Search Advanced

Mathematical Biosciences and Engineering

2007, Volume 4, Issue 2: 261-286. doi: 10.3934/mbe.2007.4.261
Previous Article Next Article

The role of delays in innate and adaptive immunity to intracellular bacterial infection

  • 1.
    Dept. of Microbiology and Immunology, University of Michigan Medical School, 6730 Med. Sci. Bldg. II, Ann Arbor, MI 48109-0620
  • 2.
    Institute of Biomathematics, University of Urbino
  • 3.
    Dept. of Microbiology and Immunology, University of Michigan Medical School, 6730 Med. Sci. Bldg. II, Ann Arbor, MI 48109-062
  • Received: 01 September 2006 Accepted: 29 June 2018 Published: 01 February 2007

  • MSC : 92B99.

  • The immune response in humans is complex and multi-fold. Initially an innate response attempts to clear any invasion by microbes. If it fails to clear or contain the pathogen, an adaptive response follows that is specific for the microbe and in most cases is successful at eliminating the pathogen. In previous work we developed a delay differential equations (DDEs) model of the innate and adaptive immune response to intracellular bacteria infection. We addressed the relevance of known delays in each of these responses by exploring different kernel and delay functions and tested how each affected infection outcome. Our results indicated how local stability properties for the two infection outcomes, namely a boundary equilibrium and an interior positive equilibrium, were completely dependent on the delays for innate immunity and independent of the delays for adaptive immunity. In the present work we have three goals. The first is to extend the previous model to account for direct bacterial killing by adaptive immunity. This reflects, for example, active killing by a class of cells known as macrophages, and will allow us to determine the relevance of delays for adaptive immunity. We present analytical results in this setting. Second, we implement a heuristic argument to investigate the existence of stability switches for the positive equilibrium in the manifold defined by the two delays. Third, we apply a novel analysis in the setting of DDEs known as uncertainty and sensitivity analysis. This allows us to evaluate completely the role of all parameters in the model. This includes identifying effects of stability switch parameters on infection outcome.

    Citation: Simeone Marino, Edoardo Beretta, Denise E. Kirschner. The role of delays in innate and adaptive immunity to intracellular bacterial infection[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 261-286. doi: 10.3934/mbe.2007.4.261

    Related Papers:

    [1] Eduardo Ibargüen-Mondragón, M. Victoria Otero-Espinar, Miller Cerón Gómez . A within-host model on the interaction dynamics between innate immune cells and Mycobacterium tuberculosis. Mathematical Biosciences and Engineering, 2025, 22(3): 511-527. doi: 10.3934/mbe.2025019
    [2] H. M. Srivastava, Khaled M. Saad, J. F. Gómez-Aguilar, Abdulrhman A. Almadiy . Some new mathematical models of the fractional-order system of human immune against IAV infection. Mathematical Biosciences and Engineering, 2020, 17(5): 4942-4969. doi: 10.3934/mbe.2020268
    [3] Yu Yang, Gang Huang, Yueping Dong . Stability and Hopf bifurcation of an HIV infection model with two time delays. Mathematical Biosciences and Engineering, 2023, 20(2): 1938-1959. doi: 10.3934/mbe.2023089
    [4] Xuejuan Lu, Lulu Hui, Shengqiang Liu, Jia Li . A mathematical model of HTLV-I infection with two time delays. Mathematical Biosciences and Engineering, 2015, 12(3): 431-449. doi: 10.3934/mbe.2015.12.431
    [5] Pensiri Yosyingyong, Ratchada Viriyapong . Global dynamics of multiple delays within-host model for a hepatitis B virus infection of hepatocytes with immune response and drug therapy. Mathematical Biosciences and Engineering, 2023, 20(4): 7349-7386. doi: 10.3934/mbe.2023319
    [6] Edgar Alberto Vega Noguera, Simeón Casanova Trujillo, Eduardo Ibargüen-Mondragón . A within-host model on the interactions of sensitive and resistant Helicobacter pylori to antibiotic therapy considering immune response. Mathematical Biosciences and Engineering, 2025, 22(1): 185-224. doi: 10.3934/mbe.2025009
    [7] A. M. Elaiw, A. S. Shflot, A. D. Hobiny . Stability analysis of general delayed HTLV-I dynamics model with mitosis and CTL immunity. Mathematical Biosciences and Engineering, 2022, 19(12): 12693-12729. doi: 10.3934/mbe.2022593
    [8] Jian Ren, Rui Xu, Liangchen Li . Global stability of an HIV infection model with saturated CTL immune response and intracellular delay. Mathematical Biosciences and Engineering, 2021, 18(1): 57-68. doi: 10.3934/mbe.2021003
    [9] Avner Friedman, Edward M. Lungu . Can malaria parasite pathogenesis be prevented by treatment with tumor necrosis factor-alpha?. Mathematical Biosciences and Engineering, 2013, 10(3): 609-624. doi: 10.3934/mbe.2013.10.609
    [10] Farzad Fatehi, Yuliya N. Kyrychko, Konstantin B. Blyuss . Time-delayed model of autoimmune dynamics. Mathematical Biosciences and Engineering, 2019, 16(5): 5613-5639. doi: 10.3934/mbe.2019279
  • The immune response in humans is complex and multi-fold. Initially an innate response attempts to clear any invasion by microbes. If it fails to clear or contain the pathogen, an adaptive response follows that is specific for the microbe and in most cases is successful at eliminating the pathogen. In previous work we developed a delay differential equations (DDEs) model of the innate and adaptive immune response to intracellular bacteria infection. We addressed the relevance of known delays in each of these responses by exploring different kernel and delay functions and tested how each affected infection outcome. Our results indicated how local stability properties for the two infection outcomes, namely a boundary equilibrium and an interior positive equilibrium, were completely dependent on the delays for innate immunity and independent of the delays for adaptive immunity. In the present work we have three goals. The first is to extend the previous model to account for direct bacterial killing by adaptive immunity. This reflects, for example, active killing by a class of cells known as macrophages, and will allow us to determine the relevance of delays for adaptive immunity. We present analytical results in this setting. Second, we implement a heuristic argument to investigate the existence of stability switches for the positive equilibrium in the manifold defined by the two delays. Third, we apply a novel analysis in the setting of DDEs known as uncertainty and sensitivity analysis. This allows us to evaluate completely the role of all parameters in the model. This includes identifying effects of stability switch parameters on infection outcome.


