A delayed fractional-order tumor virotherapy model: Stability and Hopf bifurcation

A fractional-order tumor virotherapy model with two time delays is presented and analyzed in this paper. The existence, positivity and bounbdedness of solutions under positive initial conditions will be proved. The basic reproduction number R-0 and the immune response reproduction number R-1 are given. The virus-free equilibrium E-0 and the therapy partial success equilibrium E* are presented depending on the value of R-0 and R-1. Sufficient conditions to ensure the local stability of both the virus-free equilibrium and the therapy partial success equilibrium according to different values of two time delays are established. By considering the time delay as a bifurcation parameter, it was appeared that the model undergoes a Hopf bifurcation when the delay passes through a critical value. Finally, numerical simulations were carried out to support the theoretical results and to show the effect of both the fractional-order derivative and the time delays. (C) 2022 Elsevier Ltd. All rights reserved.