.  doi: 10.3934/mbe.2015.12.937.  Google Scholar

[3]

S. Bonhoeffer, M. Rembiszewski, G. Ortiz and D. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection, AIDS, 14 (2000), 2313-2322.  doi: 10.1097/00002030-200010200-00012.  Google Scholar

[4]

M. Catalfamo, C. Wilhelm, L. Tcheung, M. Proscha and T. Friesen, et al., CD4 and CD8 T-cell immune activation during chronic HIV infection: Roles of homeostasis, HIV, Type Ⅰ IFN, and IL-7, J. Immunol., 186 (2011), 2106–2116. doi: 10.4049/jimmunol.1002000.  Google Scholar

[5]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905.  Google Scholar

[6]

A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263.  doi: 10.1016/j.nonrwa.2009.07.001.  Google Scholar

[7]

Y. Endo, T. Igarashi, Y. Nishimura, C. Buckler and A. Buckler-White, et al., Short- and long-term clinical outcomes in rhesus monkeys inoculated with a highly pathogenic chimeric simian/human immunodeficiency virus, J. Virology, 74 (2000), 6935–6945. doi: 10.1128/JVI.74.15.6935-6945.2000.  Google Scholar

[8]

X. Fan, C.-M. Brauner and L. Wittkop, Mathematical analysis of a HIV model with quadratic logistic growth term, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2359-2385.  doi: 10.3934/dcdsb.2012.17.2359.  Google Scholar

[9]

A. S. Fauci, G. Pantaleo and S. Stanley, et al., Immunopathogenic mechanisms of HIV infection, Ann. Intern. Med., 124 (1996), 654–663. doi: 10.7326/0003-4819-124-7-199604010-00006.  Google Scholar

[10]

T. C. Greenough, D. B. Brettler and F. Kirchhoff, et al., Long-term non-progressive infection with human immunodeficiency virus type in a hemophilia cohort, J. Infect. Dis., 180 (1999), 1790–1802. doi: 10.1086/315128.  Google Scholar

[11]

M. Hadjiandreou, R. Conejeros and V. Vassiliadis, Towards a long-term model construction for the dynamic simulation of HIV infection, Math. Biosci. Eng., 4 (2007), 489-504.  doi: 10.3934/mbe.2007.4.489.  Google Scholar

[12]

E. Hernandez-Vargas and R. Middleton, Modeling the three stages in HIV infection, J. Theoret. Biol., 320 (2013), 33-40.  doi: 10.1016/j.jtbi.2012.11.028.  Google Scholar

[13]

T. Igarashi, Y. Endo, Y. Nishimura, C. Buckler and R. Sadjadpour, et al., Early control of highly pathogenic simian immunodeficiency virus/human immunodeficiency virus chimeric virus infections in rhesus monkeys usually results in long-lasting asymptomatic clinical outcomes, J. Virology, 77 (2003), 10829–10840. doi: 10.1128/JVI.77.20.10829-10840.2003.  Google Scholar

[14]

T. Igarashi, Y. Endo, G. Englund, R. Sadjadpour and T. Matano, et al., Emergence of a highly pathogenic simian/human immunodeficiency virus in a rhesus macaque treated with anti-CD8 mAb during a primary infection with a nonpathogenic virus, PNAS, 96 (1999), 14049–14054. doi: 10.1073/pnas.96.24.14049.  Google Scholar

[15]

E. Jones, P. Roemer, S. Pankavich and M. Raghupathi, Analysis and simulation of the three-component model of HIV dynamics, SIURO, 2 (2014), 308-331.  doi: 10.1137/13S012698.  Google Scholar

[16]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[17]

J. Liu, B. Keele and H. Li, et al., Low-dose mucosal simian immunodeficiency virus infection restricts early replication kinetics and transmitted virus variants in rhesus monkeys, J. Virology, 84 (2010), 10406–10412. doi: 10.1128/JVI.01155-10.  Google Scholar

[18]

C. Mackall, F. Hakim and R. Gress, Restoration of T-cell homeostasis after T-cell depletion, Semin. Immunol., 9 (1997), 339-346.  doi: 10.1006/smim.1997.0091.  Google Scholar

[19]

H. Miao, X. Xia, A. Perelson and H. Wu, On identifiability of nonlinear ODE models and applications in viral dynamics, SIAM Rev., 53 (2011), 3-39.  doi: 10.1137/090757009.  Google Scholar

