A Mathematical Analysis of Diabetes Progression Using Differential Equation Models
DOI:
https://doi.org/10.63665/swz5hc07Keywords:
Differential equations; diabetes progression; Bergman minimal model; insulin sensitivity index; compartmental epidemiological modellingAbstract
Diabetes mellitus is among the fastest-growing chronic metabolic disorders globally, with 589 million adults affected as of 2024. This paper presents a systematic mathematical analysis of diabetes progression employing ordinary differential equations (ODEs) and compartmental modelling frameworks. The primary objective is to simulate glucose-insulin dynamics and disease transition states using the Bergman Minimal Model alongside a population-level compartmental model calibrated to International Diabetes Federation (IDF) and ICMR-INDIAB data. The methodology integrates parametric estimation against real-world clinical and epidemiological datasets sourced from peer-reviewed literature through 2024. The hypothesis posits that ODE-based frameworks can accurately replicate disease progression stages from normoglycaemia through prediabetes to type 2 diabetes with complications and identify mathematically precise intervention thresholds. Results demonstrate that the basic reproduction number (R₀) governs stability of disease-free and endemic equilibria, while insulin sensitivity index (Sᵢ) is the primary physiological predictor. Sensitivity analysis confirms that treatment recovery rates exert the strongest inverse influence on diabetes prevalence. Findings are contextualised within the Indian epidemiological landscape, where over 101 million individuals are currently diabetic. The conclusions affirm ODE models as indispensable predictive instruments for evidence-based national diabetes policy.
Downloads
References
1. Genitsaridi, I., Salpea, P., Salim, A., Sajjadi, S. F., Tomic, D., James, S., & Magliano, D. J. (2026). 11th edition of the IDF Diabetes Atlas: global, regional, and national diabetes prevalence estimates for 2024 and projections for 2050. The Lancet Diabetes & Endocrinology, 14(2), 149–156. https://doi.org/10.1016/S2213-8587(25)00299-2
2. Sun, H., Saeedi, P., Karuranga, S., Pinkepank, M., Ogurtsova, K., Duncan, B. B., Stein, C., Basit, A., Chan, J. C. N., Mbanya, J. C., & Magliano, D. J. (2023). IDF Diabetes Atlas: Global, regional and country-level diabetes prevalence estimates for 2021 and projections for 2045. Diabetes Research and Clinical Practice, 204, 110945. https://doi.org/10.1016/j.diabres.2023.110945
3. Anjana, R. M., Pradeepa, R., Das, A. K., Deepa, M., Bhansali, A., Joshi, S. R., Joshi, P. P., Dhandhania, V. K., Rao, P. V., Sudha, V., Unnikrishnan, R., Madhu, S. V., Kaur, T., Priya, M., Nirmal, E., Subashini, R., Venkatesan, U., Ranjit Mohan, A., & Mohan, V. (2023). Metabolic non-communicable disease health report of India: The ICMR-INDIAB national cross-sectional study (ICMR-INDIAB-17). The Lancet Diabetes & Endocrinology, 11(7), 474–489. https://doi.org/10.1016/S2213-8587(23)00119-5
4. Kumar, A., Gangwar, R., Zargar, A. A., Kumar, R., & Sharma, A. (2024). Prevalence of diabetes in India: A review of IDF Diabetes Atlas 10th Edition. Current Diabetes Reviews, 20(1), e130423215752. https://doi.org/10.2174/1573399819666230413094529
5. AlShurbaji, M., Abdul Kader, L., & Hannan, H. (2023). Comprehensive study of a diabetes mellitus mathematical model using numerical methods with stability and parametric analysis. International Journal of Environmental Research and Public Health, 20(2), 939. https://doi.org/10.3390/ijerph20020939
6. Ahmad, A., El-Ameen, B. B. M., & Raza, N. (2024). On a fractional-order mathematical model to assess the impact of diabetes and its associated complications in the United Arab Emirates. Mathematical Methods in the Applied Sciences. https://doi.org/10.1002/mma.9947
7. Althobaiti, N., Helmi, M. M., Malik, K., & Althobaiti, S. (2024). Deterministic mathematical model with Holling type II treatment function for diabetes mellitus. AIP Advances, 14(5), 055022. https://doi.org/10.1063/5.0206379
8. Bergman, R. N. (2021). Origins and history of the minimal model of glucose regulation. Frontiers in Endocrinology, 11, 583016. https://doi.org/10.3389/fendo.2020.583016
9. De Gaetano, A., & Arino, O. (2000). Mathematical modelling of the intravenous glucose tolerance test. Journal of Mathematical Biology, 40(2), 136–168. https://doi.org/10.1007/s002850050007
10. Makroglou, A., Li, J., & Kuang, Y. (2006). Mathematical models and software tools for the glucose-insulin regulatory system and diabetes: An overview. Applied Numerical Mathematics, 56(3–4), 559–573. https://doi.org/10.1016/j.apnum.2005.04.023
11. Topp, B., Promislow, K., de Vries, G., Miura, R. M., & Finegood, D. T. (2000). A model of beta-cell mass, insulin, and glucose kinetics: pathways to diabetes. Journal of Theoretical Biology, 206(4), 605–619. https://doi.org/10.1006/jtbi.2000.2150
12. Boutayeb, A., & Chetouani, A. (2006). A critical review of mathematical models and data used in diabetology. BioMedical Engineering OnLine, 5, 43. https://doi.org/10.1186/1475-925X-5-43
13. World Health Organization. (2023). Global report on diabetes. WHO Press. https://www.who.int/publications/i/item/9789241547734
14. Rihan, F. A., Sheek-Hussein, M., & Al-Mdallal, Q. M. (2024). Time delayed fractional diabetes mellitus model and consistent numerical algorithm. Scientific Reports, 14, 23988. https://doi.org/10.1038/s41598-024-74767-w
15. Yang, B., Zhao, J., & Bhatt, D. L. (2023). Modeling the progression of Type 2 diabetes with underlying obesity. PLOS Computational Biology, 19(8), e1010914. https://doi.org/10.1371/journal.pcbi.1010914
16. De Gaetano, A., Hardy, T., Beck, B., El-Aoufy, A., van Kammen, D. P., & Batzel, J. J. (2024). A simplified longitudinal model for the development of Type 2 Diabetes Mellitus. Journal of Theoretical Biology, 111822. https://doi.org/10.1016/j.jtbi.2024.111822
17. Bergman, R. N., Ider, Y. Z., Bowden, C. R., & Cobelli, C. (1979). Quantitative estimation of insulin sensitivity. American Journal of Physiology, 236(6), E667–E677. https://doi.org/10.1152/ajpendo.1979.236.6.E667
18. Boutayeb, A., Twizell, E. H., Achouayb, K., & Chetouani, A. (2004). A mathematical model for the burden of diabetes and its complications. BioMedical Engineering OnLine, 3, 20. https://doi.org/10.1186/1475-925X-3-20
19. Kouidere, A., Balatif, O., Ferjouchia, H., Boutayeb, A., & Rachik, M. (2021). Optimal control strategy for a discrete time to the dynamics of a population of diabetics with highlighting the impact of living environment. Discrete Dynamics in Nature and Society, 2021, 6638615. https://doi.org/10.1155/2021/6638615
