Logistic Tumor Growth with Delay and Impulsive Treatment

Stability and zero-Hopf bifurcation analysis of a tumour and T-helper cells interaction model in the case of HIV infection

“…The difference between the study in [29] and the present paper is that the model we examined here involves two different discrete delays, and its stability and bifurcation analysis is given for the full model, i.e., without reducing the dimension of the model proposed. In addition, the difference between the study in [12] and this work is that the helper T-cells that we consider are exactly the effector helper T-cells, not general ones. HIV attacks the helper T-cells but then each helper T-cell is target to the virus.…”

“…Therefore, it is important to investigate the dynamics of the immune system in this situation. A few of such studies can be found in [5,12,21,22,29].…”

“…To do this, we use a system of delay differential equations where the time lag describes the time needed by helper T-cells to find (or it can be said 'recognize') tumour cells and virus. Inspired by the studies in [12,29], we present a mathematical model involving four populations: tumour cells, uninfected effector helper T-cells, infected helper T-cells and free virus. The models presented in this manuscript and also [12,29] are improved from the model introduced in the papers by Lou et al [21,22].…”

“…Inspired by the studies in [12,29], we present a mathematical model involving four populations: tumour cells, uninfected effector helper T-cells, infected helper T-cells and free virus. The models presented in this manuscript and also [12,29] are improved from the model introduced in the papers by Lou et al [21,22]. In [22], Lou et al presented a model of cell-to-cell spread of HIV together with tumour in tissue cultures (in vitro).…”

“…In [5], Bodnar et al proposed a model to describe the HIV related tumour-immune system interactions in vitro. Moreover, in [12], Forys and Poleszczuk considered a similar model. However, they considered the issue of immune reaction against tumour and the second way for HIV to disseminate in vivo (circulating free viral particles to T-cells directly).…”