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2012, 9(4): 809-817. doi: 10.3934/mbe.2012.9.809

Low viral persistence of an immunological model

Suqi Ma 1,

1. 

Department of Mathematics, China Agricultural University, Beijing 100083

Received  July 2011 Revised  May 2012 Published  October 2012

Hepatitis B virus can persist at very low levels in the body in the face of host immunity, and reactive during immunosuppression and sustain the immunological memory to lead to the possible state of 'infection immunity'. To analyze this phenomena quantitatively, a mathematical model which is described by DDEs with relative to cytotoxic T lymphocyte (CTL) response to Hepatitis B virus is used. Using the knowledge of DDEs and the numerical bifurcation analysis techniques, the dynamical behavior of Hopf bifurcation which may lead to the periodic oscillation of populations is analyzed. Domains of low level viral persistence which is possible, either as a stable equilibrium or a stable oscillatory pattern, are identified in parameter space. The virus replication rate appears to have influence to the amplitude of the persisting oscillatory population densities.
Keywords: model of HBV infection, delay, bifurcation., Viral persistence.
Mathematics Subject Classification: Primary: 34K20, 92C50; Secondary: 92D2.
Citation: Suqi Ma. Low viral persistence of an immunological model. Mathematical Biosciences & Engineering, 2012, 9 (4) : 809-817. doi: 10.3934/mbe.2012.9.809
References:
[1]

P. M. Argium, P. E. Kozarsky and C. Reed, "CDC Health Information for International Travel 2008," Elsevier, Philadelphia, 2007. Google Scholar

[2]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biological Dynamic, 2 (2008), 140-153. doi: 10.1080/17513750701769873.  Google Scholar

[3]

R. M. Zinkernagel, What is missing in immunology to understand immunity?, Nat. Immunol., 1 (2000), 181-185. doi: 10.1038/79712.  Google Scholar

[4]

B. Rehermann, C. Ferrari, C. Pasquinelli and F. V. Chisari, The hepatitis B virus persists for decades after patient's recovery from acute viral hepatitis despite active maintenance of a cytotoxic T-lymphocyte response, Nat. Med., 2 (1996), 1104-1108. doi: 10.1038/nm1096-1104.  Google Scholar

[5]

L. Tatyana, R. Dirk and B. Gennady, Numerical bifurcation analysis of immunological models with time delays, Journal of Computational and Applied Mathematics, 184 (2005), 165-176. doi: 10.1016/j.cam.2004.08.019.  Google Scholar

[6]

L. Tatyana and E. Koen, Low level viral persistence after infection with LCMV: A quantitative insight through numerical bifurcation analysis, Mathematical Biosciences, 173 (2001), 1-23. doi: 10.1016/S0025-5564(01)00072-4.  Google Scholar

[7]

G. Bocharov and B. Ludewig, etc., Underwhelming the immune response: Effect of slow virus growth rates on $CD8^+ T$ lymphocyte responses, J. Virol., 78 (2004), 2247-2254. doi: 10.1128/JVI.78.5.2247-2254.2004.  Google Scholar

[8]

C. T. H. Baker, Retarded differential equations, J. Comput. Appl. Math., 125 (2000), 309-335. doi: 10.1016/S0377-0427(00)00476-3.  Google Scholar

[9]

G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, J. Comput. Appl. Math., 125 (2000), 183-199. doi: 10.1016/S0377-0427(00)00468-4.  Google Scholar

[10]

Z. H. Wang and H. Y. Hu, Stability switches of time-delayed dynamic systems with unknown parameters, Journal of Sound and Vibration, 233 (2000), 215-233. doi: 10.1006/jsvi.1999.2817.  Google Scholar

[11]

Z. H. Wang and H. Y. Hu, Delay independent stability of retarded dynamic system of multiple degrees of freedom, Journal of Sound and Vibration, 226 (1999), 57-81. doi: 10.1006/jsvi.1999.2282.  Google Scholar

[12]

S. Q. Ma, Z. S. Feng and Q. S. Lu, The double Hopf bifurcation of a neuron model with time delay, Int. J. Bifurcation and Chaos, 19 (2009), 3733-3751. doi: 10.1142/S0218127409025080.  Google Scholar

[13]

S. Q. Ma and Z. S. Feng, Fold-Hopf Bifurcation of the Rose-Hindmarsh model with time delay, Int. J. Bifurcation and Chaos, 19 (2011), 437-452. doi: 10.1142/S0218127411028490.  Google Scholar

[14]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations, J. Comput. Appl. Math., 125 (2000), 265-275. doi: 10.1016/S0377-0427(00)00472-6.  Google Scholar

[15]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21. doi: 10.1145/513001.513002.  Google Scholar

show all references

References:
[1]

P. M. Argium, P. E. Kozarsky and C. Reed, "CDC Health Information for International Travel 2008," Elsevier, Philadelphia, 2007. Google Scholar

[2]

S. A. Gourley, Y. Kuang and J. D. Nagy, Dynamics of a delay differential equation model of hepatitis B virus infection, J. Biological Dynamic, 2 (2008), 140-153. doi: 10.1080/17513750701769873.  Google Scholar

[3]

R. M. Zinkernagel, What is missing in immunology to understand immunity?, Nat. Immunol., 1 (2000), 181-185. doi: 10.1038/79712.  Google Scholar

[4]

B. Rehermann, C. Ferrari, C. Pasquinelli and F. V. Chisari, The hepatitis B virus persists for decades after patient's recovery from acute viral hepatitis despite active maintenance of a cytotoxic T-lymphocyte response, Nat. Med., 2 (1996), 1104-1108. doi: 10.1038/nm1096-1104.  Google Scholar

[5]

L. Tatyana, R. Dirk and B. Gennady, Numerical bifurcation analysis of immunological models with time delays, Journal of Computational and Applied Mathematics, 184 (2005), 165-176. doi: 10.1016/j.cam.2004.08.019.  Google Scholar

[6]

L. Tatyana and E. Koen, Low level viral persistence after infection with LCMV: A quantitative insight through numerical bifurcation analysis, Mathematical Biosciences, 173 (2001), 1-23. doi: 10.1016/S0025-5564(01)00072-4.  Google Scholar

[7]

G. Bocharov and B. Ludewig, etc., Underwhelming the immune response: Effect of slow virus growth rates on $CD8^+ T$ lymphocyte responses, J. Virol., 78 (2004), 2247-2254. doi: 10.1128/JVI.78.5.2247-2254.2004.  Google Scholar

[8]

C. T. H. Baker, Retarded differential equations, J. Comput. Appl. Math., 125 (2000), 309-335. doi: 10.1016/S0377-0427(00)00476-3.  Google Scholar

[9]

G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, J. Comput. Appl. Math., 125 (2000), 183-199. doi: 10.1016/S0377-0427(00)00468-4.  Google Scholar

[10]

Z. H. Wang and H. Y. Hu, Stability switches of time-delayed dynamic systems with unknown parameters, Journal of Sound and Vibration, 233 (2000), 215-233. doi: 10.1006/jsvi.1999.2817.  Google Scholar

[11]

Z. H. Wang and H. Y. Hu, Delay independent stability of retarded dynamic system of multiple degrees of freedom, Journal of Sound and Vibration, 226 (1999), 57-81. doi: 10.1006/jsvi.1999.2282.  Google Scholar

[12]

S. Q. Ma, Z. S. Feng and Q. S. Lu, The double Hopf bifurcation of a neuron model with time delay, Int. J. Bifurcation and Chaos, 19 (2009), 3733-3751. doi: 10.1142/S0218127409025080.  Google Scholar

[13]

S. Q. Ma and Z. S. Feng, Fold-Hopf Bifurcation of the Rose-Hindmarsh model with time delay, Int. J. Bifurcation and Chaos, 19 (2011), 437-452. doi: 10.1142/S0218127411028490.  Google Scholar

[14]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations, J. Comput. Appl. Math., 125 (2000), 265-275. doi: 10.1016/S0377-0427(00)00472-6.  Google Scholar

[15]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), 1-21. doi: 10.1145/513001.513002.  Google Scholar

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