Torus and Homoclinic Bifurcations on a Cells Repair Regulations Model of the Metastatic Nasopharyngeal Carcinoma

Nasopharyngeal carcinoma (NPC) is a tumor that grows on nasopharyngeal epithelial cells. There are some proteins that play important roles in the regulation of cell repair
for metastatic NPC, i.e., ATM, p53, MDM2, and DSB. In this paper, we construct a mathematical model that shows the interactions between such proteins and study the behavior of NPC in a long time period. The model is a four-dimensional system of first-order ODE that has a 16-dimensional parameter space. In this case, we consider the appearance of attracting patterns of the solutions near the steady-state conditions. We use the codimension 1 and codimension 2 bifurcations analysis to study the role of several important parameters for the metastasis of NPC and study the possibilities of the system to have chaotic solutions. The appearance of a chaotic solution shows the irregularity of the system due to the changes of the initial conditions. It is important to understand the metastasis behavior NPC and to determine the treatment strategies.