September  2013, 12(5): 1813-1844. doi: 10.3934/cpaa.2013.12.1813

Two-dimensional stability analysis in a HIV model with quadratic logistic growth term

1. 

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

2. 

College of Science, Guizhou University, 550025 Guiyang, China

3. 

Dipartimento di Matematica e Informatica, Università degli Studi di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

Received  July 2012 Revised  November 2012 Published  January 2013

We consider a Human Immunode ciency Virus (HIV) model with a logistic growth term and continue the analysis of the previous article [6]. We now take the viral di usion in a two-dimensional environment. The model consists of two ODEs for the concentrations of the target T cells, the infected cells, and a parabolic PDE for the virus particles. We study the stability of the uninfected and infected equilibria, the occurrence of Hopf bifurcation and the stability of the periodic solutions.
Citation: Claude-Michel Brauner, Xinyue Fan, Luca Lorenzi. Two-dimensional stability analysis in a HIV model with quadratic logistic growth term. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1813-1844. doi: 10.3934/cpaa.2013.12.1813
References:
[1]

C.-M. Brauner, D. Jolly, L. Lorenzi and R. Thiébaut, Heterogeneous viral environment in a HIV spatial model, Discr. Contin. Dyn. Syst. B, 15 (2011), 545-572. doi: 10.3934/dcdsb.2011.15.545.

[2]

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), 1-27. doi: 10.1016/j.mbs.2005.12.006.

[3]

G. Da Prato and A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Arch. Ration. Mech. Anal., 101 (1988), 115-142.

[4]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis, and Interpretation,'' John Wiley & Sons, Ltd., Chichester, 2000.

[5]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Graduate Texts in Mathematics, 194 (). 

[6]

X. Y. Fan, C.-M. Brauner and L. Wittkop, Mathematical analysis of a HIV model with quadratic logistic growth term, Discr. Cont. Dyn. Syst. B, 17 (2012), 2359-2385. doi: 10.3934/dcdsb.2012.17.2359.

[7]

G. A. Funk, V. A. A. Jansen, S. Bonhoeffer and T. Killingback, Spatial models of virus-immune dynamics, J. Theor. Biol., 233 (2005), 221-236. doi: 10.1016/j.jtbi.2004.10.004.

[8]

F. R. Gantmakher, "The Theory of Matrices,'' Reprint of the 1959 translation. AMS Chelsea Publishing, Providence, RI, 1998.

[9]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,'' sixth edition, Oxford University Press, Oxford, 2008.

[10]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'' Cambridge University Press, Cambridge, 1981.

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'', Lect. Notes. Math. 61, 61 (). 

[12]

T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlin-New York, 1976.

[13]

H. B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems, Bull. Amer. Math Soc., 74 (1968), 887-891. doi: 10.1090/S0002-9904-1968-12067-1.

[14]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'' Birkh\"auser, Basel, 1995.

[15]

J. E. Marsden and M. McCracken, "The Hopf Bifurcation and its Applications,'' Springer-Verlag, New York, 1976.

[16]

A. S. Perelson, D. E. 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.

[17]

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

[18]

K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95. doi: 10.1016/j.mbs.2007.05.004.

show all references

References:
[1]

C.-M. Brauner, D. Jolly, L. Lorenzi and R. Thiébaut, Heterogeneous viral environment in a HIV spatial model, Discr. Contin. Dyn. Syst. B, 15 (2011), 545-572. doi: 10.3934/dcdsb.2011.15.545.

[2]

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), 1-27. doi: 10.1016/j.mbs.2005.12.006.

[3]

G. Da Prato and A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Arch. Ration. Mech. Anal., 101 (1988), 115-142.

[4]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis, and Interpretation,'' John Wiley & Sons, Ltd., Chichester, 2000.

[5]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Graduate Texts in Mathematics, 194 (). 

[6]

X. Y. Fan, C.-M. Brauner and L. Wittkop, Mathematical analysis of a HIV model with quadratic logistic growth term, Discr. Cont. Dyn. Syst. B, 17 (2012), 2359-2385. doi: 10.3934/dcdsb.2012.17.2359.

[7]

G. A. Funk, V. A. A. Jansen, S. Bonhoeffer and T. Killingback, Spatial models of virus-immune dynamics, J. Theor. Biol., 233 (2005), 221-236. doi: 10.1016/j.jtbi.2004.10.004.

[8]

F. R. Gantmakher, "The Theory of Matrices,'' Reprint of the 1959 translation. AMS Chelsea Publishing, Providence, RI, 1998.

[9]

G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers,'' sixth edition, Oxford University Press, Oxford, 2008.

[10]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation,'' Cambridge University Press, Cambridge, 1981.

[11]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'', Lect. Notes. Math. 61, 61 (). 

[12]

T. Kato, "Perturbation Theory for Linear Operators," Second edition, Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlin-New York, 1976.

[13]

H. B. Keller, Nonexistence and uniqueness of positive solutions of nonlinear eigenvalue problems, Bull. Amer. Math Soc., 74 (1968), 887-891. doi: 10.1090/S0002-9904-1968-12067-1.

[14]

A. Lunardi, "Analytic Semigroups and Optimal Regularity in Parabolic Problems,'' Birkh\"auser, Basel, 1995.

[15]

J. E. Marsden and M. McCracken, "The Hopf Bifurcation and its Applications,'' Springer-Verlag, New York, 1976.

[16]

A. S. Perelson, D. E. 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.

[17]

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

[18]

K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95. doi: 10.1016/j.mbs.2007.05.004.

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