Bifurcation analysis of a diffuse large b-cell lymphoma growth model in germinal center

In this paper, a new mathematical model of diffuse large B-cell lymphoma (DLBCL) in the germinal center and its microenvironment has been considered. The model is a five-dimensional system of first-order nonlinear ordinary differential equations that consists of interactions between centroblasts, centrocyts, plasmablasts, DLBCL cells, and effector cells. Our analysis focuses on understanding the long-term behavior of the DLBCL from a mathematical perspective. The cycle characteristics of DLBCL growth that can be used to detect the duration of the dormant states of the cancer cells and to choose the treatment methods are important to study. By using codimension-one and codimension-two bifurcations, we found Hopf bifurcations that show the appearance of the cycle and some bifurcations of the periodic solutions that are able to be used to characterize the cycle of the disease. In our case, by varying the carrying capacity parameter and the decay rate of effector cells due to the competition with DLBCL, the system undergoes a Hopf bifurcation and then is followed by a generalized Hopf bifurcation, a limit point bifurcation, and a branch point bifurcation. The occurrence of these bifurcations is crucial for understanding the role of effector cells in the regulation of the DLBCL cycle. Furthermore, the appearance of chaotic solutions reflects the irregularity of the system due to changes in initial conditions, highlighting potential uncertainty in the progression of DLBCL metastasis.