[20]

M. Moreno-Fernandez, P. Presiccea and C. Chougneta, Homeostasis and function of regulatory T-cells in HIV/SIV infection, J. Virol., 86 (2012), 10262-10269.  doi: 10.1128/JVI.00993-12.  Google Scholar

[21]

C. Noecker, K. Schaefer, K. Zaccheo, Y. Yang, J. Day and V. Ganusov, Simple mathematical models do not accurately predict early SIV dynamics, Viruses, 7 (2015), 1189-1217.  doi: 10.3390/v7031189.  Google Scholar

[22] M. A. Nowak and R. M. May, Virus Dynamics. Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.   Google Scholar
[23]

S. Pankavich, The effects of latent infection on the dynamics of HIV, Differ. Equ. Dyn. Syst., 24 (2016), 281-303.  doi: 10.1007/s12591-014-0234-6.  Google Scholar

[24]

S. Pankavich and T. Loudon, Mathematical analysis and dynamic active subspaces for a long term model of HIV, Math. Biosci. Eng., 14 (2017), 709-733.  doi: 10.3934/mbe.2017040.  Google Scholar

[25]

S. Pankavich and C. Parkinson, Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1237-1257.  doi: 10.3934/dcdsb.2016.21.1237.  Google Scholar

[26]

S. Pankavich and D. Shutt, An In-Host Model of HIV Incorporating Latent Infection and Viral Mutation, 10th AIMS Conference. Suppl., 2015,913–922. doi: 10.3934/proc.2015.0913.  Google Scholar

[27]

A. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T-cells, Math. Biosci., 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[28]

A. S. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[29]

L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theoret. Biol., 260 (2009), 308-331.  doi: 10.1016/j.jtbi.2009.06.011.  Google Scholar

[30]

C. Tanchot, M. Rosado, F. Agenes, A. Freitas and B. Rocha, Lymphocyte homeostasis, Semin. Immunol., 9 (1997), 331-337.  doi: 10.1006/smim.1997.0090.  Google Scholar

[31]

M. T. Wentworth, R. C. Smith and H. T. Banks, Parameter selection and verification techniques based on global sensitivity analysis illustrated for an HIV model, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 266-297.  doi: 10.1137/15M1008245.  Google Scholar

[32]

H. Wu, H. Zhu, H. Miao and A. Perelson, Parameter identifiability and estimation of HIV/AIDS dynamic models, Bull. Math. Biol., 70 (2008), 785-799.  doi: 10.1007/s11538-007-9279-9.  Google Scholar

[33]

X. Xia and C. Moog, Identifiability of nonlinear systems with application to HIV/AIDS models, IEEE Trans. Automat. Control, 48 (2003), 330-336.  doi: 10.1109/TAC.2002.808494.  Google Scholar

show all references

References:
[1]

B. Adams, H. T. Banks, M. Davidian and E. Rosenberg, Estimation and prediction with HIV-treatment interruption data, Bull. Math. Biol., 69 (2007), 563-584.  doi: 10.1007/s11538-006-9140-6.  Google Scholar

[2]

H. T. Banks, R. Baraldi, K. Cross, K. Flores, C. McChesney, L. Poag and E. Thorpe, Uncertainty quantification in modeling HIV viral mechanics, Math. Biosci. Eng., 12 (2015), 937-964.  doi: 10.3934/mbe.2015.12.937.  Google Scholar

[3]

S. Bonhoeffer, M. Rembiszewski, G. Ortiz and D. Nixon, Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection, AIDS, 14 (2000), 2313-2322.  doi: 10.1097/00002030-200010200-00012.  Google Scholar

[4]

M. Catalfamo, C. Wilhelm, L. Tcheung, M. Proscha and T. Friesen, et al., CD4 and CD8 T-cell immune activation during chronic HIV infection: Roles of homeostasis, HIV, Type Ⅰ IFN, and IL-7, J. Immunol., 186 (2011), 2106–2116. doi: 10.4049/jimmunol.1002000.  Google Scholar

[5]

P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis, SIAM J. Appl. Math., 63 (2003), 1313-1327.  doi: 10.1137/S0036139902406905.  Google Scholar

[6]

A. M. Elaiw, Global properties of a class of HIV models, Nonlinear Anal. Real World Appl., 11 (2010), 2253-2263.  doi: 10.1016/j.nonrwa.2009.07.001.  Google Scholar

[7]

