December  2011, 15(3): 545-572. doi: 10.3934/dcdsb.2011.15.545

Heterogeneous viral environment in a HIV spatial model

1. 

School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China, and Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence cedex, France

2. 

Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence cedex, France

3. 

Dipartimento di Matematica, Università degli Studi di Parma, Viale Parco Area delle Scienze 53/A, I-43124 Parma

4. 

(M.D.) Equipe Biostatistique de l'U897 INSERM ISPED, Université de Bordeaux, 33076 Bordeaux cedex, France

Received  February 2010 Revised  March 2010 Published  February 2011

We consider a basic model of virus dynamics in the modeling of Human Immunodeficiency Virus (HIV), in a two-dimensional heterogenous environment. It consists of two ODEs for the uninfected and infected CD4$^+$ T lymphocytes, $T$ and $I$, and a parabolic PDE for the free virus particles $V$. We introduce a new parameter $\lambda_0$ which is the largest eigenvalue of some Sturm-Liouville problem and takes the heterogenous reproductive ratio into account. For $\lambda_0<0$ the uninfected steady state is the only equilibrium. When $\lambda_0>0$, it becomes unstable and there is a unique positive infected equilibrium. Considering the model as a dynamical system, we prove the existence of a positively invariant region. Finally, in the case of an alternating structure of viral sources, we define a homogenized limiting environment which justifies the classical approach via ODE systems.
Citation: Claude-Michel Brauner, Danaelle Jolly, Luca Lorenzi, Rodolphe Thiebaut. Heterogeneous viral environment in a HIV spatial model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 545-572. doi: 10.3934/dcdsb.2011.15.545
References:
[1]

S. Agmon, "Lectures on Elliptic Boundary Value Problems,'', Van Nostrand Mathematical Studies, 2 (1965).   Google Scholar

[2]

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

[3]

J. M. Brenchley, D. A. Price and D. C. Douek, HIV disease: fallout from a mucosal catastrophe?, Nat. Immunol., 7 (2006), 235.  doi: 10.1038/ni1316.  Google Scholar

[4]

D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bull. Math. Biol., 64 (2002), 29.  doi: 10.1006/bulm.2001.0266.  Google Scholar

[5]

M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1.  doi: 10.1016/j.mbs.2005.12.006.  Google Scholar

[6]

T. W. Chun , L. Carruth, D. Finzi, X. Shen, J. A. DiGiuseppe, H. Taylor, M. Hermankova, K. Chadwick, J. Margolick, T. C. Quinn, Y. H. Kuo, R. Brookmeyer, M. A. Zeiger, P. Barditch-Crovo and R. F. Siliciano, Quantification of latent tissue reservoirs and total body viral load in HIV-1 infection,, Nature, 387 (1997), 183.  doi: 10.1038/387183a0.  Google Scholar

[7]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[8]

E. S. Daar, T. Moudgil, R. D. Meyer and D. D. Ho, Transient high levels of viremia in patients with primary human immunodeficiency virus type 1,, New Engl. J. Med., 324 (1991), 961.  doi: 10.1056/NEJM199104043241405.  Google Scholar

[9]

G. A. Funk, V. A. A. Jansen, S. Bonhoeffer and T. Killingback, Spatial models of virus-immune dynamics,, J. Theoret. Biol., 233 (2005), 221.  doi: 10.1016/j.jtbi.2004.10.004.  Google Scholar

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'', Monographs and Studies in Mathematics, 24 (1985).   Google Scholar

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'', Lecture Notes in Mathematics, 840 (1981).   Google Scholar

[12]

D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection,, Nature, 373 (1995), 123.  doi: 10.1038/373123a0.  Google Scholar

[13]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, "Homogenization of Differential Operators and Integral Functionals,'', Springer-Verlag, (1994).   Google Scholar

[14]

H. B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems,, Bull. Amer. Math Soc., 74 (1968), 887.  doi: 10.1090/S0002-9904-1968-12067-1.  Google Scholar

[15]

J. L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal,, Lect. Notes in Math., 323 (1973).   Google Scholar

[16]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'', Birkhäuser, (1995).   Google Scholar

[17]

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

[18]

M. A. Nowak and R. May, "Virus Dynamics: Mathematical Principles of Immunology and Virology,'', Oxford University Press, (2000).   Google Scholar

[19]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'', Plenum Press, (1992).   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

A. N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response,, Science, 271 (1996), 497.  doi: 10.1126/science.271.5248.497.  Google Scholar

[24]

H. H. Schaefer and M. P. Wolff, "Topological Vector Spaces. Second Edition,'', Graduate Texts in Mathematics, 3 (1999).   Google Scholar

[25]

R. Temam, "Infinite-Dimensional Dynamical Systems In Mechanics And Physics. Second Edition,'', Applied Mathematical Sciences, 68 (1997).   Google Scholar

[26]

K. Wang and W. Wang, Propagation of HBV with spatial dependence,, Math. Biosci., 210 (2007), 78.  doi: 10.1016/j.mbs.2007.05.004.  Google Scholar

[27]

