Hopf bifurcation without parameters in deterministic and stochastic modeling of cancer virotherapy, part I

In this research, we propose deterministic and stochastic models to explain the complexity of interactions in cancer virotherapy and outcomes of current preclinical and clinical trials of oncolytic viral treatments. In Part I, we analyze the deterministic model. The model incorporates both innate and adaptive immune responses which have opposite effects on the outcome of the therapy. According to relative immune clearance rates, the model can be reduced to two subsystems, one with only innate immunity and one with only adaptive immunity, which provide detailed dynamical properties for the full model. The full system shows many different asymptotic behaviors which correspond to outcomes of the therapy. It undergoes classical Hopf bifurcation when the infectivity constant passes through a particular value and, interestingly, it also undergoes Hopf bifurcation without parameters when the tumor cell number passes through some particular value. We conduct numerical simulations to demonstrate our analytical results and provide detailed medical interpretations.