Abstract
In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system’s parameters. Some numerical simulations are presented to verify the obtained mathematical results.
Highlights
Cancer can be classified as abnormal growth and uncontrolled division of normal cells
The effects of the delay term on the oscillatory behavior were not considered in their model
A delay term was included in our model, and we investigated the system behavior with varying system parameters
Summary
Cancer can be classified as abnormal growth and uncontrolled division of normal cells. Considering a reaction-diffusion system, including time delay under the Neumann boundary conditions, Kayan et al [30] modified the model [11] which described tumor-immune competitions. They studied the Hopf bifurcation analysis and found that the effect of diffusion of tumor-immune interaction can significantly change the dynamics of the model. This suggest that for k > 0 the co-axial equilibrium E2 is unstable This implies that the tumor cells can proliferate faster if the interaction time delay crosses a given critical value and the system loses its stability at E2.
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