hopf bifurcation analysis in a delayed model of tumor therapy with oncolytic viruses

The stability and hopf bifurcation of a nonlinear mathematical model are described by the delay differential equation proposed by wodarz for interaction between uninfected tumor cells and infected tumor cells with the virus. by choosing τ as a bifurcation parameter, we show that the hopf bifurcation can occur for a critical value τ. using the normal form theory and the center manifold theory, formulas are given to determine the stability and the direction of bifurcation and other properties of bifurcating periodic solutions. then, by changing the infection rate to two nonlinear infection rates, we investigate the stability and existence of a limit cycle for the appropriate value of τ, numerically. lastly, we present some numerical simulations to justify our theoretical results.