The filled Julia set of the complex map is defined as the set of initial points such that trajectories starting from this point will be bounded. Its boundary, known as the Julia set, characterizes the intricate interface between stable and unstable dynamics, while its complement forms the Fatou set. These sets encode essential information about the global behavior of nonlinear systems and provide a geometric framework for understanding boundedness and stability. We examine the fractal behavior in a nonlinear discrete fractional disease model. We take a specific one that depicts a fractional tumor-immune interaction, and the same analysis will work for other suitable types of fractional disease models. Firstly, 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. In our study, 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 them to 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.