Finding Hopf bifurcation islands and identifying thresholds for success or failure in oncolytic viral therapy

We model interactions between cancer cells and viruses during oncolytic viral therapy. One of our primary goals is to identify parameter regions that yield treatment failure or success. We show that the tumor size under therapy at a particular time is less than the size without therapy. Our analysis demonstrates two thresholds for the horizontal transmission rate: a “failure threshold” below which treatment fails, and a “success threshold” above which infection prevalence reaches 100% and the tumor shrinks to its smallest size. Moreover, we explain how changes in the virulence of the virus alter the success threshold and the minimum tumor size. Our study suggests that the optimal virulence of an oncolytic virus depends on the timescale of virus dynamics. We identify a threshold for the virulence of the virus and show how this threshold depends on the timescale of virus dynamics. Our results suggest that when the timescale of virus dynamics is fast, administering a more virulent virus leads to a greater reduction in the tumor size. Conversely, when the viral timescale is slow, higher virulence can induce oscillations with high amplitude in the tumor size. Furthermore, we introduce the concept of a “Hopf bifurcation Island” in the parameter space, an idea that has applications far beyond the results of this paper and is applicable to many mathematical models. We elucidate what a Hopf bifurcation Island is, and we prove that small Islands can imply very slowly growing oscillatory solutions.