Stability and Hopf bifurcation analysis for the hypothalamic-pituitary-adrenal axis model with memory.

This article generalizes the existing minimal model of the hypothalamic-pituitary-adrenal (HPA) axis in a realistic way, by including memory terms: distributed time delays, on one hand and fractional-order derivatives, on the other hand. The existence of a unique equilibrium point of the mathematical models is proved and a local stability analysis is undertaken for the system with general distributed delays. A thorough bifurcation analysis for the distributed delay model with several types of delay kernels is provided. Numerical simulations are carried out for the distributed delays models and for the fractional-order model with discrete delays, which substantiate the theoretical findings. It is shown that these models are able to capture the vital mechanisms of the HPA system.

Deterministic and stochastic model for the role of the immune response time delay in periodic therapy of the tumors.

We consider the deterministic model of interaction between the immune system and tumor cells including a memory function that reflects the influence of the past states, to simulate the time needed by the latter to develop a chemical and cell mediated response to the presence of the tumor. The memory function is called delay kernel. The results are compared with those from other papers, concluding that the memory function introduces new instabilities in the system leading to an uncontrollable growth of the tumor. If the coefficient of the memory function is used as a bifurcation parameter, it is found that Hopf bifurcation occurs for kernel. The direction and stability of the bifurcating periodic solutions are determined. The deterministic model with delay allows stochastic perturbation. Mean value and square mean value of the linearized model are analyzed for the variables of the stochastic model. Some numerical simulations for justifying the theoretical analysis are also given.

The p53 network modeling - current state and future prospects.

p53 gene is a central hub in a network that have the role to protect the cells against carcinogenesis. A lot of research work has been done in this field, a huge amount of data was collected and many experiments have been performed. For a better understanding of the processes and for the help of experiments design some mathematical models were proposed in the literature. In this paper we make a brief overview of these models, pointing out their strong points and weak points. Some improvements of the models were also presented with some graphical representations of numerical simulation. Finally some conclusions were made regarding future prospects of p53 network modeling.