Y. Endo, T. Igarashi, Y. Nishimura, C. Buckler and A. Buckler-White, et al., Short- and long-term clinical outcomes in rhesus monkeys inoculated with a highly pathogenic chimeric simian/human immunodeficiency virus, J. Virology, 74 (2000), 6935–6945. doi: 10.1128/JVI.74.15.6935-6945.2000.  Google Scholar

[8]

X. Fan, C.-M. Brauner and L. Wittkop, Mathematical analysis of a HIV model with quadratic logistic growth term, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2359-2385.  doi: 10.3934/dcdsb.2012.17.2359.  Google Scholar

[9]

A. S. Fauci, G. Pantaleo and S. Stanley, et al., Immunopathogenic mechanisms of HIV infection, Ann. Intern. Med., 124 (1996), 654–663. doi: 10.7326/0003-4819-124-7-199604010-00006.  Google Scholar

[10]

T. C. Greenough, D. B. Brettler and F. Kirchhoff, et al., Long-term non-progressive infection with human immunodeficiency virus type in a hemophilia cohort, J. Infect. Dis., 180 (1999), 1790–1802. doi: 10.1086/315128.  Google Scholar

[11]

M. Hadjiandreou, R. Conejeros and V. Vassiliadis, Towards a long-term model construction for the dynamic simulation of HIV infection, Math. Biosci. Eng., 4 (2007), 489-504.  doi: 10.3934/mbe.2007.4.489.  Google Scholar

[12]

E. Hernandez-Vargas and R. Middleton, Modeling the three stages in HIV infection, J. Theoret. Biol., 320 (2013), 33-40.  doi: 10.1016/j.jtbi.2012.11.028.  Google Scholar

[13]

T. Igarashi, Y. Endo, Y. Nishimura, C. Buckler and R. Sadjadpour, et al., Early control of highly pathogenic simian immunodeficiency virus/human immunodeficiency virus chimeric virus infections in rhesus monkeys usually results in long-lasting asymptomatic clinical outcomes, J. Virology, 77 (2003), 10829–10840. doi: 10.1128/JVI.77.20.10829-10840.2003.  Google Scholar

[14]

T. Igarashi, Y. Endo, G. Englund, R. Sadjadpour and T. Matano, et al., Emergence of a highly pathogenic simian/human immunodeficiency virus in a rhesus macaque treated with anti-CD8 mAb during a primary infection with a nonpathogenic virus, PNAS, 96 (1999), 14049–14054. doi: 10.1073/pnas.96.24.14049.  Google Scholar

[15]

E. Jones, P. Roemer, S. Pankavich and M. Raghupathi, Analysis and simulation of the three-component model of HIV dynamics, SIURO, 2 (2014), 308-331.  doi: 10.1137/13S012698.  Google Scholar

[16]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.  Google Scholar

[17]

J. Liu, B. Keele and H. Li, et al., Low-dose mucosal simian immunodeficiency virus infection restricts early replication kinetics and transmitted virus variants in rhesus monkeys, J. Virology, 84 (2010), 10406–10412. doi: 10.1128/JVI.01155-10.  Google Scholar

[18]

C. Mackall, F. Hakim and R. Gress, Restoration of T-cell homeostasis after T-cell depletion, Semin. Immunol., 9 (1997), 339-346.  doi: 10.1006/smim.1997.0091.  Google Scholar

[19]

H. Miao, X. Xia, A. Perelson and H. Wu, On identifiability of nonlinear ODE models and applications in viral dynamics, SIAM Rev., 53 (2011), 3-39.  doi: 10.1137/090757009.  Google Scholar

[20]

M. Moreno-Fernandez, P. Presiccea and C. Chougneta, Homeostasis and function of regulatory T-cells in HIV/SIV infection, J. Virol., 86 (2012), 10262-10269.  doi: 10.1128/JVI.00993-12.  Google Scholar

[21]

C. Noecker, K. Schaefer, K. Zaccheo, Y. Yang, J. Day and V. Ganusov, Simple mathematical models do not accurately predict early SIV dynamics, Viruses, 7 (2015), 1189-1217.  doi: 10.3390/v7031189.  Google Scholar

[22] M. A. Nowak and R. M. May, Virus Dynamics. Mathematical Principles of Immunology and Virology, Oxford University Press, Oxford, 2000.   Google Scholar
[23]

