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On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime
Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection
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 |
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.
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. |







Quantity | Values / Initial Values | References | |
Original Populations | |||
Uninfected CD4 |
[12] | ||
Infected CD4 |
[12] | ||
Wild-type HIV virions | [12] | ||
Dimensionless Populations ( |
|||
Original Parameters | |||
Rate of supply of T-cells | [27] | ||
Maximum homeostatic growth rate | [12] | ||
Homeostatic half-velocity | [12] | ||
Infection rate | [12,11] | ||
Death rate of uninfected T-cells | [11,12] | ||
Death rate of infected T-cells | [27,12] | ||
Rate of viral production | [12,11] | ||
Clearance rate of free virus | [12,11] | ||
Dimensionless Parameters | |||
Quantity | Values / Initial Values | References | |
Original Populations | |||
Uninfected CD4 |
[12] | ||
Infected CD4 |
[12] | ||
Wild-type HIV virions | [12] | ||
Dimensionless Populations ( |
|||
Original Parameters | |||
Rate of supply of T-cells | [27] | ||
Maximum homeostatic growth rate | [12] | ||
Homeostatic half-velocity | [12] | ||
Infection rate | [12,11] | ||
Death rate of uninfected T-cells | [11,12] | ||
Death rate of infected T-cells | [27,12] | ||
Rate of viral production | [12,11] | ||
Clearance rate of free virus | [12,11] | ||
Dimensionless Parameters | |||
Noise level |
Calculated |
||||
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 |
Noise level |
Calculated |
||||
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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