The monkeypox virus causes a respiratory illness called monkeypox, which belongs to the Poxviridae virus family and the Orthopoxvirus genus. Although initially endemic in Africa, it has recently become a global threat with cases worldwide. Using the Antangana–Baleanu fractional order approach, this study aims to propose a new monkeypox transmission model that represents the interaction between the infected human and rodent populations. An iterative method and the fixed-point theorem are used to prove the existence and uniqueness of the symmetry model’s system of solutions. It shows that the symmetry model has equilibrium points when there are epidemics and no diseases. As well as the local asymptotic stability of the disease-free equilibrium point, conditions for the endemic equilibrium point’s existence have also been demonstrated. For this purpose, the existence of optimal control is first ensured. The aim of the proposed optimal control problem is to minimize both the treatment and prevention costs, and the number of infected individuals. Optimal conditions are acquired Pontryagin’s maximum principle is used. Then, the stability of the symmetry model is discussed at monkeypox-free and endemic equilibrium points with treatment strategies to control the spread of the disease. Numerical simulations clearly show how necessary and successful the proposed combined control strategy is in preventing the disease from becoming epidemic.

This paper utilizes a mathematical model based on the Atangana–Baleanu fractal-fractional derivative to investigate different epidemiological aspects of monkeypox virus infection. The goal is to evaluate the effects of treatment and vaccination on the transmission dynamics of the virus. Initially, the model utilizes integer-order nonlinear differential equations, integrating imperfect vaccination and treatment as control strategies within the human population. Subsequently, the model is reformulated using a fractal fractional-order derivative based on a power law to offer a more comprehensive insight into disease dynamics. Conditions are established for the basic reproduction number and equilibrium points, and the feasible region of the model is identified. Stability analysis of the endemic equilibrium is conducted using the Lyapunov function approach. The fixed-point method is used to explore the existence and uniqueness of solutions in the Atangana–Baleanu model with fractal-fractional derivative order. Additionally, the study examines Ulam–Hyer’s stability. The numerical scheme applies Lagrange’s interpolation polynomial, enabling precise model interpolation. We present numerical comparisons for various fractional and fractal orders to showcase the accuracy of our approach.

This study explores the spread of malware within a digital framework by introducing a unique fractional-order model that employs the Atangana–Baleanu–Caputo (ABC) derivative. As cyber threats grow increasingly sophisticated and widespread, traditional models using classical differential equations often prove inadequate, particularly in capturing long-term memory effects and historical dependencies inherent in real-world systems. To address these challenges, the proposed approach utilizes the non-local characteristics of fractional calculus, offering a more comprehensive framework for understanding malware behavior. The model includes the derivation of the basic reproduction number, ${\Re }_0$, to evaluate conditions for malware persistence or elimination, sensitivity analysis and examines equilibrium states to assess overall system stability. Theoretical analysis ensures the existence and uniqueness of solutions through fixed-point techniques. Through numerical simulations, the theoretical results are validated, emphasizing the significant impact of antidotal and recovery measures in controlling malware spread. These findings provide essential guidance for enhancing the protection and robustness of sophisticated cyber-physical and humanoid infrastructures.