S. Pankavich, The effects of latent infection on the dynamics of HIV, Differ. Equ. Dyn. Syst., 24 (2016), 281-303.  doi: 10.1007/s12591-014-0234-6.  Google Scholar

[24]

S. Pankavich and T. Loudon, Mathematical analysis and dynamic active subspaces for a long term model of HIV, Math. Biosci. Eng., 14 (2017), 709-733.  doi: 10.3934/mbe.2017040.  Google Scholar

[25]

S. Pankavich and C. Parkinson, Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1237-1257.  doi: 10.3934/dcdsb.2016.21.1237.  Google Scholar

[26]

S. Pankavich and D. Shutt, An In-Host Model of HIV Incorporating Latent Infection and Viral Mutation, 10th AIMS Conference. Suppl., 2015,913–922. doi: 10.3934/proc.2015.0913.  Google Scholar

[27]

A. Perelson, D. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T-cells, Math. Biosci., 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.  Google Scholar

[28]

A. S. Perelson and P. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107.  Google Scholar

[29]

L. Rong and A. S. Perelson, Modeling HIV persistence, the latent reservoir, and viral blips, J. Theoret. Biol., 260 (2009), 308-331.  doi: 10.1016/j.jtbi.2009.06.011.  Google Scholar

[30]

C. Tanchot, M. Rosado, F. Agenes, A. Freitas and B. Rocha, Lymphocyte homeostasis, Semin. Immunol., 9 (1997), 331-337.  doi: 10.1006/smim.1997.0090.  Google Scholar

[31]

M. T. Wentworth, R. C. Smith and H. T. Banks, Parameter selection and verification techniques based on global sensitivity analysis illustrated for an HIV model, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 266-297.  doi: 10.1137/15M1008245.  Google Scholar

[32]

H. Wu, H. Zhu, H. Miao and A. Perelson, Parameter identifiability and estimation of HIV/AIDS dynamic models, Bull. Math. Biol., 70 (2008), 785-799.  doi: 10.1007/s11538-007-9279-9.  Google Scholar

[33]

X. Xia and C. Moog, Identifiability of nonlinear systems with application to HIV/AIDS models, IEEE Trans. Automat. Control, 48 (2003), 330-336.  doi: 10.1109/TAC.2002.808494.  Google Scholar

Table 1, and a comparison with the full three-stage model in [11] - T-cell count (left) and viral load (right)">Figure 1.  A representative simulation of (3CM) with parameter values given in Table 1, and a comparison with the full three-stage model in [11] - T-cell count (left) and viral load (right)
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Figure 2.  Regions of existence (i.e. real and positive values) for the uninfected state, $ E_u $, and infected states, $ E_i^{+} $ and $ E_i^{-} $ in the $ (R_m, R_0) $ plane
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Table 1">Figure 3.  Regions of stability in the $ (R_m, R_0) $ plane when $ \alpha_1, \alpha_2, \beta $ are fixed to the values in Table 1
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Table 1">Figure 4.  Changes to stability of equilibria given differing values of $ \beta $ where $ \beta^* = 0.73 $ is the fitted value for $ \beta $ in Table 1
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Figure 5.  Small changes in initial data, $ T_0 $ (top) and $ V_0 $ (bottom), yield changes in asymptotic behavior within the bistable parameter region - T-cell count (left) and viral load (right). Within both simulations, dimensionless parameters are fixed to $ R_m = 0.6 $ and $ R_0 = 3 $. Recall that $ \overline{V}_u \equiv 0 $ is the equilibrium viral load for $ E_u $, and note that $ V(t) $ is represented on a log scale
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Figure 6.  Simulation results for $ R_m, R_0 $ values within the bistable region. A simulation was conducted at each point in the displayed parameter space with initial conditions $ T_0 = 1, I_0 = 0, V_0 = \overline{V}_+ $, where $ \overline{V}_+ $ is the equilibrium viral load of the $ E_i^+ $ infective equilibrium evaluated at $ (R_m, R_0) $. Green dots indicate solutions which tend toward the uninfected steady state $ E_u $, while red crosses indicate solutions tending to $ E_i^{+} $, the infected steady state
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Figure 8">Figure 7.  The dependence of equilibria on initial values of healthy T-cells and the initial viral load at arbitrary points in the $ (R_m, R_0) $ plane. Corresponding results are shown in Figure 8
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Figure 7">Figure 8.  Basins of attraction within the bistable region. For each $ (R_0, R_m) $ point, the initial T-cell count and viral load (both dimensionless) were set to the $ E_i^+ $ steady state evaluated at $ (R_0, R_m) $. This forms the central point in each plot, and then both initial conditions are shifted $ \pm 40\% $. Green dots denote simulation convergence to $ E_u $, and red crosses to $ E_i^+ $. A visual summary of the points chosen in the bistable region is provided by Figure 7
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Table 1">Figure 9.  Real and imaginary parts of eigenvalues $ \eta_2 $ and $ \eta_3 $ as functions of the bifurcation parameter $ R_m $ with $ R_0 = 1.25 $ fixed. These curves cross the imaginary axis at $ R_m = R_m^{*} \approx 9 $ when using parameter values from Table 1
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Figure 3. Hence, complex eigenvalues of the system linearized about $ E_i^+ $ are of the form $ \alpha \pm i \beta $ where $ \alpha < 0 $, yielding a stable steady state. Green and red horizontal lines indicate corresponding population values of the $ E_i^+ $ and $ E_u $ steady states, respectively">Figure 10.  Simulation of the dimensionless system for $ R_m = 6.5 $ and $ R_0 = 1.25 $. The parameter location is in the (dark) red region of Figure 3. Hence, complex eigenvalues of the system linearized about $ E_i^+ $ are of the form $ \alpha \pm i \beta $ where $ \alpha < 0 $, yielding a stable steady state. Green and red horizontal lines indicate corresponding population values of the $ E_i^+ $ and $ E_u $ steady states, respectively
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Figure 3. Hence, complex eigenvalues of the system linearized about $ E_i^+ $ are of the form $ \pm i \beta $, signaling a transition in the asymptotic behavior of solutions and the emergence of a stable orbit">Figure 11.  Simulation of the dimensionless system for $ R_m = 9 $ and $ R_0 = 1.25 $. The parameter location is at the border of the (dark) red and unshaded regions of Figure 3. Hence, complex eigenvalues of the system linearized about $ E_i^+ $ are of the form $ \pm i \beta $, signaling a transition in the asymptotic behavior of solutions and the emergence of a stable orbit
