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–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–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—that is, a condition wherein the pancreatic $\beta$ -cell population holds and there are no antigen cells labeled by the activated macrophages.

Blood glucose concentration shows the quality and health of an individual. High glucose levels in the blood flow indicate atypical behavior of the human system. Currently, there is an acceptable test method to estimate glucose level; it consists in taking out a blood sample by the finger with a pin prick. However, the blood sample test has some disadvantages because it can produce some non-desired issues, such as infections, especially a typical case in people living with diabetes. Therefore, this work is motivated by applying reflective Photoplethysmography (PPG), a non-invasive technique based on a light source and a photodetector near the human body's skin. In this case, we propose two different blood glucose sensors using the infrared diodes LTE-302 and COM-09349 with a photodetector FDS-100. It is worth noting that both designs have multiple development stages, and all of them are self-designed and validated through various test samples with three healthy volunteers, one woman and two men. The results present an error of about 14 % for the first sensor, while for the second sensor is about 7.9 %. Moreover, the sensor measurements are compared against a commercial glucometer, the One Touch Select Plus Flex, and the specifications of the NORM ISO 15197:2013 standard. Finally, the reflective PPG technique could be applied as a non-invasive method to estimate the blood glucose level in the human body.

In this work, a Thau observer is designed based on a nonlinear third-order mathematical model described by ODEs, which captures the dynamics among insulin levels, $\beta$-cells, and glucose concentration. The novelty of this research lies in its interdisciplinary approach to understanding a complex biological system. The observer’s mathematical validation is established using the Localization of Compact Invariant Sets to determine the domain of attraction and global knowledge about the system’s dynamic bounds. These bounds are used to compute the Lipschitz constant and the elements of the free gain matrix that satisfy the constraints for designing a Thau observer, such as the stability matrix and asymptotic stability. This analysis provides insights into how insulin levels evolve over time at various glucose concentrations, an essential step toward hardware implementation due to the system’s chaotic behavior. It also establishes a mathematical background that could contribute to treatment planning in future Digital Twins studies. Numerical simulations demonstrate that the observer can accurately track the dynamic behavior of the Diabetes Mellitus model analyzed in this work through in silico methods.

En este trabajo de investigación, se presenta el desarrollo de una interfaz gráfica en Python para la simulación del modelo mínimo de Bergman, el cual describe la interacción entre la glucosa y la insulina en pacientes con diabetes tipo 1. El modelo fue implementado en una interfaz que permite la simulación de diferentes escenarios de la enfermedad, facilitando el análisis de cómo diversos parámetros influyen en las dinámicas glucosa-insulina. La validación del modelo se llevó a cabo mediante comparaciones con estudios previos, confirmando su precisión en la representación de estas interacciones. Como resultado, se logró crear una herramienta que facilita la exploración y el estudio de las dinámicas de glucosa e insulina en diferentes condiciones, proporcionando una base para futuras investigaciones en el control de la glucosa en diabetes tipo 1.