Advertisement

Fractional Model for Type 1 Diabetes

  • Ana R. M. Carvalho
  • Carla M. A. PintoEmail author
  • João M. de Carvalho
Chapter
  • 43 Downloads
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 30)

Abstract

Type 1 diabetes (T1D) is an autoimmune disease characterized by the destruction of β-cells, which are responsible for the production of insulin. T1D develops from an abnormal immune response, where specific clones of cytotoxic T-cells invade the pancreatic islets of Langerhans. Other immune cells, such as macrophages and dendritic cells, are also involved in the onset of T1D. In this paper, we generalize an integer-order model for T1D to include a non-integer order (also known as, fractional order (FO)) derivative. We study the local and the global stabilities of the disease-free equilibrium. Then, we discuss the results of the simulations of the FO model and investigate the role of macrophages from non-obese diabetic (NOD) mice and from control (Balb/c) mice in triggering autoimmune T1D. We observe that, for a value of the order of the fractional derivative equal to 1 (α = 1), an apoptotic wave can trigger T1D in NOD but not in Balb/c mice. The apoptotic wave is cleared efficiently in Balb/c mice preventing the onset of T1D. For smaller values of α, the inflammation persists for NOD and control mice. This alludes to a specific role of the order of the fractional derivative α in disease progression.

Notes

Acknowledgements

The authors were partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/ UI0144/2013. The research of AC was partially supported by a FCT grant with reference SFRH/BD/96816/2013.

References

  1. 1.
  2. 2.
    A.R.M. Carvalho, C.M.A. Pinto, Within-host and synaptic transmissions: contributions to the spread of HIV infection. Math. Methods Appl. Sci. 40, 1231–1264 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    C. Castillo-Chavez, Z. Feng, W. Huang, On the computation of RO and its role in global stability, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An 388, ed. by S. Tennenbaum, T.G. Kassem, S. Roudenko, C. Castillo-ChavezGoogle Scholar
  4. 4.
    M. Goharimanesh, A. Lashkaripour, A.A. Mehrizi, Fractional order PID controller for diabetes patients. J. Comput. Appl. Mech. 46(1), 69–76 (2015)Google Scholar
  5. 5.
    W. Lin, Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl. 332, 709–726 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    G. Magombedze, P. Nduru, C.P. Bhunu, S. Mushayabasa, Mathematical modelling of immune regulation of type 1 diabetes. BioSystems 102, 88–98 (2010)CrossRefGoogle Scholar
  7. 7.
    J.M. Mahaffy, L. Edelstein-Keshet, Modeling cyclic waves of circulating T cells in autoimmune diabetes. SIAM J. Appl. Math. 67(4), 915–937 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Matignon, Stability results for fractional differential equations with applications to control processing, in Computational Engineering in Systems Applications, vol. 2 (Lille, France, 1996), p. 963Google Scholar
  9. 9.
    A.F.M. Maree, R. Kublik, D.T. Finegood, L. Edelstein-Keshet, Modelling the onset of type 1 diabetes: can impaired macrophage phagocytosis make the difference between health and disease? Philos. Trans. R. Soc. Lond. A 364, 1267–1282 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    T. Marinkovic, M. Sysi-Aho, M. Oresic, Integrated model of metabolism and autoimmune response in β-cell death and progression to type 1 diabetes. PLoS One 7(12), e51909 (2012)Google Scholar
  11. 11.
    K.H.M. Nielsen, F.M. Pociot, J.T. Ottesen, Bifurcation analysis of an existing mathematical model reveals novel treatment strategies and suggests potential cure for type 1 diabetes. Math. Med. Biol. 31(3), 205–225 (2014). https://doi.org/0.1093/imammb/dqt006 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Z.M. Odibat, N.T. Shawagfeh, Generalized Taylor’s formula. Appl. Math. Comput. 186, 286–293 (2007)MathSciNetzbMATHGoogle Scholar
  13. 13.
    K. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order (Academic Press, New York, 1974)zbMATHGoogle Scholar
  14. 14.
    C.M.A. Pinto, A.R.M. Carvalho, The role of synaptic transmission in a HIV model with memory. Appl. Math. Comput. 292, 76–95 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    C.M.A. Pinto, A.R.M. Carvalho, Persistence of low levels of plasma viremia and of the latent reservoir in patients under ART: a fractional-order approach. Commun. Nonlinear Sci. Numer. Simul. 43, 251–260 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    S. Sakulrang, E.J. Moore, S. Sungnul, A. Gaetano, A fractional differential equation model for continuous glucose monitoring data. Adv. Difference Equ. (2017), 150 (2017). https://doi.org/10.1186/s13662-017-1207-1
  17. 17.
    S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, London, 1993)zbMATHGoogle Scholar
  18. 18.
    Sociedade Portuguesa de Diabetologia: Diabetes: Factos e Números—O Ano de 2015—Relatório Anual do Observatório Nacional da Diabetes (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Ana R. M. Carvalho
    • 1
  • Carla M. A. Pinto
    • 2
    • 3
    Email author
  • João M. de Carvalho
    • 1
  1. 1.Faculty of SciencesUniversity of PortoPortoPortugal
  2. 2.School of EngineeringPolytechnic of PortoPortoPortugal
  3. 3.Center for Mathematics of the University of PortoPortoPortugal

Personalised recommendations