Deterministic and stochastic models for analyzing the dynamics of diabetes mellitus

Diabetes mellitus has become a global health threat as well as a financial burden. According to the International Diabetes Federation (IDF) Atlas (10th edition, 2021), approximately 537 million adults live with diabetes globally, which is anticipated to rise to 643 million in 2030 and 783 million by 2045. The report shows 6.7 million deaths due to diabetes in 2021 (1 every 5 s) and health expenditure of at least 966 billion USD (316% increase over the past 15 years). This research focuses on mathematical modeling and analysis of diabetes mellitus using deterministic and stochastic models. The study is conducted without considering genetic factors. First, we construct a deterministic diabetes mellitus model and transform it into a stochastic model by incorporating Brownian motions and stochastic environmental factor intensities. We provide qualitative results for both models, including the positivity of the solution, equilibrium points, basic reproduction numbers, local stability results, and sensitivity analysis. We show that the disease-free equilibrium is locally asymptotically stable via the Routh–Hurwitz criterion. Again, the sensitivity analysis result indicates that the transmission and birth parameters at a given period have a significant role in the increase of diabetes mellitus in the population if their values increase. We further establish the existence and uniqueness of the global positive solution by employing the random Lyapunov function theory. Using the Milstein method, the numerical scheme for the stochastic model is presented, and the approximate solution using the scheme is discussed. Additionally, we simulate the dynamics of the deterministic model using the Euler–Maruyama method. The simulation results indicate that by prioritizing policies aimed at minimizing exposure to diabetes mellitus, the strain on healthcare systems can be alleviated, leading to reduced hospitalization rates and enhanced quality of life for individuals.