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

Abstract

In this paper, we present a mathematical model governing the dynamics of tumourimmune cells interaction under HIV infection. The interactions between tumour cells, helper T-cells, infected helper T-cells and virus cells are explained by using delay differential equations including two different discrete time delays. In the model, these time lags describe the time needed by the helper T-cells to find (or recognize) tumour cells and virus, respectively. First, we analyze the dynamics of the model without delays. We prove the positivity of the solution, analyze the local and global stabilities of the steady states of the model. Second, we study the effects of two discrete time delays on the stability of the endemically infected equilibrium point. We determine the conditions on parameters at which the system undergoes a zero-Hopf bifurcation. Choosing one of the delay terms as a bifurcation parameter and fixing the other, we show that a zero-Hopf bifurcation arises as the bifurcation parameter passes through a critical value. Finally, we perform numerical simulations to support and extend our theoretical results. The results concluded help to better understand the links between the immune system and the tumour development in the case of HIV infection.