Analysis of a stochastic IVGTT glucose-insulin model with time delay.

Diabetes mellituse has been one of the major diseases in the world due to the high percentage of diabetics in the global population and the increasing growth rate of its onset. Identifying individual physiological characteristics, e.g., insulin sensitivity and glucose effectiveness and others, is extremely important in developing effective drugs and investigating genetic pathways causing the defects in these physiological responses. Intravenous glucose tolerance test (IVGTT) is such a protocol to determine an individual insulin sensitivity and glucose effectiveness indices. In this paper, we propose a stochastic delay differential equation model for the IVGTT protocol attempting to develop a method to increase the accuracy of parameter estimation. We first study the existence and uniqueness of the global positive solution and its asymptotic behavior of the stochastic path close to the steady state of the corresponding deterministic model. Then we develop a maximum likelihood estimation method to estimate the parameters involved in the proposed model. Our simulation studies numerically confirm our theoretical findings and demonstrate that the proposed model with estimated parameters can improve the fitness of clinical data.

A mathematical model with pulse effect for three populations of the giant panda and two kinds of bamboo.

A mathematical model for the relationship between the populations of giant pandas and two kinds of bamboo is established. We use the impulsive perturbations to take into account the effect of a sudden collapse of bamboo as a food source. We show that this system is uniformly bounded. Using the Floquet theory and comparison techniques of impulsive equations, we find conditions for the local and global stabilities of the giant panda-free periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent. The results provide a theoretical basis for giant panda habitat protection.