We propose two novel mathematical models with impulsive injection of insulin or its analogues for type 1 and type 2 diabetes mellitus. One model incorporates the periodic impulsive injection of insulin. We analytically show the existence and uniqueness of a positive globally asymptotically stable periodic solution for type 1 diabetes, which implies that the perturbation caused by insulin injection will not disturb homeostasis but amend the glucose from a pathological level toward a healthier level. We also show that the system is uniformly permanent for type 2 diabetes, that is, the glucose concentration level is uniformly bounded above and below. The other model determines the insulin injection by closely monitoring the glucose level when it reaches or passes a predefined but adjustable threshold value. We analytically prove the existence and stability of the order one periodic solution, which ensures that the perturbation by the injection in such an automated way can keep the blood glucose concentration under control. Our numerical analysis confirms and further enhances the usefulness and robustness of our models. The first model has clinical implications in that the glucose level of a diabetic can be controlled within a desired level by adjusting the values of two model parameters, injection period and injection dose. A method for choosing these two parameters is described. The second model is, to the best of our knowledge, the first attempt to conquer some critical issues in the design of an artificial pancreas with closed-loop approach. For the former case, our numerical analysis reveals that a regime with a smaller insulin dose and more frequent deliveries is more efficient and effective, but surprisingly, for the latter case, a larger dose is more effective. This can be significant in the design and development of the insulin pump and artificial pancreas.

To study the impact of releasing sterile mosquitoes on mosquito-borne disease transmissions, we propose two mathematical models with impulsive releases of sterile mosquitoes. We consider periodic impulsive releases in the first model and obtain the existence, uniqueness, and globally stability of a wild-mosquito-eradication periodic solution. We also establish thresholds for the control of the wild mosquito population by selecting the release rate and the release period. In the second model, the impulsive releases are determined by the closely monitored wild mosquito density, or the state feedback. We prove the existence of an order one periodic solution and find a relatively small attraction region, which ensures the wild mosquito population is under control. We provide numerical analysis which shows that a smaller release rate and more frequent releases are more efficient in controlling the wild mosquito population for the periodic releases, but an early release of sterile mosquitoes is more effective for the state feedback releases.

We continue our attempt of modeling the open-loop control of glucose level by impulsive insulin injections. Several time delays exist in the system. The delays include the time needed for insulin from injection depot to transport to the interstitial compartment, the time for the slow inhibition of the hepatic glucose production (HGP), and the time for insulin secretion caused by the elevated glucose concentration level from remaining functional pancreatic $\beta$-cells. None of them are negligible. The model proposed in this paper incorporates these time delays. Our analytical studies show that all solutions are permanent, a periodic solution exists, and for the case of type 1 diabetes mellitus the periodic solution is unique and globally asymptotically stable. Numerically it has been shown that moderate time delays in the system are beneficial in lowering blood sugar level rather than harmful. In contrast, these time delays cause that the open-loop control takes a longer time to lower glucose concentration level. Our studies also elucidate the noticeable inhibitory effect on HGP by the remaining functional $\beta$-cells. Similarly to our previous work, we demonstrated that a smaller dose with higher delivery frequency has a better effect on continuous subcutaneous insulin injection administration. We expect that our findings are helpful for clinical therapies.