# American Institute of Mathematical Sciences

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

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##### 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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