Bifurcations and multistability in a virotherapy model with two time delays

Abstract

In this paper, we establish a delayed virotherapy model including infected tumor cells, uninfected tumor cells and free virus. In this model, both infected and uninfected tumor cells have special growth patterns, and there are at most two positive equilibria. We mainly analyze the stability and Hopf bifurcation of the model under different time delays. For the model without delay, we study the Hopf and Bogdanov–Takens bifurcations. For the delayed model, by center manifold theorem and normal form theory of functional differential equation, we study the direction of Hopf bifurcation and stability of the bifurcated periodic solution. Moreover, we prove the existence of Zero-Hopf bifurcation. Finally, some numerical simulations show the results of our theoretical calculations, and the dynamic behaviors near Zero-Hopf and Bogdanov–Takens point of the system are also observed in the simulations, such as bistability, periodic coexistence and chaotic behavior.