Skip to main content
SpringerLink
Log in

Dynamics and Biological Thresholds

  • Chapter
  • First Online:
Dynamics, Games and Science I

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 1))

Abstract

Our main interest is to study the relevant biological thresholds that appear in epidemic and immunological dynamical models. We compute the thresholds, of the SIRI epidemic models, that determine the appearance of an epidemic disease. We compute the thresholds, of a Tregs immunological model, that determine the appearance of an immune response.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bailey, N.T.J.: The Elements of Stochastic Processes with Applications to the Natural Sciences. Wiley, New York (1964)

    MATH  Google Scholar 

  2. Boer, R.J., Hogeweg, P.: Immunological discrimination between self and non-self by precursor depletion and memory accumulation. J. Theor. Biol. 124, 343 (1987)

    Article  Google Scholar 

  3. Burroughs, N.J., Oliveira, B.M.P.M., Pinto, A.A.: Autoimmunity arising from bystander proliferation of T cells in an immune response model. Math. Comput. Model. (2010)

    Google Scholar 

  4. Burroughs, N.J., Oliveira, B.M.P.M., Pinto, A.A.: Regulatory T cell adjustment of quorum growth thresholds and the control of local immune responses. J. Theor. Biol. 241, 134–141 (2006)

    Article  Google Scholar 

  5. Burroughs, N.J., Oliveira, B.M.P.M., Pinto, A.A., Ferreira, M.: A transcritical bifurcation in an immune response model. J. Differ. Eqn. Appl. (to appear)

    Google Scholar 

  6. Burroughs, N.J., Oliveira, B.M.P.M., Pinto, A.A., Ferreira, M.: Immune responses Dynamics. Math. Comput. Model (2010)

    Google Scholar 

  7. Burroughs, N.J., Oliveira, B.M.P.M., Pinto, A.A., Sequeira, H.J.T.: Sensibility of the quorum growth thresholds controlling local immune responses. Math. Comput. Model. 47, 714–725 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dammer, S.M., Hinrichsen, H.: Spreading with immunization in high dimensions. J. Stat. Mech. (2004)

    Google Scholar 

  9. Furtado, Glaucia C., Curotto de Lafaille, Maria A., Kutchukhidze, Nino L., Juan, J.: IInterleukin 2 signaling is required for CD4 +  regulatory T cell function. J. Exp. Med. 6, 851–857 (2002)

    Google Scholar 

  10. Grassberger, P., Chaté, H., Rousseau, G.: Spreading in media with long-time memory. Phys. Rev. E 55, 2488–2495 (1997)

    Article  Google Scholar 

  11. Hsieh, C.-S., Liang, Y., Tyznik, A.J., Self, S.G., Liggitt, D., Rudensky, A.Y.: Recognition of the peripheral self by naturally arising CD25 +  CD4 +  T cell receptors. Immunity 21, 267–277 (2004)

    Article  Google Scholar 

  12. Joo, J., Lebowitz, J.L.: Pair approximation of the stochastic susceptible-infected-recovered-susceptible epidemic model on the hypercubic lattice. Phys. Rev. E 70, 036114 (2004)

    Article  MathSciNet  Google Scholar 

  13. Kryscio, R., Lefevre, C.: On the extinction of the S-I-S stochastic logistic epidemic. J. Appl. Prob. 26, 685–694 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Leon, K., Faro, J., Lage, A., Carneiro, J.: Inverse correlation between the incidences of autoimmune disease and infection predicted by a model of T cell mediated tolerance. J. Autoimmun. 31–42 (2004)

    Google Scholar 

  15. Leon, K., Lage, A., Carneiro, J.: Tolerance and immunity in a mathematical model of T-cell mediated suppression. J. Theor. Biol. 225, 107–126 (2003)

    Article  MathSciNet  Google Scholar 

  16. Leon, K., Perez, R., Lage, A., Carneiro, J.: Modelling T-cell-mediated suppression dependent on interactions in multicellular conjugates. J. Theor. Biol. 207, 231–254 (2000)

    Article  Google Scholar 

  17. Martins, J., Aguiar, M., Pinto, A.A., Stollenwerk, N.: On the series expansion of the spatial SIS evolution operator. J. Differ. Eqn. Appl. 1–13 (2010)

    Google Scholar 

  18. Martins, J., Pinto, A.A., Stollenwerk, N.: A scaling analysis in the SIRI epidemiological model. J. Biol. Dyn. 3(5), 479–496 (2009)

    Article  MathSciNet  Google Scholar 

  19. Martins, J., Pinto, A.A., Stollenwerk, N.: Stationarity in moment closure and quasi-stationarity of the SIS model. (submitted)

    Google Scholar 

  20. Maurus de la Rosa, S., Rutz, S., Dorninger, H., Scheffold, A.: Interleukin-2 is essential for CD4 + CD25 +  regulatory T cells function. Eur. J. Immunol. 34, 2480–2488 (2004)

    Google Scholar 

  21. Mudur, G.S.: Maths for movies, medicine & markets. The Telegraph Calcutta, India, 20/09/2010

    Google Scholar 

  22. Nåsell, I.: On the quasi-stationary distribution of the stochastic logistic epidemic. Math. Biosci. 156, 21–40 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nåsell, I.: The quasi-stationary distribution of the closed endemic SIS model. Adv. Appl. Prob. 28, 895–932 (1996)

    Article  MATH  Google Scholar 

  24. Nagata, S.: Fas ligand-induced apoptosis. Annu. Rev. Genet. 29–55 (1999)

    Google Scholar 

  25. Pinto, A.A., Aguiar, M., Martins, J., Stollenwerk, N.: Dynamics of Epidemiological Models. Acta Biotheor. 1–15 (2010)

    Google Scholar 

  26. Pinto, A.A., Burroughs, N.J., Ferreira, M., Oliveira, B.M.P.M.: Dynamics of Immunological Models. Acta Biotheor. (to appear)

    Google Scholar 

  27. Pinto, A.A., Martins, J., Stollenwerk, N.: The higher moments dynamic on SIS model. Numerical Analysis and Applied Mathematics. In: Simos, T.E., et al. (eds.) AIP Conference Proceedings, vol. 1168, pp. 1527–1530 (2009)

    Google Scholar 

  28. Rand, D.A.: Correlation equations and pair approximations for spatial ecologies. In: Jacqueline McGlade (ed.) Advanced Ecological Theory, pp. 100–142. Blackwell Science, Oxford, London, Edinburgh, Paris (1999)

    Google Scholar 

  29. Rogers, Paul R., Dubey, Caroline, Swain and Susan L.: Qualitative changes accompany memory t cell generation: faster, more effective responses at lower doses of antigen. J. Immunol. 164, 2338–2346 (2000)

    Google Scholar 

  30. Sakaguchi, S.: Naturally arising CD4 +  regulatory T cells for immunological self-tolerance and negative control of immune responses. Annu. Rev. Immunol. 22, 531–562 (2004)

    Article  Google Scholar 

  31. Shevach, E.M., McHugh, R.S., Piccirillo, C.A., Thornton, A.M.: Control of T-cell activation by CD4( + ) CD25( + ) suppressor T cells. Immunol. Rev. 182, 58–67 (2001)

    Article  Google Scholar 

  32. Schluns, K.S., Kieper, W.C., Jameson, S.C., Lefrancois, L.: Interleukin-7 mediates the homeostasis of naive and memory CD8 T cells. Nat. Immunol. 1, 426–432 (2000)

    Article  Google Scholar 

  33. Stollenwerk, N., Aguiar, M.: The SIRI stochastic model with creation and annihilation operators. Quant. Methods 1, 1–10 (2008)

    Google Scholar 

  34. Stollenwerk, N., Martins, J., Pinto, A.A.: The phase transition lines in pair approximation for the basic reinfection model SIRI. Phys. Lett. A 371, 379–388 (2007)

    Article  Google Scholar 

  35. Stollenwerk, N., van Noort, S., Martins, J., Aguiar, M., Hilker, F., Pinto, A.A., Gomes, G.: A spatially stochastic epidemic model with partial immunization shows in mean field approximation the reinfection threshold. J. Biol. Dyn. 1–17 (2010)

    Google Scholar 

  36. Tanchot, C., Vasseur, F., Pontoux, C., Garcia, C., Sarukhan, A.: Immune regulation by self-reactive T cells is antigen specific. J. Immunol. 172, 4285–4291 (2004)

    Google Scholar 

  37. Thornton, A.M., Donovan, E.E., Piccirillo, C.A., Shevach, E.M.: Cutting edge: IL-2 is critically required for the in vitro activation of CD4( + )CD25( + ) T cell suppressor function. J. Immunol. 172, 6519–6523 (2004)

    Google Scholar 

  38. Thornton, A.M., Shevach, E.M.: CD4 + CD25 +  immunoregulatory T cells suppress polyclonal T cell activation in vitro by inhibiting interleukine 2 production. J. Exp. Med. 188, 287–296 (1998)

    Article  Google Scholar 

  39. Thornton, A.M., Shevach, E.M.: Suppressor effector function of CD4 + CD25 +  immunoregulatory T cells is antigen nonspecific. J. Immunol. 164, 183–190 (2000)

    Google Scholar 

  40. van Kampen, N.G.: Stochastic processes in physics and chemistry. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

Download references

Acknowledgements

Previous versions of this work were presented in the International Congresses of Mathematicians ICM 2006 and 2010, European Conference on Mathematical and Theoretical Biology ECMTB 2008 and ICDEA 2010. This work was presented in the Encontro Ciência 2008, organized by Alexande Quintanilha, João Sentieiro, Luís Magalhães, Joaquim Cabral e Alberto Pinto. We thank LIAAD-INESC Porto LA, Calouste Gulbenkian Foundation, Programs FEDER, POCTI and POCI by FCT and Ministério da Ciência, Tecnologia e Ensino Superior and Centro de Matemática da Universidade do Minho and Centro de Matemática e Aplicações Fundamentais da Universidade de Lisboa for their financial support. José Martins also acknowledges the financial support from the FCT grant with reference SFRW/BD/37433/2007.14.5pc]Please update references “[5, 6, 8, 19, 26]”. Miguel Ferreira also acknowledges the financial support from the FCT grant with reference SFRH/BD/27706/2006.

Author information

Authors and Affiliations

  1. University of Warwick, Coventry, CV4 7AL, UK

    N. J. Burroughs

  2. Escola Superior de Estudos Industriais e de Gestão do Instituto Politécnico do Porto (IPP), LIAAD-INESC Porto LA, Escola de Ciências, Universidade do Minho, Braga, Portugal

    M. Ferreira

  3. LIAAD-INESC Porto LA, Porto, Portugal

    J. Martins

  4. Department of Mathematics, School of Technology and Management, Polytechnic Institute of Leiria, Campus 2, Morro do Lena – Alto do Vieiro, 2411-901, Leiria, Portugal

    J. Martins

  5. Faculdade de Ciências da Nutrição e Alimentação da, Universidade do Porto, LIAAD-INESC Porto LA, Porto, Portugal

    B. M. P. M. Oliveira

  6. LIAAD-INESC Porto LA e Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Porto, Portugal

    Alberto A. Pinto

  7. Centro de Matemática e Departamento de Matemática e Aplicações, Escola de Ciências, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Alberto A. Pinto

  8. Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Lisbon, Portugal

    N. Stollenwerk

Authors
  1. N. J. Burroughs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Martins
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. B. M. P. M. Oliveira
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alberto A. Pinto
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. N. Stollenwerk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Ferreira .

Editor information

Editors and Affiliations

  1. Pura e Aplicada (IMPA), Instituto de Matemática, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Rio de Janeiro, Brazil

    Mauricio Matos Peixoto

  2. Faculdade de Ciências, Departamento de Matemática, Universidade do Porto, Rua do Campo Alegre 687, Porto, 4169-007, Portugal

    Alberto Adrego Pinto

  3. Warwick Systems Biology, University of Warwick, Coventry House, Coventry, CV4 7AL, United Kingdom

    David A. Rand

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Burroughs, N.J., Ferreira, M., Martins, J., Oliveira, B.M.P.M., Pinto, A.A., Stollenwerk, N. (2011). Dynamics and Biological Thresholds. In: Peixoto, M., Pinto, A., Rand, D. (eds) Dynamics, Games and Science I. Springer Proceedings in Mathematics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11456-4_12

Download citation

Publish with us

Policies and ethics

Search

Navigation