# American Institute of Mathematical Sciences

• Previous Article
Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes
• MBE Home
• This Issue
• Next Article
The impact of migrant workers on the tuberculosis transmission: General models and a case study for China
2012, 9(4): 809-817. doi: 10.3934/mbe.2012.9.809

## Low viral persistence of an immunological model

 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.
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, (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.  doi: 10.1080/17513750701769873.  Google Scholar [3] R. M. Zinkernagel, What is missing in immunology to understand immunity?,, Nat. Immunol., 1 (2000), 181.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, (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.  doi: 10.1080/17513750701769873.  Google Scholar [3] R. M. Zinkernagel, What is missing in immunology to understand immunity?,, Nat. Immunol., 1 (2000), 181.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1145/513001.513002.  Google Scholar
 [1] Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749 [2] Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074 [3] Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143 [4] Ting Guo, Haihong Liu, Chenglin Xu, Fang Yan. Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4223-4242. doi: 10.3934/dcdsb.2018134 [5] Hong Yang, Junjie Wei. Dynamics of spatially heterogeneous viral model with time delay. Communications on Pure & Applied Analysis, 2020, 19 (1) : 85-102. doi: 10.3934/cpaa.2020005 [6] Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006 [7] Stephen Pankavich, Deborah Shutt. An in-host model of HIV incorporating latent infection and viral mutation. Conference Publications, 2015, 2015 (special) : 913-922. doi: 10.3934/proc.2015.0913 [8] Yu Ji. Global stability of a multiple delayed viral infection model with general incidence rate and an application to HIV infection. Mathematical Biosciences & Engineering, 2015, 12 (3) : 525-536. doi: 10.3934/mbe.2015.12.525 [9] Zhixing Hu, Weijuan Pang, Fucheng Liao, Wanbiao Ma. Analysis of a CD4$^+$ T cell viral infection model with a class of saturated infection rate. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 735-745. doi: 10.3934/dcdsb.2014.19.735 [10] Don A. Jones, Hal L. Smith, Horst R. Thieme. Spread of viral infection of immobilized bacteria. Networks & Heterogeneous Media, 2013, 8 (1) : 327-342. doi: 10.3934/nhm.2013.8.327 [11] Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215 [12] Yu Ji, Lan Liu. Global stability of a delayed viral infection model with nonlinear immune response and general incidence rate. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 133-149. doi: 10.3934/dcdsb.2016.21.133 [13] Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Taza Gul, Fawad Hussain. A fractional order HBV model with hospitalization. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 957-974. doi: 10.3934/dcdss.2020056 [14] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [15] Hui Cao, Yicang Zhou, Zhien Ma. Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1399-1417. doi: 10.3934/mbe.2013.10.1399 [16] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [17] Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002 [18] Sukhitha W. Vidurupola, Linda J. S. Allen. Basic stochastic models for viral infection within a host. Mathematical Biosciences & Engineering, 2012, 9 (4) : 915-935. doi: 10.3934/mbe.2012.9.915 [19] Steffen Eikenberry, Sarah Hews, John D. Nagy, Yang Kuang. The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 2009, 6 (2) : 283-299. doi: 10.3934/mbe.2009.6.283 [20] Songbai Guo, Wanbiao Ma. Global behavior of delay differential equations model of HIV infection with apoptosis. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 103-119. doi: 10.3934/dcdsb.2016.21.103

2018 Impact Factor: 1.313