American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Pattern formation in a predator-mediated coexistence model with prey-taxis
  • DCDS-B Home
  • This Issue
  • Next Article
    On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Table Options

Download as excel

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
Table Options

Download as excel

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
[1]

Miaoran Yao, Yongxin Zhang, Wendi Wang. Bifurcation analysis for an in-host Mycobacterium tuberculosis model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2299-2322. doi: 10.3934/dcdsb.2020324
[2]

Stephen Pankavich, Deborah Shutt. An in-host model of HIV incorporating latent infection and viral mutation. Conference Publications, 2015, 2015 (special) : 913-922. doi: 10.3934/proc.2015.0913
[3]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237
[4]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[5]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[6]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[7]

Cameron J. Browne, Sergei S. Pilyugin. Minimizing $\mathcal R_0$ for in-host virus model with periodic combination antiviral therapy. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3315-3330. doi: 10.3934/dcdsb.2016099
[8]

Tzy-Wei Hwang, Yang Kuang. Host Extinction Dynamics in a Simple Parasite-Host Interaction Model. Mathematical Biosciences & Engineering, 2005, 2 (4) : 743-751. doi: 10.3934/mbe.2005.2.743
[9]

Bing Zeng, Pei Yu. A hierarchical parametric analysis on Hopf bifurcation of an epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022069
[10]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[11]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[12]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[13]

Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111
[14]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
[15]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[16]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[17]

Samantha Erwin, Stanca M. Ciupe. Germinal center dynamics during acute and chronic infection. Mathematical Biosciences & Engineering, 2017, 14 (3) : 655-671. doi: 10.3934/mbe.2017037
[18]

Adam Sullivan, Folashade Agusto, Sharon Bewick, Chunlei Su, Suzanne Lenhart, Xiaopeng Zhao. A mathematical model for within-host Toxoplasma gondii invasion dynamics. Mathematical Biosciences & Engineering, 2012, 9 (3) : 647-662. doi: 10.3934/mbe.2012.9.647
[19]

Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060
[20]

Yongli Cai, Weiming Wang. Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 989-1013. doi: 10.3934/dcdsb.2015.20.989

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (342)
  • HTML views (340)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy