Time-delayed model of autoimmune dynamics

Farzad Fatehi; Yuliya N. Kyrychko; Konstantin B. Blyuss; Farzad Fatehi; Yuliya N. Kyrychko; Konstantin B. Blyuss

doi:10.3934/mbe.2019279

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5613-5639. doi: 10.3934/mbe.2019279

Previous Article Next Article

Research article Special Issues

Time-delayed model of autoimmune dynamics

Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom

Received: 30 December 2018 Accepted: 30 May 2019 Published: 17 June 2019

Among various environmental factors associated with triggering or exacerbating autoimmune response, an important role is played by infections. A breakdown of immune tolerance as a byproduct of immune response against these infections is one of the major causes of autoimmune disease. In this paper we analyse the dynamics of immune response with particular emphasis on the role of time delays characterising the infection and the immune response, as well as on interactions between different types of T cells and cytokines that mediate their behaviour. Stability analysis of the model provides insights into how different model parameters affect the dynamics. Numerical stability analysis and simulations are performed to identify basins of attraction of different dynamical states, and to illustrate the behaviour of the model in different regimes.
- immune response,
- mathematical model,
- time delay,
- autoimmunity,
- multi-stability
Citation: Farzad Fatehi, Yuliya N. Kyrychko, Konstantin B. Blyuss. Time-delayed model of autoimmune dynamics[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5613-5639. doi: 10.3934/mbe.2019279

Related Papers:

Abstract

Among various environmental factors associated with triggering or exacerbating autoimmune response, an important role is played by infections. A breakdown of immune tolerance as a byproduct of immune response against these infections is one of the major causes of autoimmune disease. In this paper we analyse the dynamics of immune response with particular emphasis on the role of time delays characterising the infection and the immune response, as well as on interactions between different types of T cells and cytokines that mediate their behaviour. Stability analysis of the model provides insights into how different model parameters affect the dynamics. Numerical stability analysis and simulations are performed to identify basins of attraction of different dynamical states, and to illustrate the behaviour of the model in different regimes.

References

[1]	A. K. Abbas, A. H. H. Lichtman and S. Pillai, Cellular and Molecular Immunology, Elsevier Health Sciences, 2015.
[2]	D. Mason, A very high level of crossreactivity is an essential feature of the T-cell receptor, Immunol. Today, 19 (1998), 395–404.
[3]	E. M. Shevach, R. S. McHugh, C. A. Piccirillo, et al., Control of T-cell activation by CD4⁺CD25⁺ suppressor T cells, Immunol. Rev., 182 (2001), 58–67.
[4]	A. M. Thornton and E. M. Shevach, CD4⁺CD25⁺ immunoregulatory T cells suppress polyclonal T cell activation in vitro by inhibiting interleukin 2 production, J. Exp. Med., 188 (1998), 287–296.
[5]	A. Corthay, How do regulatory T cells work?, Scand. J. Immunol., 70 (2009), 326–336.
[6]	D. Buljevac, H. Z. Flach, W. C. J. Hop, et al., Prospective study on the relationship between infections and multiple sclerosis exacerbations, Brain, 125 (2002), 952–960.
[7]	D. Germolic, D. H. Kono, J. C. Pfau, et al., Animal models used to examine the role of environment in the development of autoimmune disease: findings from an NIEHS Expert Panel Workshop, J. Autoimmun., 39 (2012), 285–293.
[8]	M. P. Mallampalli, E. Davies, D. Wood, et al., Role of environment and sex differences in the development of autoimmune disease: a roundtable meeting report, J. Womens Health, 22 (2013), 578–586.
[9]	B. Krone and J. M. Grange, Multiple sclerosis: are protective immune mechanisms compromised by a complex infectious background?, Autoimmune Dis., 2011 (2010), 708750.
[10]	M. Ohashi, N. Orlova, C. Quink, et al., Cloning of the Epstein-Barr virus-related rhesus lymphocryptovirus as a bacterial artificial chromosome: a loss-of-function mutation of the rhBARF1 immune evasion gene, J. Virol., 85 (2011), 1330–1339.
[11]	D. Hober and P. Sauter, Pathogenesis of type 1 diabetes mellitus: interplay between enterovirus and host, Nat. Rev. Endocrinol., 6 (2010), 279–289.
[12]	S. E. Myers, L. Brewer, D. P. Shaw, et al., Prevalent human coxsackie B-5 virus infects porcine islet cells primarily using the coxsackie-adenovirus receptor, Xenotransplantation, 11 (2004), 536–546.
[13]	K. Döhner, K. Radtke, S. Schmidt, et al., Eclipse phase of herpes simplex virus type 1 infection: Efficient dynein-mediated capsid transport without the small capsid protein VP26, J. Virol., 80 (2006), 8211–8224.
[14]	U. Maurer, B. Sodeik and K. Gruenewald, Native 3D intermediates of membrane fusion in herpes simplex virus 1 entry, Proc. Natl. Acad. Sci. USA, 105 (2008), 10559–10564.
[15]	S. Manfredo Vieira, M. Hiltensperger, V. Kumar, et al., Translocation of a gut pathobiont drives autoimmunity in mice and humans, Science, 359 (2018), 1156–1161.
[16]	R. S. Fujinami, Can virus infections trigger autoimmune disease?, J. Autoimmun., 16 (2001), 229–234.
[17]	A. M. Ercolini and S. D. Miller, The role of infections in autoimmune disease, Clin. Exp. Immunol., 155 (2009), 1–15.
[18]	R. S. Fujinami, M. G. von Herrath, U. Christen, et al., Molecular mimicry, bystander activation, or viral persistence: infections and autoimmune disease, Clin. Microbiol. Rev., 19 (2006), 80–94.
[19]	R. S. Fujinami, M. B. Oldstone, Z. Wroblewska, et al., Molecular mimicry in virus infection: crossreaction of measles virus phosphoprotein or of herpes simplex virus protein with human intermediate filaments, Proc. Natl. Acad. Sci. USA, 80 (1983), 2346–2350.
[20]	M. G. von Herrath and M. B. A. Oldstone, Virus-induced autoimmune disease, Curr. Opin. Immunol., 8 (1996), 878–885.
[21]	C. Münz, J. D. Lünemann, M. T. Getts, et al., Antiviral immune responses: triggers of or triggered by autoimmunity?, Nat. Rev. Immunol., 9 (2009), 246.
[22]	S. Bonhoeffer, R. M. May, G. M. Shaw, et al., Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971–6976.
[23]	M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent viruses, Science-AAAS-Weekly Paper Edition, 272 (1996), 74–79.
[24]	A. S. Perelson, Viral kinetics and mathematical models, Am. J. Med., 107 (Suppl 2) (1999), 49–52.
[25]	A. S. Perelson, Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2 (2002), 28–36.
[26]	P. Baccam, C. Beauchemin, C. A. Macken, et al., Kinetics of influenza A infection in humans, J. Virol., 80 (2006), 7590–7599.
[27]	C. A. A. Beauchemin, J. J. McSharry, G. L. Drusano, et al., Modeling amantadine treatment of influenza A virus in vitro, J. Theor. Biol., 254 (2008), 439–451.
[28]	C. A. A. Beauchemin and A. Handel, A review of mathematical models of influenza A infections within a host or cell culture: lessons learned and challenges ahead, BMC Public Health, 11 (Suppl 1) (2011), S7.
[29]	A. S. Perelson, A. Neumann, M. Markowitz, et al., HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582–1586.
[30]	A. S. Perelson, P. Essunger, Y. Cao, et al., Decay characteristics of HIV-1 infected compartments during combination therapy, Nature, 387 (1997), 188–191.
[31]	M. A. Nowak, S. Bonhoeffer, A. M. Hill, et al., Viral dynamics in hepatitis b virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398–4402.
[32]	A. U. Neumann, N. P. Lam, H. Dahari, et al., Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy, Science, 282 (1998), 103–107.
[33]	A. U. Neumann, N. P. Lam, H. Dahari, et al., Differences in viral dynamics between genotypes 1 and 2 of hepatitis C virus, J. Infect. Dis., 182 (2000), 28–35.
[34]	R. M. Ribeiro, J. Layden-Almer, K. A. Powers, et al., Dynamics of alanine aminotransferase during hepatitis C virus treatment, Hepatology, 38 (2003), 509–517.
[35]	L. A. Segel, E. Jäger, D. Elias, et al., A quantitative model of autoimmune disease and T-cell vaccination: does more mean less?, Immunol. Today, 16 (1995), 80–84.
[36]	J. A. M. Borghans and R. J. De Boer, A minimal model for T-cell vaccination, Proc. R. Soc. Lond. B Biol. Sci., 259 (1995), 173–178.
[37]	J. A. M. Borghans, R. J. De Boer, E. Sercarz, et al., T cell vaccination in experimental autoimmune encephalomyelitis: a mathematical model, J. Immunol., 161 (1998), 1087–1093.
[38]	S. Iwami, Y. Takeuchi, Y. Miura, et al., Dynamical properties of autoimmune disease models: tolerance, flare-up, dormancy, J. Theor. Biol., 246 (2007), 646–659.
[39]	S. Iwami, Y. Takeuchi, K. Iwamoto, et al., A mathematical design of vector vaccine against autoimmune disease, J. Theor. Biol., 256 (2009), 382–392.
[40]	K. León, R. Perez, A. Lage, et al., Modelling T-cell-mediated suppression dependent on interactions in multicellular conjugates, J. Theor. Biol., 207 (2000), 231–254.
[41]	K. León, A. Lage and J. Carneiro, Tolerance and immunity in a mathematical model of T-cell mediated suppression, J. Theor. Biol., 225 (2003), 107–126.
[42]	K. León, J. Faro, A. Lage, et al., Inverse correlation between the incidences of autoimmune disease and infection predicted by a model of T cell mediated tolerance, J. Autoimmun., 22 (2004), 31–42.
[43]	J. Carneiro, T. Paixão, D. Milutinovic, et al., Immunological self-tolerance: lessons from mathematical modeling, J. Comput. Appl. Math., 184 (2005), 77–100.
[44]	D. Wodarz and V. A. A. Jansen, A dynamical perspective of CTL cross-priming and regulation: implications for cancer immunology, Immunol. Lett., 86 (2003), 213–227.
[45]	R. Root-Bernstein, Theories and modeling of autoimmunity, J. Theor. Biol., 375 (2015), 1–124.
[46]	J. D. Fontenot, M. A. Gavin and A. Y. Rudensky, Foxp3 programs the development and function of CD4⁺CD25⁺ regulatory T cells, Nat. Immunol., 4 (2003), 330–336.
[47]	S. Sakaguchi, Naturally arising CD4⁺ regulatory T cells for immunologic self-tolerance and negative control of immune responses, Annu. Rev. Immunol., 22 (2004), 531–562.
[48]	S. Z. Josefowicz, L. F. Lu and A. Y. Rudensky, Regulatory T cells: Mechanisms of differentiation and function, Annu. Rev. Immunol., 30 (2012), 531–564.
[49]	H. K. Alexander and L. M. Wahl, Self-tolerance and autoimmunity in a regulatory T cell model, Bull. Math. Biol., 73 (2011), 33–71.
[50]	N. J. Burroughs, M. Ferreira, B. M. P. M. Oliveira, et al., A transcritical bifurcation in an immune response model, J. Differ. Equ. Appl., 17 (2011), 1101–1106.
[51]	N. J. Burroughs, M. Ferreira, B. M. P. M. Oliveira, et al., Autoimmunity arising from bystander proliferation of T cells in an immune response model, Math. Comput. Model., 53 (2011), 1389–1393.
[52]	Z. Grossman and W. E. Paul, Adaptive cellular interactions in the immune system: the tunable activation threshold and the significance of subthreshold responses, Proc. Natl. Acad. Sci. USA, 89 (1992), 10365–10369.
[53]	Z. Grossman and A. Singer, Tuning of activation thresholds explains flexibility in the selection and development of T cells in the thymus, Proc. Natl. Acad. Sci. USA, 93 (1996), 14747–14752.
[54]	Z. Grossman and W. E. Paul, Self-tolerance: context dependent tuning of T cell antigen recognition, Semin. Immunol., 12 (2000), 197–203.
[55]	A. J. Noest, Designing lymphocyte functional structure for optimal signal detection: voilá, T cells, J. Theor. Biol., 207 (2000), 195–216.
[56]	D. A. Peterson, R. J. DiPaolo, O. Kanagawa, et al., Cutting edge: negative selection of immature thymocytes by a few peptide-MHC complexes: differential sensitivity of immature and mature T cells, J. Immunol., 162 (1999), 3117–3120.
[57]	L. B. Nicholson, A. C. Anderson and V. K. Kuchroo, Tuning T cell activation threshold and effector function with cross-reactive peptide ligands, Int. Immunol., 12 (2000), 205–213.
[58]	P. Wong, G. M. Barton, K. A. Forbush et al., Dynamic tuning of T cell reactivity by self-peptide-major histocompatibility complex ligands, J. Exp. Med., 193 (2001), 1179–1187.
[59]	A. D. Bitmansour, D. C. Douek, V. C. Maino, et al., Direct ex vivo analysis of human CD4⁺ memory T cell activation requirements at the single clonotype level, J. Immunol., 169 (2002), 1207–1218.
[60]	I. Stefanová, J. R. Dorfman and R. N. Germain, Self-recognition promotes the foreign antigen sensitivity of naive T lymphocytes, Nature, 420 (2002), 429–434.
[61]	G. Altan-Bonnet and R. N. Germain, Modeling T cell antigen discrimination based on feedback control of digital ERK responses, PLoS Biol., 3 (2005), e356.
[62]	H. A. van den Berg and D. A. Rand, Dynamics of T cell activation threshold tuning, J. Theor. Biol., 228 (2004), 397–416.
[63]	A. Scherer, A. Noest and R. J. de Boer, Activation-threshold tuning in an affinity model for the T-cell repertoire, Proc. R. Soc. Lond. B Biol. Sci., 271 (2004), 609–616.
[64]	A. R. McLean, Modelling T cell memory, J. Theor. Biol., 170 (1994), 63–74.
[65]	C. Utzny and N. J. Burroughs, Perturbation theory analysis of competition in a heterogeneous population, Physica D, 175 (2003), 109–126.
[66]	N. J. Burroughs, B. M. P. M. de Oliveira and A. A. Pinto, Regulatory T cell adjustment of quorum growth thresholds and the control of local immune responses, J. Theor. Biol., 241 (2006), 134–141.
[67]	N. J. Burroughs, B. M. P. M. Oliveira, A. A. Pinto, et al., Sensibility of the quorum growth thresholds controlling local immune responses, Math. Comput. Model., 47 (2008), 714–725.
[68]	A. L. DeFranco, R. M. Locksley and M. Robertson, Immunity: The immune response in infectious and inflammatory disease, New Science Press Ltd., 2007.
[69]	A. Toda and C. A. Piccirillo, Development and function of naturally occurring CD4⁺CD25⁺ regulatory T cells, J. Leukoc. Biol., 80 (2006), 458–470.
[70]	P. S. Kim, P. P. Lee and D. Levy, Modeling regulation mechanisms in the immune system, J. Theor. Biol., 246 (2007), 33–69.
[71]	B. M. P. M. Oliveira, R. Trinchet, M. V. O. Espinar, et al., Modelling the suppression of autoimmunity after pathogen infection, Math. Meth. Appl. Sci., 41 (2018), 8565–8570.
[72]	J. Tam, Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16 (1999), 29–37.
[73]	R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4⁺ T-cells, Math. Biosci., 165 (2000), 27–39.
[74]	P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73–94.
[75]	X. Zhou, X. Song and X. Shi, Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. Math. Comput., 199 (2008), 23–38.
[76]	A. J. Yates, M. Van Baalen and R. Antia, Virus replication strategies and the critical CTL numbers required for the control of infection, PLoS Comput. Biol., 7 (2011), e1002274.
[77]	G. J. M. Webster, S. Reignat, M. K. Maini, et al., Incubation phase of acute hepatitis B in man: dynamic of cellular immune mechanisms, Hepatology, 32 (2000), 1117–1124.
[78]	M. S. Ciupe, B. L. Bivort, D. M. Bortz, et al., Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1–27.
[79]	M. P. Davenport, R. M. Ribeiro and A. S. Perelson, Kinetics of virus-specific CD8⁺ T cells and the control of human immunodeficiency virus infection, J. Virol., 78 (2004), 10096–10103.
[80]	R. Thimme, J. Bukh, H. C. Spangenberg, et al., Viral and immunological determinants of hepatitis C virus clearance, persistence, and disease, Proc. Natl. Acad. Sci. USA, 99 (2002), 15661–15668.
[81]	K. B. Blyuss and L. B. Nicholson, The role of tunable activation thresholds in the dynamics of autoimmunity, J. Theor. Biol., 308 (2012), 45–55.
[82]	K. B. Blyuss and L. B. Nicholson, Understanding the roles of activation threshold and infections in the dynamics of autoimmune disease, J. Theor. Biol., 375 (2015), 13–20.
[83]	D. Ben Ezra and J. V. Forrester, Fundal white dots: the spectrum of a similar pathological process, Br. J. Ophthalmol., 79 (1995), 856–860.
[84]	T. F. Davies, D. C. Evered, B. Rees Smith, et al., Value of thyroid-stimulating-antibody determination in predicting the short-term thyrotoxic relapse in Graves' disease, Lancet, 309 (1997), 1181–1182.
[85]	A. Nylander and D. A. Hafler, Multiple sclerosis, J. Clin. Invest., 122 (2012), 1180–1188.
[86]	F. Fatehi, Y. N. Kyrychko, R. Molchanov, et al., Bifurcations and multi-stability in a model of cytokine-mediated autoimmunity, Int. J. Bif. Chaos, 29 (2019), 1950034.
[87]	F. Fatehi, Y. N. Kyrychko and K. B. Blyuss, Effects of viral and cytokine delays on dynamics of autoimmunity, Mathematics, 6 (2018), 66.
[88]	F. Fatehi, S. N. Kyrychko, A. Ross, et al., Stochastic effects in autoimmune dynamics, Front. Physiol., 9 (2018), 45.
[89]	D. A. Copland, M. S. Wertheim, W. J. Armitage, et al., The clinical time-course of Experimental Autoimmune Uveoretinitis using topical endoscopic fundal imaging with histologic and cellular infiltrate correlation, Invest. Ophthalmol. Vis. Sci., 49 (2008), 5458–5465.
[90]	J. Boldison, T. K. Khera, D. A. Copland, et al., A novel pathogenic RBP-3 peptide reveals epitope spreading in persistent experimental autoimmune uveoretinitis, Immunology, 146 (2015), 301–311.
[91]	P. Krishnapriya and M. Pitchaimani, Analysis of time delay in viral infection model with immune impairment, J. Appl. Math. Comput., 55 (2017), 421–453.
[92]	S. D. Wolf, B. N. Dittel, F. Hardardottir, et al., Experimental autoimmune encephalomyelitis induction in genetically B cell-deficient mice, J. Exp. Med., 184 (1996), 2271–2278.
[93]	H. J. Wu, I. I. Ivanov, J. Darce, et al., Gut-residing segmented filamentous bacteria drive autoimmune arthritis via T helper 17 cells, Immunity, 32 (2010), 815–827.
[94]	I. Baltcheva, L. Codarri, G. Pantaleo, et al., Lifelong dynamics of human CD4⁺CD25⁺ regulatory T cells: Insights from in vivo data and mathematical modeling, J. Theor. Biol., 266 (2010), 307–322.
[95]	J. Li, L. Zhang and Z. Wang, Two effective stability criteria for linear time-delay systems with complex coefficients, J. Syst. Sci. Complex., 24 (2011), 835–849.
[96]	B. Rahman, K. B. Blyuss and Y. N. Kyrychko, Dynamics of neural systems with discrete and distributed time delays, SIAM J. Appl. Dyn. Syst., 14 (2015), 2069–2095.
[97]	A. Skapenko, J. Leipe, P. E. Lipsky, et al., The role of the T cell in autoimmune inflammation, Arthritis Res. Ther., 7(Suppl 2) (2005), S4–S14.
[98]	R. Antia, V. V. Ganusov and R. Ahmed, The role of models in understanding CD8⁺ T-cell memory, Nat. Rev. Immunol., 5 (2005), 101–111.

Reader Comments

Your name:*

Email:*
© 2019 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)