American Institute of Mathematical Sciences

Advanced
July  2007, 8(1): 241-259. doi: 10.3934/dcdsb.2007.8.241

Immune system memory realization in a population model

Jianhong Wu 1, , Weiguang Yao 2, and  Huaiping Zhu 3,

1. 

Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3
2. 

LAboratory of Mathematical Parallel systems (LAMPS), Laboratory for Industrial and Applied Mathematics (LIAM), Department of Mathematics and Statistics, York University, Toronto M3J 1P3, Canada
3. 

Department of Mathematics and Statistics, Laboratory of Mathematical Parallel systems (LAMPS) and CDM, York University, Toronto M3J 1P3, Canada

Received  December 2005 Revised  October 2006 Published  April 2007

A general process of the immune system consists of effector stage and memory stage. Current theoretical studies of the immune system often focus on the memory stage and pay less attention on the function of non-immune system substances such as tissue cells in adjusting the dynamical behavior of the immune system. We propose a mathematical population model to investigate the interaction between influenza A virus(IAV) susceptible tissue cells and generic immune cells when the tissue is invaded by IAV. We carry out a linear stability analysis and numerically study the Neimark-Sacker bifurcation of the models. The behavior of the model system agrees with some important experimental or clinical observations for IAV. However, we show that without considering the space between tissue cells, the expected memory stage does not form. By considering the space which allows antibodies to bind antigens, the memory stage then forms without missing the property of the system in the effector stage.
Keywords: age-dependent mathematical model., IAV, memory stage, Immune system, effector stage.
Mathematics Subject Classification: Primary: 92D; Secondary: 34K11, 37N2.
Citation: Jianhong Wu, Weiguang Yao, Huaiping Zhu. Immune system memory realization in a population model. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 241-259. doi: 10.3934/dcdsb.2007.8.241
[1]

Dan Liu, Shigui Ruan, Deming Zhu. Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. Mathematical Biosciences & Engineering, 2012, 9 (2) : 347-368. doi: 10.3934/mbe.2012.9.347
[2]

Youssef Amal, Martin Campos Pinto. Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 517-535. doi: 10.3934/dcdsb.2010.13.517
[3]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005
[4]

Xinyu Song, Liming Cai, U. Neumann. Ratio-dependent predator-prey system with stage structure for prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 747-758. doi: 10.3934/dcdsb.2004.4.747
[5]

Ruijun Zhao, Jemal Mohammed-Awel. A mathematical model studying mosquito-stage transmission-blocking vaccines. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1229-1245. doi: 10.3934/mbe.2014.11.1229
[6]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931
[7]

Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233
[8]

Shanjing Ren. Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1337-1360. doi: 10.3934/mbe.2017069
[9]

Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040
[10]

Gabriella Di Blasio. An ultraparabolic problem arising from age-dependent population diffusion. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 843-858. doi: 10.3934/dcds.2009.25.843
[11]

Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691
[12]

Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067
[13]

Jaume Llibre, Claudio Vidal. Hopf periodic orbits for a ratio--dependent predator--prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1859-1867. doi: 10.3934/dcdsb.2016026
[14]

Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei, Jingxue Yin. Early and late stage profiles for a chemotaxis model with density-dependent jump probability. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1345-1385. doi: 10.3934/mbe.2018062
[15]

Jing-An Cui, Xinyu Song. Permanence of predator-prey system with stage structure. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 547-554. doi: 10.3934/dcdsb.2004.4.547
[16]

Jia Li. Malaria model with stage-structured mosquitoes. Mathematical Biosciences & Engineering, 2011, 8 (3) : 753-768. doi: 10.3934/mbe.2011.8.753
[17]

G. Buffoni, S. Pasquali, G. Gilioli. A stochastic model for the dynamics of a stage structured population. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 517-525. doi: 10.3934/dcdsb.2004.4.517
[18]

Feng-Bin Wang. A periodic reaction-diffusion model with a quiescent stage. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 283-295. doi: 10.3934/dcdsb.2012.17.283
[19]

Jinping Fang, Guang Lin, Hui Wan. Analysis of a stage-structured dengue model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4045-4061. doi: 10.3934/dcdsb.2018125
[20]

Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences