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.

Immunotherapy is a targeted therapy that can be applied to cervical cancer patients to prevent DNA damage caused by human papillomavirus (HPV). The HPV infects normal cervical cells withing a specific cell age interval, i.e., between the $G_1$ to $S$ phase of the cell cycle. In this study, we developed a new mathematical model of age-dependent immunotherapy for cervical cancer. The model is a four-dimensional first-order partial differential equation with time- and age-independent variables. The cell population is divided into four sub-populations, i.e., susceptible cells, cells infected by HPV, precancerous cells, and cancer cells. The immunotherapy term has been added to precancerous cells since these cells can experience regression if appointed by proper treatments. The immunotherapy process is closely related to the rate of T-cell division. The treatment works in the same cell cycle that stimulates and inhibits the immune system. In our model, immunotherapy is represented as a periodic function with a small amplitude. It is based on the fluctuating interaction between T-cells and precancerous cells. We have found that there are two types of steady-state conditions, i.e., infection-free and endemic. The local and global stability of an infection-free steady-state has been analyzed based on basic reproduction numbers. We have solved the Riccati differential equation to show the existence of an endemic steady-state. The stability analysis of the endemic steady-state has been determined by using the perturbation approach and solving integral equations. Some numerical simulations are also presented in this paper to illustrate the behavior of the solutions.