A Dynamically Consistent Nonstandard Difference Scheme for a Discrete-Time Immunogenic Tumors Model.

Area of Science:

• Mathematical Biology
• Computational Biology
• Dynamical Systems Theory

Background:

• Immunogenic tumor models are crucial for understanding cancer immunology and developing therapies.
• Qualitative analysis of tumor dynamics can reveal complex behaviors and potential therapeutic targets.
• Discrete-time models offer a computationally tractable approach to studying biological systems.

Purpose of the Study:

• To develop a dynamically consistent discrete-time model for immunogenic tumors.
• To analyze the stability of fixed points and identify bifurcation phenomena.
• To compare the dynamical behavior of the discrete model with its continuous counterpart.

Main Methods:

• A nonstandard difference scheme was employed to discretize the continuous immunogenic tumor model.
• Fixed point analysis was used to determine equilibrium states and their stability.
• Bifurcation theory, specifically analyzing Hopf and Neimark-Sacker bifurcations, was applied.
• Numerical simulations were conducted to validate theoretical findings.

Main Results:

• A unique positive fixed point was identified in the discrete-time model.
• The discrete model exhibits Neimark-Sacker bifurcation at the positive fixed point.
• The continuous model exhibits Hopf bifurcation at its corresponding positive fixed point.
• Period-doubling bifurcations were found to be absent in the discrete-time model.

Conclusions:

• The discrete-time immunogenic tumor model accurately captures key dynamical properties, including bifurcations.
• Neimark-Sacker bifurcation in the discrete model offers insights into complex tumor growth patterns.
• The study highlights differences in bifurcation behavior between continuous and discrete dynamical systems in tumor modeling.