K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay,, J. Theor. Biol., 253 (2008), 36.  doi: 10.1016/j.jtbi.2007.11.007.  Google Scholar

[28]

X. Wang and X. Song, Global stability and periodic solution of a model for HIV infection of CD4$^+$ T cells,, Appl. Math. Comput., 189 (2007), 1331.  doi: 10.1016/j.amc.2006.12.044.  Google Scholar

[29]

X. Wei, S. K. Ghosh, M. E. Taylor, V. A. Johnson, E. A. Emini, P. Deutsch, J. D. Lifson and S. Bonhoeffer, Viral dynamics in human immunodeficiency virus type 1 infection,, Nature, 373 (1995), 117.  doi: 10.1038/373117a0.  Google Scholar

show all references

References:
[1]

S. Agmon, "Lectures on Elliptic Boundary Value Problems,'', Van Nostrand Mathematical Studies, 2 (1965).   Google Scholar

[2]

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

[3]

J. M. Brenchley, D. A. Price and D. C. Douek, HIV disease: fallout from a mucosal catastrophe?, Nat. Immunol., 7 (2006), 235.  doi: 10.1038/ni1316.  Google Scholar

[4]

D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bull. Math. Biol., 64 (2002), 29.  doi: 10.1006/bulm.2001.0266.  Google Scholar

[5]

M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models,, Math. Biosci., 200 (2006), 1.  doi: 10.1016/j.mbs.2005.12.006.  Google Scholar

[6]

T. W. Chun , L. Carruth, D. Finzi, X. Shen, J. A. DiGiuseppe, H. Taylor, M. Hermankova, K. Chadwick, J. Margolick, T. C. Quinn, Y. H. Kuo, R. Brookmeyer, M. A. Zeiger, P. Barditch-Crovo and R. F. Siliciano, Quantification of latent tissue reservoirs and total body viral load in HIV-1 infection,, Nature, 387 (1997), 183.  doi: 10.1038/387183a0.  Google Scholar

[7]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^+$ T-cells,, Math. Biosci., 165 (2000), 27.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[8]

E. S. Daar, T. Moudgil, R. D. Meyer and D. D. Ho, Transient high levels of viremia in patients with primary human immunodeficiency virus type 1,, New Engl. J. Med., 324 (1991), 961.  doi: 10.1056/NEJM199104043241405.  Google Scholar

[9]

G. A. Funk, V. A. A. Jansen, S. Bonhoeffer and T. Killingback, Spatial models of virus-immune dynamics,, J. Theoret. Biol., 233 (2005), 221.  doi: 10.1016/j.jtbi.2004.10.004.  Google Scholar

[10]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'', Monographs and Studies in Mathematics, 24 (1985).   Google Scholar

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'', Lecture Notes in Mathematics, 840 (1981).   Google Scholar

[12]

D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection,, Nature, 373 (1995), 123.  doi: 10.1038/373123a0.  Google Scholar

[13]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, "Homogenization of Differential Operators and Integral Functionals,'', Springer-Verlag, (1994).   Google Scholar

[14]

H. B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems,, Bull. Amer. Math Soc., 74 (1968), 887.  doi: 10.1090/S0002-9904-1968-12067-1.  Google Scholar

[15]

J. L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal,, Lect. Notes in Math., 323 (1973).   Google Scholar

[16]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'', Birkhäuser, (1995).   Google Scholar

[17]

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

[18]

M. A. Nowak and R. May, "Virus Dynamics: Mathematical Principles of Immunology and Virology,'', Oxford University Press, (2000).   Google Scholar

[19]

C. V. Pao, "Nonlinear Parabolic and Elliptic Equations,'', Plenum Press, (1992).   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

A. N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response,, Science, 271 (1996), 497.  doi: 10.1126/science.271.5248.497.  Google Scholar

[24]

H. H. Schaefer and M. P. Wolff, "Topological Vector Spaces. Second Edition,'', Graduate Texts in Mathematics, 3 (1999).   Google Scholar

[25]

R. Temam, "Infinite-Dimensional Dynamical Systems In Mechanics And Physics. Second Edition,'', Applied Mathematical Sciences, 68 (1997).   Google Scholar

[26]

K. Wang and W. Wang, Propagation of HBV with spatial dependence,, Math. Biosci., 210 (2007), 78.  doi: 10.1016/j.mbs.2007.05.004.  Google Scholar

[27]

K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay,, J. Theor. Biol., 253 (2008), 36.  doi: 10.1016/j.jtbi.2007.11.007.  Google Scholar

[28]

X. Wang and X. Song, Global stability and periodic solution of a model for HIV infection of CD4$^+$ T cells,, Appl. Math. Comput., 189 (2007), 1331.  doi: 10.1016/j.amc.2006.12.044.  Google Scholar

[29]

X. Wei, S. K. Ghosh, M. E. Taylor, V. A. Johnson, E. A. Emini, P. Deutsch, J. D. Lifson and S. Bonhoeffer, Viral dynamics in human immunodeficiency virus type 1 infection,, Nature, 373 (1995), 117.  doi: 10.1038/373117a0.  Google Scholar

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