Mathematical Analysis of Malware Spread in Digital Systems Using Atangana–Baleanu–Caputo Fractional Dynamics

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.