Nonlinear Analysis for a Type-1 Diabetes Model with Focus on T-Cells and Pancreatic β-Cells Behavior
| Autor: | Carlos E. Vázquez, Diana Gamboa, Paul J. Campos |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
endocrine system diseases
Cell Population Biology behavioral disciplines and activities lcsh:QA75.5-76.95 Immune system Antigen immune system diseases medicine compact invariant sets Invariant (mathematics) education Autoimmune disease education.field_of_study Innate immune system lcsh:T57-57.97 lcsh:Mathematics Applied Mathematics General Engineering nutritional and metabolic diseases lcsh:QA1-939 medicine.disease Cell biology Computational Mathematics TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES medicine.anatomical_structure Bounded function lcsh:Applied mathematics. Quantitative methods diabetes mellitus lcsh:Electronic computers. Computer science nonlinear control |
| Zdroj: | Mathematical and Computational Applications Volume 25 Issue 2 Mathematical and Computational Applications, Vol 25, Iss 23, p 23 (2020) |
| ISSN: | 2297-8747 |
| DOI: | 10.3390/mca25020023 |
| Popis: | Type-1 diabetes mellitus (T1DM) is an autoimmune disease that has an impact on mortality due to the destruction of insulin-producing pancreatic &beta cells in the islets of Langerhans. Over the past few years, the interest in analyzing this type of disease, either in a biological or mathematical sense, has relied on the search for a treatment that guarantees full control of glucose levels. Mathematical models inspired by natural phenomena, are proposed under the prey&ndash predator scheme. T1DM fits in this scheme due to the complicated relationship between pancreatic &beta cell population growth and leukocyte population growth via the immune response. In this scenario, &beta cells represent the prey, and leukocytes the predator. This paper studies the global dynamics of T1DM reported by Magombedze et al. in 2010. This model describes the interaction of resting macrophages, activated macrophages, antigen cells, autolytic T-cells, and &beta cells. Therefore, the localization of compact invariant sets is applied to provide a bounded positive invariant domain in which one can ensure that once the dynamics of the T1DM enter into this domain, they will remain bounded with a maximum and minimum value. Furthermore, we analyzed this model in a closed-loop scenario based on nonlinear control theory, and proposed bases for possible control inputs, complementing the model with them. These entries are based on the existing relationship between cell&ndash cell interaction and the role that they play in the unchaining of a diabetic condition. The closed-loop analysis aims to give a deeper understanding of the impact of autolytic T-cells and the nature of the &beta cell population interaction with the innate immune system response. This analysis strengthens the proposal, providing a system free of this illness&mdash that is, a condition wherein the pancreatic &beta cell population holds and there are no antigen cells labeled by the activated macrophages. |
| Databáze: | OpenAIRE |
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