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Figure 3). Complex eigenvalues of the system linearized about $ E_i^+ $ are of the form $ \alpha \pm i \beta $ with $ \alpha > 0 $ yielding instability of the equilibrium. With initial conditions taken near the bifurcation, a stable periodic orbit appears">Figure 12.  Simulation of the dimensionless system for $ R_m = 9.25 $ and $ R_0 = 1.25 $. The parameters now lie within the unshaded region (Figure 3). Complex eigenvalues of the system linearized about $ E_i^+ $ are of the form $ \alpha \pm i \beta $ with $ \alpha > 0 $ yielding instability of the equilibrium. With initial conditions taken near the bifurcation, a stable periodic orbit appears
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Table 1.  Variables and Parameters
Quantity Values / Initial Values References
Original Populations
$ T $ Uninfected CD4$ ^+ $ T-cells $ 1000\; \mathrm{mm}^{-3} $ [12]
$ I $ Infected CD4$ ^+ $ T-cells $ 0\; \mathrm{mm}^{-3} $ [12]
$ V $ Wild-type HIV virions $ 10^{-2}\; \mathrm{mm}^{-3} $ [12]
Dimensionless Populations ($ ^* $ omitted in exposition)
$ T^* = \frac{pk}{d_I d_V} T $ $ T^*_0 = 1.81 $
$ I^* = \frac{pk}{d_T d_V} I $ $ I^*_0 = 0 $
$ V^* = \frac{k}{d_T} V $ $ V^*_0 = 4.57\times 10^{-5} $
$ t^* = d_T t $
Original Parameters
$ \lambda $ Rate of supply of T-cells $ 10\; \mathrm{mm}^{-3}\; \mathrm{day}^{-1} $ [27]
$ \rho $ Maximum homeostatic growth rate $ 0.01\; \mathrm{day}^{-1} $ [12]
$ C $ Homeostatic half-velocity $ 300\; \mathrm{copies}\; \mathrm{mm}^{-3} $ [12]
$ k $ Infection rate $ 4.57\times10^{-5}\; \mathrm{mm}^{3}\; \mathrm{day}^{-1} $ [12,11]
$ d_T $ Death rate of uninfected T-cells $ 0.01\; \mathrm{day}^{-1} $ [11,12]
$ d_I $ Death rate of infected T-cells $ 0.40\; \mathrm{day}^{-1} $ [27,12]
$ p $ Rate of viral production $ 38\; \mathrm{virions}\; \mathrm{per\; cell}\; \mathrm{day}^{-1} $ [12,11]
$ d_V $ Clearance rate of free virus $ 2.4\; \mathrm{day}^{-1} $ [12,11]
Dimensionless Parameters
$ R_0 = \frac{\lambda k p}{d_T d_I d_V} $ $ 1.81 $
$ R_m = \frac{\rho}{Ck} $ $ 0.73 $
$ \alpha_1 = \frac{d_I}{d_T} $ $ 40 $
$ \alpha_2 = \frac{d_V}{d_T} $ $ 240 $
$ \beta = \frac{d_T}{Ck} $ $ 0.73 $
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Quantity Values / Initial Values References
Original Populations
$ T $ Uninfected CD4$ ^+ $ T-cells $ 1000\; \mathrm{mm}^{-3} $ [12]
$ I $ Infected CD4$ ^+ $ T-cells $ 0\; \mathrm{mm}^{-3} $ [12]
$ V $ Wild-type HIV virions $ 10^{-2}\; \mathrm{mm}^{-3} $ [12]
Dimensionless Populations ($ ^* $ omitted in exposition)
$ T^* = \frac{pk}{d_I d_V} T $ $ T^*_0 = 1.81 $
$ I^* = \frac{pk}{d_T d_V} I $ $ I^*_0 = 0 $
$ V^* = \frac{k}{d_T} V $ $ V^*_0 = 4.57\times 10^{-5} $
$ t^* = d_T t $
Original Parameters
$ \lambda $ Rate of supply of T-cells $ 10\; \mathrm{mm}^{-3}\; \mathrm{day}^{-1} $ [27]
$ \rho $ Maximum homeostatic growth rate $ 0.01\; \mathrm{day}^{-1} $ [12]
$ C $ Homeostatic half-velocity $ 300\; \mathrm{copies}\; \mathrm{mm}^{-3} $ [12]
$ k $ Infection rate $ 4.57\times10^{-5}\; \mathrm{mm}^{3}\; \mathrm{day}^{-1} $ [12,11]
$ d_T $ Death rate of uninfected T-cells $ 0.01\; \mathrm{day}^{-1} $ [11,12]
$ d_I $ Death rate of infected T-cells $ 0.40\; \mathrm{day}^{-1} $ [27,12]
$ p $ Rate of viral production $ 38\; \mathrm{virions}\; \mathrm{per\; cell}\; \mathrm{day}^{-1} $ [12,11]
$ d_V $ Clearance rate of free virus $ 2.4\; \mathrm{day}^{-1} $ [12,11]
Dimensionless Parameters
$ R_0 = \frac{\lambda k p}{d_T d_I d_V} $ $ 1.81 $
$ R_m = \frac{\rho}{Ck} $ $ 0.73 $
$ \alpha_1 = \frac{d_I}{d_T} $ $ 40 $
$ \alpha_2 = \frac{d_V}{d_T} $ $ 240 $
$ \beta = \frac{d_T}{Ck} $ $ 0.73 $
Table 2.  Calculated $ ARE $ of each parameter with $ N = 1000 $ trials
Noise level $\delta$ in $\%$ Calculated $ARE$ in $\%$ of fitted value
$\alpha_1$ $\alpha_2$ $\beta$ $R_0$ $R_m$
5 2.6066 5.1185 8.5814 3.2080 4.1637
10 3.5020 6.6600 14.9092 4.7260 6.0953
15 4.2652 7.4536 20.0601 6.5109 8.0292
20 4.6269 8.5138 24.2512 7.7800 8.6645
25 5.2935 9.6199 27.6901 5.5697 9.7956
30 5.6840 9.9405 30.8474 9.9440 10.9832
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Noise level $\delta$ in $\%$ Calculated $ARE$ in $\%$ of fitted value
$\alpha_1$ $\alpha_2$ $\beta$ $R_0$ $R_m$
5 2.6066 5.1185 8.5814 3.2080 4.1637
10 3.5020 6.6600 14.9092 4.7260 6.0953
15 4.2652 7.4536 20.0601 6.5109 8.0292
20 4.6269 8.5138 24.2512 7.7800 8.6645
25 5.2935 9.6199 27.6901 5.5697 9.7956
30 5.6840 9.9405 30.8474 9.9440 10.9832
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