Hopf and backward bifurcations induced by immune effectors in a cancer oncolytic virotherapy dynamics

Oncolytic virotherapy is emerging as a promising new method for cancer treatment. In this paper, we consider a mathematical model for treatment of cancer by using oncolytic virotherapy in the presence of immune effectors. We provide a theoretical analysis of the model and derive the basic reproduction number $$\mathcal {R}_0$$ which determines the extinction and the presence of oncolytic viruses during therapy. The existence of equilibria and their stability are investigated. More precisely, we show that, depending of the values of the parameters, there exits a quantity $$\varepsilon $$ so that, when $$\varepsilon <1$$ , the tumor can be eliminated in the body. However, we also show that, if $$\mathcal {R}_0<1$$ and $$\varepsilon >1$$ , the infection-free equilibrium is stable and the model is shown to exhibit the phenomenon of backward bifurcation (where a stable infection-free equilibrium coexist with one or more stable endemic equilibria when the associated basic reproduction ratio is less than unity). Furthermore the model presents a Hopf bifurcation which is supercritical, from which birth of oscillation occurs. Numerical simulations support our theoretical results.