The Caputo-Fabrizio derivative is considered to be one of the most successful tools of fractional modeling since it has non singular nuclei, which make it a better candidate for modeling some interdisciplinary models that rely on memory effects and hereditary features such as the mathematical models in biology. In this study, tumor-macrophages interactions are modeled via the Caputo-Fabrizio derivatives. The uniqueness of the model's solutions is shown. A convergence analysis based on the Adomian decomposition method (ADM) is proved. The error analysis using the ADM is discussed. The Picard method is applied to the considered model. According to the stability theory of fractional-order systems governed by the Caputo-Fabrizio derivatives, the stability conditions of the tumor-free, the tumor-dominant, and the co-axial or the existence equilibrium points are discussed. Numerical simulations are carried out to show the rich complex dynamics in the model, including the existence of chaotic attractors.

Although fractional calculus is about three centuries old, it has become the key to understanding many complex real-world phenomena. During the past few decades, many fractional derivatives have appeared. Among these, the fractal-fractional derivatives have shown acceptance in describing some real-world problems. In this paper, the Caputo, Atangana-Baleanu, and Caputo-Fabrizio fractal-fractional operators were applied to generate complex dynamics in a 4D dynamical system. Some conditions for the exact solutions' existence and uniqueness were demonstrated when the fractal-fractional operators are implemented into the mentioned 4D dynamical system. Some Ulam-Hyers stability results were demonstrated in the indicated fractal-fractional systems. Computation processes were carried out to demonstrate some graphical results that showed the existence of several complex dynamics in the considered system as the fractal-fractional operators are implemented. Furthermore, the computations of the system's Lyapunov exponents and the bifurcation diagrams were used to illustrate the wide range of chaotic dynamics that exist in the considered fractal-fractional 4D system. Existence of hidden chaotic attractors were also found. This interesting dynamical phenomenon was validated by the bifurcation diagrams and basin set of attraction.

In this work, we present a comprehensive analysis of the spatio-temporal $\mathrm{S}\mathrm{E}\mathrm{I}\mathrm{R}$ epidemic model of fractional order. The infection dynamics in the proposed fractional order model (FOM) are described by a system of partial differential equations (PDEs) within a time-fractional order and diffusion operator in one-dimensional space, considering that the total population is split into four compartments: Susceptible, exposed, infected, and recovered individuals denoted as $\mathrm{S}$, $\mathrm{E}$, $\mathrm{I}$ and $\mathrm{R}$, respectively. Our contributions commence by establishing the existence and uniqueness of positively bounded solutions for the proposed FOM. Moreover, we determined all equilibrium points (EPs) and investigated their local stability based on the basic reproduction number (BRN) ${\mathcal{R}}_0$, which is calculated by the next-generation matrix (NGM) method. Additionally, we demonstrated global stability using an appropriate Lyapunov function with fractional LaSalle's invariance principle (LIP). Sensitivity analysis of the FOM parameters was discussed to identify the most critical parameters by which the volume of disease propagation can be measured. The theoretical findings were corroborated by numerical simulations of solutions that are displayed in 3D and 2D graphs. Graphical simulations highlight the effect of vaccination on infection severity. Changing the fractional order $\alpha$ in the proposed FOM has an influence on the speed of convergence to the steady state as a result of the memory effect. Furthermore, vaccination emerges as an effective strategy for disease control.