Fractal dimension and control of Julia set of discrete fractional tumor-immune model

The article's goal is to examine the fractal behavior in a non-linear fractional disease model. We consider a specific example that depicts a fractional tumor-immune interaction, and the same analysis will work for other suitable types of fractional disease model. The fractal point of view has not yet been applied to the discrete fractional tumor-immune model. Due to the memory effect and the hereditary property, the discrete fractional model helps in the study of the fractal behavior of the tumor-immune model and provides a degree of freedom for the study of the model. First, we take the Caputo fractional-order tumor-immune model. The corresponding discrete fractional model is then determined using the Caputo definition. The escape-time algorithm is used to generate the Julia set of the discrete fractional model. Fractal dimension analysis and graphical exploration are used to study how memory parameters interact with the Julia set. In this article, we design two novel controllers based on the fixed-point and discrete fractional models, and add these two novel controllers in distinct ways. We notice the different effects of the controllers by adding in different parts of the system. We also used coordinate transformation to design the controller, which helps to control the Julia set using the stability of the fixed point. Finally, the simulations help in the study of the effectiveness of each controller towards the control of fractal behaviors.