Fractional Model for Type 1 Diabetes

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


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.



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.


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© 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

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