Stability, Hopf bifurcation, and optimal control of a delayed tumor–immune–oncolytic virus model with chemotherapy

Our study proposes a delay differential equation (DDE) model for the dynamics of tumor cells, immune cells, oncolytic viruses, and chemotherapy. Two discrete delays associated with immune activation and chemotherapy response are incorporated into the model. Positive and bounded solutions are established, several equilibria are determined, and delay-dependent stability conditions are derived. A Hopf bifurcation may occur when the immune-activation delay crosses an equilibrium-dependent critical threshold. To identify time-dependent chemotherapy and virotherapy schedules that reduce tumor burden while preserving immune activity, we formulate an optimal control problem. Based on numerical simulations, combined chemo-virotherapy provides better tumor control than either treatment alone, particularly when treatment timing is appropriately selected.