  • This article has been cited by:

    1. Mustafa Kudu, Ilhame Amirali, Gabil M. Amiraliyev, A finite-difference method for a singularly perturbed delay integro-differential equation, 2016, 308, 03770427, 379, 10.1016/j.cam.2016.06.018
    2. M. Zarebnia, L. Shiri, The Numerical Solution of Volterra Integro-Differential Equations with State-Dependent Delay, 2017, 41, 1028-6276, 465, 10.1007/s40995-017-0268-z
    3. Ali Abdi, Jean–Paul Berrut, Seyyed Ahmad Hosseini, The Linear Barycentric Rational Method for a Class of Delay Volterra Integro-Differential Equations, 2018, 75, 0885-7474, 1757, 10.1007/s10915-017-0608-3
    4. Ömer Yapman, Gabil M. Amiraliyev, Ilhame Amirali, Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay, 2019, 355, 03770427, 301, 10.1016/j.cam.2019.01.026
    5. Mohammad Shakourifar, Wayne Enright, Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay, 2012, 52, 0006-3835, 725, 10.1007/s10543-012-0373-5
    6. Shinji Nakaoka, Mathematical analysis and classification of tumor immune dynamics in T cell transfer treatment, 2015, 6, 2185-4106, 54, 10.1587/nolta.6.54
    7. Simeone Marino, Mohammed El-Kebir, Denise Kirschner, A hybrid multi-compartment model of granuloma formation and T cell priming in Tuberculosis, 2011, 280, 00225193, 50, 10.1016/j.jtbi.2011.03.022
    8. Karthik Raman, Nagasuma Chandra, 2011, Chapter 5, 978-1-4419-7963-6, 83, 10.1007/978-1-4419-7964-3_5
    9. Ömer Yapman, Gabil M. Amiraliyev, A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation, 2020, 97, 0020-7160, 1293, 10.1080/00207160.2019.1614565
    10. Simeone Marino, Ian B. Hogue, Christian J. Ray, Denise E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, 2008, 254, 00225193, 178, 10.1016/j.jtbi.2008.04.011
    11. M. Shakourifar, W. H. Enright, Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays, 2011, 33, 1064-8275, 1134, 10.1137/100793098
    12. Simeone Marino, Denise Kirschner, A Multi-Compartment Hybrid Computational Model Predicts Key Roles for Dendritic Cells in Tuberculosis Infection, 2016, 4, 2079-3197, 39, 10.3390/computation4040039
    13. E B Megam Ngouonkadi, M Kabong Nono, H B Fotsin, M Ekonde Sone, D Yemele, Hopf and quasi-periodic Hopf bifurcations and deterministic coherence in coupled Duffing-Holmes and Van der Pol oscillators: the Arnol’d resonance web, 2022, 97, 0031-8949, 065202, 10.1088/1402-4896/ac6547
    14. Hayriye GUCKİR CAKİR, Firat CAKİR, Musa ÇAKIR, A numerical method on Bakhvalov-Shishkin mesh for Volterra integro-differential equations with a boundary layer, 2022, 71, 1303-5991, 51, 10.31801/cfsuasmas.946910
    15. Guangqing Long, Li-Bin Liu, Zaitang Huang, Richardson Extrapolation Method on an Adaptive Grid for Singularly Perturbed Volterra Integro-Differential Equations, 2021, 42, 0163-0563, 739, 10.1080/01630563.2021.1928698
    16. Ömer Yapman, Mustafa Kudu, Gabil M. Amiraliyev, Method of Exact Difference Schemes for the Numerical Solution of Parameterized Singularly Perturbed Problem, 2023, 20, 1660-5446, 10.1007/s00009-023-02360-y
    17. Li-Bin Liu, Yige Liao, Guangqing Long, Error estimate of BDF2 scheme on a Bakhvalov-type mesh for a singularly perturbed Volterra integro-differential equation, 2023, 18, 1556-1801, 547, 10.3934/nhm.2023023
    18. Ömer Yapman, Gabil M. Amiraliyev, Convergence analysis of the homogeneous second order difference method for a singularly perturbed Volterra delay-integro-differential equation, 2021, 150, 09600779, 111100, 10.1016/j.chaos.2021.111100
    19. Sunil Kumar, Shashikant Kumar, , A priori and a posteriori error estimation for singularly perturbed delay integro-differential equations, 2023, 1017-1398, 10.1007/s11075-023-01620-y
    20. Yige Liao, Xianbing Luo, Li-Bin Liu, A second order difference scheme on a Bakhvalov-type mesh for the singularly perturbed Volterra delay-integro-differential equation, 2024, 43, 2238-3603, 10.1007/s40314-024-02873-6
  • Reader Comments
  • © 2007 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(3003) PDF downloads(612) Cited by(20)

Preview PDF
Download XML
Export Citation
Article outline
Abstract

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog