American Institute of Mathematical Sciences

Advanced
2007, 4(2): 261-286. doi: 10.3934/mbe.2007.4.261

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

Simeone Marino 1, , Edoardo Beretta 2, and  Denise E. Kirschner 3,

1. 

Dept. of Microbiology and Immunology, University of Michigan Medical School, 6730 Med. Sci. Bldg. II, Ann Arbor, MI 48109-0620, United States
2. 

Institute of Biomathematics, University of Urbino, Italy
3. 

Dept. of Microbiology and Immunology, University of Michigan Medical School, 6730 Med. Sci. Bldg. II, Ann Arbor, MI 48109-062, United States

Received  September 2006 Revised  November 2006 Published  February 2007

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.
Keywords: Delay differential equations model, uncertainty and sensitivity analysis., bacterial infections, innate and adaptive immunity.
Mathematics Subject Classification: 92B9.
Citation: Simeone Marino, Edoardo Beretta, Denise E. Kirschner. The role of delays in innate and adaptive immunity to intracellular bacterial infection. Mathematical Biosciences & Engineering, 2007, 4 (2) : 261-286. doi: 10.3934/mbe.2007.4.261
[1]

H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301
[2]

Chunhua Shan, Hongjun Gao, Huaiping Zhu. Dynamics of a delay Schistosomiasis model in snail infections. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1099-1115. doi: 10.3934/mbe.2011.8.1099
[3]

Mudassar Imran, Hal L. Smith. The dynamics of bacterial infection, innate immune response, and antibiotic treatment. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 127-143. doi: 10.3934/dcdsb.2007.8.127
[4]

Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527
[5]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445
[6]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[7]

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
[8]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020042
[9]

Yuhua Zhu. A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method. Networks & Heterogeneous Media, 2019, 14 (4) : 677-707. doi: 10.3934/nhm.2019027
[10]

Tatiana Filippova. Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty. Conference Publications, 2011, 2011 (Special) : 410-419. doi: 10.3934/proc.2011.2011.410
[11]

Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355
[12]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[13]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[14]

Hedia Fgaier, Hermann J. Eberl. Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation. Conference Publications, 2009, 2009 (Special) : 230-239. doi: 10.3934/proc.2009.2009.230
[15]

Michael Dellnitz, Mirko Hessel-Von Molo, Adrian Ziessler. On the computation of attractors for delay differential equations. Journal of Computational Dynamics, 2016, 3 (1) : 93-112. doi: 10.3934/jcd.2016005
[16]

Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751
[17]

Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827
[18]

Jia Li. A malaria model with partial immunity in humans. Mathematical Biosciences & Engineering, 2008, 5 (4) : 789-801. doi: 10.3934/mbe.2008.5.789
[19]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[20]

Teresa Faria, José J. Oliveira. On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2451-2472. doi: 10.3934/dcdsb.2016055

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences