Evolutionary approaches have a critical role in different disciplines such as real-world problems, computer programming, machine learning, biological sciences, and many more. The design of the stochastic model is based on transition probabilities and non-parametric techniques. Positivity, boundedness, and equilibria are investigated in deterministic and stochastic senses. An essential tool, Euler–Maruyama, is studied for the solution of said model. Standard and nonstandard evolutionary approaches are presented for the stochastic model in terms of efficiency and low-cost approximations. The standard evolutionary procedures like stochastic Euler–Maruyama and stochastic Runge–Kutta fail to restore the essential features of biological problems. On the other hand, the proposed method is efficient, of meager cost, and adopts all the desired feasible properties. At the end of this paper the comparison section is presented to support efficient analysis.

This work introduces a new mathematical model designed to describe the glucose–insulin dynamics associated with a glucose-responsive smart microneedle patch reported in the literature. The model captures the complete sequence of the patch behavior, from detecting glucose changes to controlled transdermal insulin delivery and gradually restoring blood glucose levels to the normal range. Our simulations show that the patch can effectively manage glucose not only during fasting conditions but also after single and multiple meals, restoring glucose levels to healthy levels within a short period. The model predictions are consistent with experimentally reported trends in previously published studies, which strengthens confidence in the biological realism of the proposed mechanism. Because some parameters in such systems are difficult to measure directly, we also performed a comprehensive sensitivity analysis to understand how variations in key parameters influence system stability. The results highlight the central role of the insulin release rate and the five glucose–regulation parameters examined in the sensitivity analysis, providing clear guidance on the most critical aspects of patch design for reliable performance. Overall, this study provides a simplified yet robust mathematical framework that makes the behavior of a glucose-responsive microneedle patch easy to understand and analyze. It lays the groundwork for future refinement of control strategies and optimization of patch design, improving control strategies, and developing more advanced systems that can maintain healthy glucose levels naturally and intuitively.

Type 1 diabetes (T1D) is a chronic autoimmune disease caused by the immune-mediated reduction of β cells, resulting in lifelong dependence on exogenous insulin administration. We studied the impact of M1/M2 macrophages on the β cell level. We obtained the healthy state (absence of T1D) and unhealthy state (presence of T1D). We found that in the unhealthy state, β cell levels are decreased with a continuous alternative switch between M1 and M2 macrophages. However, in the healthy state, the β cell levels increase with a high level of M2 macrophages.