Dynamics of fractional-order delay differential model for tumor-immune system

Time-delays and fractional-order play a vital role in modeling biological systems with memory. In this paper, we propose a novel delay differential model with fractional-order for tumor immune system with external treatments. Non-negativity of the solution of such model has been investigated. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to two differenttumor time-delays τ₁ and τ₂. The occurrence of Hopf bifurcation is captured when any of the time-delay passes through critical value τ₁ ^*, or τ₂ ^*. Theoretical results are validated numerically by solving the governing system, using the modified Adams–Bashforth–Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model.