Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

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.
  Article Metrics

Keywords Delay differential equations model; uncertainty and sensitivity analysis.; bacterial infections; innate and adaptive immunity

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


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, Journal of Computational and Applied Mathematics, 2016, 308, 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, Iranian Journal of Science and Technology, Transactions A: Science, 2017, 41, 2, 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, Journal of Scientific Computing, 2018, 75, 3, 1757, 10.1007/s10915-017-0608-3
  • 4. Mohammad Shakourifar, Wayne Enright, Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay, BIT Numerical Mathematics, 2012, 52, 3, 725, 10.1007/s10543-012-0373-5
  • 5. Shinji Nakaoka, Mathematical analysis and classification of tumor immune dynamics in T cell transfer treatment, Nonlinear Theory and Its Applications, IEICE, 2015, 6, 1, 54, 10.1587/nolta.6.54
  • 6. Simeone Marino, Mohammed El-Kebir, Denise Kirschner, A hybrid multi-compartment model of granuloma formation and T cell priming in Tuberculosis, Journal of Theoretical Biology, 2011, 280, 1, 50, 10.1016/j.jtbi.2011.03.022
  • 7. Simeone Marino, Ian B. Hogue, Christian J. Ray, Denise E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology, 2008, 254, 1, 178, 10.1016/j.jtbi.2008.04.011
  • 8. M. Shakourifar, W. H. Enright, Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays, SIAM Journal on Scientific Computing, 2011, 33, 3, 1134, 10.1137/100793098
  • 9. Simeone Marino, Denise Kirschner, A Multi-Compartment Hybrid Computational Model Predicts Key Roles for Dendritic Cells in Tuberculosis Infection, Computation, 2016, 4, 4, 39, 10.3390/computation4040039
  • 10. Ömer Yapman, Gabil M. Amiraliyev, Ilhame Amirali, Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay, Journal of Computational and Applied Mathematics, 2019, 10.1016/j.cam.2019.01.026
  • 11. Karthik Raman, Nagasuma Chandra, , Understanding the Dynamics of Biological Systems, 2011, Chapter 5, 83, 10.1007/978-1-4419-7964-3_5
  • 12. Ömer Yapman, Gabil M. Amiraliyev, A novel second-order fitted computational method for a singularly perturbed Volterra integro-differential equation, International Journal of Computer Mathematics, 2019, 1, 10.1080/00207160.2019.1614565

Reader Comments

your name: *   your email: *  

Copyright Info: 2007, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved