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June  2016, 21(4): 1237-1257. doi: 10.3934/dcdsb.2016.21.1237

Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity

Stephen Pankavich 1, and  Christian Parkinson 1,

1. 

Colorado School of Mines, 1500 Illinois St, Golden, CO 80401, United States, United States

Received  May 2015 Revised  December 2015 Published  March 2016

We consider a spatially-heterogeneous generalization of a well-established model for the dynamics of the Human Immunodeficiency Virus-type 1 (HIV) within a susceptible host. The model consists of a nonlinear system of three coupled reaction-diffusion equations with parameters that may vary spatially. Upon formulating the model, we prove that it preserves the positivity of initial data and construct global-in-time solutions that are both bounded and smooth. Finally, additional results concerning the local and global asymptotic behavior of these solutions are also provided.
Keywords: longtime asymptotic behavior., nonlinear diffusion, HIV, Global existence.
Mathematics Subject Classification: Primary: 35K51, 92C17, 35Q92; Secondary: 35K45, 35K57, 92C50, 35B4.
Citation: 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
References:
[1]

S. Bonhoeffer, R. May, G. Shaw and M. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971.

[2]

C.-M. Brauner, D. Jolly, L. Lorenzi and R. Thiebaut, Heterogeneous viral environment in a HIV spatial model, Discrete and Continuous Dynamical Systems - Series B, 15 (2011), 545-572. doi: 10.3934/dcdsb.2011.15.545.

[3]

R. Cheynier, S. Henrichwark, F. Hadida, E. Pelletier, E. Oksenhendler, B. Autran and S. Wain-Hobson, HIV and T cell expansion in splenic white pulps is accompanied by infiltration of HIV-specific cytotoxic T lymphocytes, Cell, 78 (1994), 373-387. doi: 10.1016/0092-8674(94)90417-0.

[4]

R. DeBoer, Understanding the failure of CD8+ vaccination against Simian/Human Immunodeficiency, J. Virol., 81 (2007), 2838-2848.

[5]

M. Escobedo and M. A. Herrero, A semilinear parabolic system in a bounded domain, Annali di Matematica pura ed applicata, 165 (1993), 315-336. doi: 10.1007/BF01765854.

[6]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[7]

S. Frost, M. Dumaurier, S. Wain-Hobson and A. Leigh Brown, Genetic drift and within-host metapopulation dynamics of HIV-1 infection, Proc. Natl Acad. Sci. USA, 98 (2001), 6975-6980. doi: 10.1073/pnas.131056998.

[8]

G. Funk, V. Jansen, S. Bonhoeffer and T. Killingback, Spatial model of virus-immune dynamics, Journal of Theoretical Biology, 233 (2005), 221-236. doi: 10.1016/j.jtbi.2004.10.004.

[9]

F. Graw and A. Perelson, Spatial aspects of HIV infection, in Mathematical Methods and Models in Biomedicine, Ledzewicz, U., Sch\"attler, H., Friedman, A., Kashdan, E. (Eds.), Springer, New York, (2013) 3-31. doi: 10.1007/978-1-4614-4178-6_1.

[10]

Z. Grossman, M. Feinberg and W. Paul, Multiple modes of cellular activation and virus transmission in HIV infection: a role for chronically and latently infected cells in sustaining viral replication, Proc. Natl Acad. Sci. USA, 95 (1998), 6314-6319. doi: 10.1073/pnas.95.11.6314.

[11]

A. Haase, K. Henry, M. Zupancic, G. Sedgewick, R. Faust, H. Melrose, W. Cavert, K. Gebhard, K. Staskus, Z. Zhang, P. Dailey, H. Balfour, A. Erice and A. Perelson, Quantitative image analysis of HIV-1 infection in lymphoid tissue, Science, 274 (1996), 985-989. doi: 10.1126/science.274.5289.985.

[12]

T. H. Harris, E. J. Banigan, D. A. Christian, C. Konradt, E. D. Tait Wojno, K. Norose, E. H. Wilson, B. John, W. Weninger, A. D. Luster, A. J. Liu and C. A. Hunter, Generalized Levy walks and the role of chemokines in migration of effector CD8+ T cells, Nature, 486 (2012), 545-548. doi: 10.1038/nature11098.

[13]

E. Jones, P. Roemer, M. Raghupathi and S. Pankavich, Analysis and simulation of the three-component model of HIV dynamics, SIAM Undergraduate Research Online, 7 (2014), 89-105. doi: 10.1137/13S012698.

[14]

K. Kreith, Criteria for positive green's functions, Illinois J. Math., 12 (1968), 475-478.

[15]

J. McKeating, P. Balfe, P. Clapham and R. Weiss, Recombinant CD4-selected human immunodeficiency virus type 1 variants with reduced gp120 affinity for CD4 and increased cell fusion capacity, J Virol., 65 (1991), 4777-4785.

[16]

M. J. Miller, S. H. Wei, M. D. Cahalan and I. Parker, Autonomous T cell trafficking examined in vivo with intravital two-photon microscopy, Proc. Nat. Acad. Sci. USA, 100 (2003), 2604-2609. doi: 10.1073/pnas.2628040100.

[17]

S. Miller, R. Levenson, C. Aldridge, S. Hester, D. Kenan and D. Howell, Identification of focal viral infections by confocal microscopy for subsequent ultrastructural analysis, Ultrastructural Pathology, 21 (1997), 183-193. doi: 10.3109/01913129709021317.

[18]

J. Murray, G. Kaufman, A. Kelleher and D. Cooper, A model of primary HIV-1 infection, Math. Biosci., 154 (1998), 57-85. doi: 10.1016/S0025-5564(98)10046-9.

[19]

M. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses, Science 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[20]

M. Nowak and R. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University Press, 2000.

[21]

M. Nowak and A. McMichael, How HIV defeats the immune system, Scientific American, 273 (1995), 58-65. doi: 10.1038/scientificamerican0895-58.

[22]

S. Pankavich and N. Michalowski, Global classical solutions for the "one and one-half" dimensional relativistic vlasov-maxwell-fokker-planck system, Kinetic and Related Models, 8 (2015), 169-199. doi: 10.3934/krm.2015.8.169.

[23]

S. Pankavich and N. Michalowski, A short proof of increased parabolic regularity, Electronic Journal of Differential Equations, 205 (2015), 1-9.

[24]

A. Perelson, Modeling Viral and Immune System Dynamics, Nature Reviews, 2 (2002), 28-36.

[25]

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

[26]

A. Perelson and R. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), p96. doi: 10.1186/1741-7007-11-96.

[27]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.

[28]

T. Reinhart, M. Rogan, A. Amedee, M. Murphey-Corb, D. Rausch, L. Eiden and A. Haase, Tracking members of the simian immunodeficiency virus delta b670 quasispecies population in vivo at single-cell resolution, J. Virol., 72 (1998), 113-120.

[29]

R. Ribeiro, L. Qin, L. Chavez, D. Li, S. Self and A. Perelson, Estimation of the initial viral growth rate and basic reproductive number during acute HIV-1 infection, J. Virol., 84 (2010), 6096-6102. doi: 10.1128/JVI.00127-10.

[30]

O. Stancevic, C. N. Angstmann, J. M. Murray and B. I. Henry, Turing patterns from dynamics of early HIV infection, Bulletin of Mathematical Biology, 75 (2013), 774-795. doi: 10.1007/s11538-013-9834-5.

[31]

M. C. Strain, D. D. Richman, J. K. Wong and H. Levine, Spatiotemporal dynamics of HIV propagation, Journal of Theoretical Biology, 218 (2002), 85-96. doi: 10.1006/jtbi.2002.3055.

[32]

M. X. Wang, Global existence and finite time blow up for a reaction-diffusion system, Z. Angew. Math. Phys., 51 (2000), 160-167. doi: 10.1007/PL00001504.

[33]

W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942.

show all references

References:
[1]

S. Bonhoeffer, R. May, G. Shaw and M. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. doi: 10.1073/pnas.94.13.6971.

[2]

C.-M. Brauner, D. Jolly, L. Lorenzi and R. Thiebaut, Heterogeneous viral environment in a HIV spatial model, Discrete and Continuous Dynamical Systems - Series B, 15 (2011), 545-572. doi: 10.3934/dcdsb.2011.15.545.

[3]

R. Cheynier, S. Henrichwark, F. Hadida, E. Pelletier, E. Oksenhendler, B. Autran and S. Wain-Hobson, HIV and T cell expansion in splenic white pulps is accompanied by infiltration of HIV-specific cytotoxic T lymphocytes, Cell, 78 (1994), 373-387. doi: 10.1016/0092-8674(94)90417-0.

[4]

R. DeBoer, Understanding the failure of CD8+ vaccination against Simian/Human Immunodeficiency, J. Virol., 81 (2007), 2838-2848.

[5]

M. Escobedo and M. A. Herrero, A semilinear parabolic system in a bounded domain, Annali di Matematica pura ed applicata, 165 (1993), 315-336. doi: 10.1007/BF01765854.

[6]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[7]

S. Frost, M. Dumaurier, S. Wain-Hobson and A. Leigh Brown, Genetic drift and within-host metapopulation dynamics of HIV-1 infection, Proc. Natl Acad. Sci. USA, 98 (2001), 6975-6980. doi: 10.1073/pnas.131056998.

[8]

G. Funk, V. Jansen, S. Bonhoeffer and T. Killingback, Spatial model of virus-immune dynamics, Journal of Theoretical Biology, 233 (2005), 221-236. doi: 10.1016/j.jtbi.2004.10.004.

[9]

F. Graw and A. Perelson, Spatial aspects of HIV infection, in Mathematical Methods and Models in Biomedicine, Ledzewicz, U., Sch\"attler, H., Friedman, A., Kashdan, E. (Eds.), Springer, New York, (2013) 3-31. doi: 10.1007/978-1-4614-4178-6_1.

[10]

Z. Grossman, M. Feinberg and W. Paul, Multiple modes of cellular activation and virus transmission in HIV infection: a role for chronically and latently infected cells in sustaining viral replication, Proc. Natl Acad. Sci. USA, 95 (1998), 6314-6319. doi: 10.1073/pnas.95.11.6314.

[11]

A. Haase, K. Henry, M. Zupancic, G. Sedgewick, R. Faust, H. Melrose, W. Cavert, K. Gebhard, K. Staskus, Z. Zhang, P. Dailey, H. Balfour, A. Erice and A. Perelson, Quantitative image analysis of HIV-1 infection in lymphoid tissue, Science, 274 (1996), 985-989. doi: 10.1126/science.274.5289.985.

[12]

T. H. Harris, E. J. Banigan, D. A. Christian, C. Konradt, E. D. Tait Wojno, K. Norose, E. H. Wilson, B. John, W. Weninger, A. D. Luster, A. J. Liu and C. A. Hunter, Generalized Levy walks and the role of chemokines in migration of effector CD8+ T cells, Nature, 486 (2012), 545-548. doi: 10.1038/nature11098.

[13]

E. Jones, P. Roemer, M. Raghupathi and S. Pankavich, Analysis and simulation of the three-component model of HIV dynamics, SIAM Undergraduate Research Online, 7 (2014), 89-105. doi: 10.1137/13S012698.

[14]

K. Kreith, Criteria for positive green's functions, Illinois J. Math., 12 (1968), 475-478.

[15]

J. McKeating, P. Balfe, P. Clapham and R. Weiss, Recombinant CD4-selected human immunodeficiency virus type 1 variants with reduced gp120 affinity for CD4 and increased cell fusion capacity, J Virol., 65 (1991), 4777-4785.

[16]

M. J. Miller, S. H. Wei, M. D. Cahalan and I. Parker, Autonomous T cell trafficking examined in vivo with intravital two-photon microscopy, Proc. Nat. Acad. Sci. USA, 100 (2003), 2604-2609. doi: 10.1073/pnas.2628040100.

[17]

S. Miller, R. Levenson, C. Aldridge, S. Hester, D. Kenan and D. Howell, Identification of focal viral infections by confocal microscopy for subsequent ultrastructural analysis, Ultrastructural Pathology, 21 (1997), 183-193. doi: 10.3109/01913129709021317.

[18]

J. Murray, G. Kaufman, A. Kelleher and D. Cooper, A model of primary HIV-1 infection, Math. Biosci., 154 (1998), 57-85. doi: 10.1016/S0025-5564(98)10046-9.

[19]

M. Nowak and C. Bangham, Population dynamics of immune responses to persistent viruses, Science 272 (1996), 74-79. doi: 10.1126/science.272.5258.74.

[20]

M. Nowak and R. May, Virus dynamics: Mathematical principles of immunology and virology, Oxford University Press, 2000.

[21]

M. Nowak and A. McMichael, How HIV defeats the immune system, Scientific American, 273 (1995), 58-65. doi: 10.1038/scientificamerican0895-58.

[22]

S. Pankavich and N. Michalowski, Global classical solutions for the "one and one-half" dimensional relativistic vlasov-maxwell-fokker-planck system, Kinetic and Related Models, 8 (2015), 169-199. doi: 10.3934/krm.2015.8.169.

[23]

S. Pankavich and N. Michalowski, A short proof of increased parabolic regularity, Electronic Journal of Differential Equations, 205 (2015), 1-9.

[24]

A. Perelson, Modeling Viral and Immune System Dynamics, Nature Reviews, 2 (2002), 28-36.

[25]

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

[26]

A. Perelson and R. Ribeiro, Modeling the within-host dynamics of HIV infection, BMC Biology, 11 (2013), p96. doi: 10.1186/1741-7007-11-96.

[27]

A. Perelson, A. Neumann, M. Markowitz, J. Leonard and D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586. doi: 10.1126/science.271.5255.1582.

[28]

T. Reinhart, M. Rogan, A. Amedee, M. Murphey-Corb, D. Rausch, L. Eiden and A. Haase, Tracking members of the simian immunodeficiency virus delta b670 quasispecies population in vivo at single-cell resolution, J. Virol., 72 (1998), 113-120.

[29]

R. Ribeiro, L. Qin, L. Chavez, D. Li, S. Self and A. Perelson, Estimation of the initial viral growth rate and basic reproductive number during acute HIV-1 infection, J. Virol., 84 (2010), 6096-6102. doi: 10.1128/JVI.00127-10.

[30]

O. Stancevic, C. N. Angstmann, J. M. Murray and B. I. Henry, Turing patterns from dynamics of early HIV infection, Bulletin of Mathematical Biology, 75 (2013), 774-795. doi: 10.1007/s11538-013-9834-5.

[31]

M. C. Strain, D. D. Richman, J. K. Wong and H. Levine, Spatiotemporal dynamics of HIV propagation, Journal of Theoretical Biology, 218 (2002), 85-96. doi: 10.1006/jtbi.2002.3055.

[32]

M. X. Wang, Global existence and finite time blow up for a reaction-diffusion system, Z. Angew. Math. Phys., 51 (2000), 160-167. doi: 10.1007/PL00001504.

[33]

W. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942.

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