Rapid computer network expansion in the digital age has made systems more vulnerable to malware, especially viruses. Antivirus techniques and robust network topologies depend on understanding viral transmission mechanisms. Mathematics provides a formal framework for virus dynamics, outbreak thresholds, and intervention evaluation. Classical models, based on integer-order differential equations, focus on virus spread but cannot account for memory and hereditary effects in complex digital and biological systems [1–3].

Fractional calculus has developed into a robust extension of conventional modelling methods, enabling derivatives of non-integer order to include non-locality, memory, and hereditary characteristics [4]. Initial formulations like the Riemann–Liouville and Caputo derivatives are defined by singular kernels, limiting their use for modelling smooth memory processes in practical scenarios [5]. To address these constraints, Caputo and Fabrizio proposed a fractional derivative including a nonsingular exponential kernel [6–8].

Fractional calculus has emerged as a powerful framework for modeling complex systems with memory and hereditary properties that cannot be adequately captured by classical integer-order derivatives. A comprehensive overview of its real-world relevance is provided by Baleanu et al. [9], who demonstrated the effectiveness of fractional operators in diverse applications spanning physics, engineering, and applied sciences. Building on these foundations, Atangana [10] introduced the concept of fractal–fractional differentiation, which unifies fractal geometry with fractional calculus and enables a more faithful representation of multi-scale and irregular dynamics. This formulation has proven particularly useful in the analysis of chaotic systems. For instance, Alsulami et al. [11] investigated the controlled chaos of the Newton–Leipnik system using fractal–fractional operators, revealing enhanced control performance and richer dynamical behavior compared to classical models. Similarly, Yan et al. [12] applied Adomian decomposition techniques in conjunction with fractal–fractional operators to a Lorenz-type system, highlighting the significant impact of memory effects on chaotic trajectories and convergence properties. In parallel, Alhazmi et al. [13] employed the Caputo–Fabrizio operator to study chaotic systems with nonsingular kernels, demonstrating improved numerical stability and an accurate depiction of fading memory effects. Collectively, these studies underscore the growing importance of fractal–fractional and nonsingular fractional operators in the analysis, control, and numerical approximation of nonlinear and chaotic dynamical systems. The use of fractional-order differential equations in dynamical system modeling has a long and well-established history, particularly in the context of mechanical systems exhibiting damping and memory effects [14, 15].

More recently, comprehensive investigations of fractional glucose–insulin systems have been conducted, addressing both theoretical and numerical aspects. Saber and Alahmari [16] analyzed existence, stability, and numerical behavior using residual power series and generalized Runge–Kutta methods, offering detailed comparisons among various fractional operators. The application of fractal–fractional operators to nonlinear glucose–insulin models was further advanced by Althubyani et al. [17], who derived semi-analytical solutions via the Adomian decomposition method and demonstrated the strong influence of fractal memory on system dynamics. In parallel, Saber and Mirgani [18] developed and analyzed several numerical schemes based on Laplace residual power series methods and Atangana–Baleanu as well as Caputo derivatives, revealing rich dynamical behaviors, including stability transitions and chaos control, particularly under variable-order formulations. Saber et al. [19] employed the Jumarie–Stancu collocation series method alongside a multi-step generalized differential transform approach to construct efficient numerical solutions. Furthermore, the robustness of fractional glucose–insulin systems was rigorously examined through Hyers–Ulam stability analysis and control strategies in [20], ensuring reliability of approximate solutions under perturbations. Collectively, these studies demonstrate that fractional, fractal–fractional, and variable-order modeling frameworks, together with advanced numerical and stability analyses, provide a powerful and flexible foundation for capturing the complex dynamics of glucose–insulin regulation in diabetic systems. Further contributions encompass the creation of novel numerical approximations utilising non-local and non-singular kernels [21], and the application of fractal–fractional operators in chaotic and biological systems [22,23]. Additionally, numerical techniques tailored for Caputo–Fabrizio (CF) operators, including multi-step Adams–Bashforth (AB) methods, Laplace–Adomian decomposition methods (LADM), and Newton polynomial-based methods (NPM), have been suggested to guarantee stability and precision in addressing these systems [24,25].

Choice of Caputo–Fabrizio Operator. The selection of the Caputo–Fabrizio derivative for this cyber-epidemiological model is motivated by several key considerations specific to computer virus dynamics. First, the exponential kernel of the CF operator exp⁡[−α(t−τ)1−α] naturally represents fading memory effects, which align with the gradual decay of influence from past infection states in digital systems—such as delayed patch deployments, gradual antivirus signature updates, and persistent latent infections. This contrasts with the Mittag-Leffler kernel of the Atangana–Baleanu (AB) operator, which exhibits power-law decay that may overemphasize distant past states in cyber contexts where memory effects typically diminish exponentially. Second, the CF derivative’s non-singular kernel avoids computational singularities at t=τ, enabling more stable numerical simulations crucial for real-time cybersecurity applications. Third, the CF formulation facilitates the development of efficient numerical schemes (like our adapted Adams–Bashforth method) with proven convergence properties, whereas AB operators often require more complex numerical handling of the Mittag-Leffler function. Finally, the CF parameter α provides a clear interpretation as a memory strength index: α→1 recovers memoryless classical dynamics, while α→0 corresponds to strong hereditary effects—an intuitive mapping for cybersecurity practitioners designing memory-aware defense strategies.

The modelling of computer viruses has progressed from deterministic compartmental frameworks to stochastic and fractional-order models. Classical models classify the computer population into susceptible, infected, and recovered compartments [26,27], whereas more sophisticated systems incorporate latent stages, antivirus activation, and network feedback effects [28]. The integration of fractional derivatives facilitates a more accurate depiction of memory-dependent processes in digital contexts, enhancing predictions regarding virus survival and control efficacy. Furthermore, fractional-order models have lately been utilised in various applied sciences, including thermoelasticity and nanofluid dynamics, underscoring its transdisciplinary potential [29–32]. Mathematical modeling has long played a pivotal role in understanding the dynamics of computer virus propagation and the effectiveness of control strategies in networked systems. Classical integer-order epidemic models have been widely employed to describe virus transmission mechanisms; however, such models often fail to account for inherent delay effects, long-term memory, and hereditary characteristics observed in complex cyber environments. To address these limitations, nonstandard computational approaches incorporating delay and memory effects have been proposed, demonstrating improved realism and analytical depth in cyber-epidemiological modeling [33]. In recent years, fractional calculus has emerged as a powerful extension of classical modeling frameworks, offering enhanced capability to represent nonlocal and memory-dependent dynamics through derivatives of non-integer order. Efficient numerical methods for fractional differential equations have further strengthened their applicability across physics, engineering, and applied sciences by enabling accurate simulation of systems governed by memory effects [34]. Moreover, nonlinear and chaotic behaviors have been identified in virus-related and biological models, highlighting the necessity of advanced mathematical tools to capture complex dynamical phenomena that arise in both biological and cyber systems [35]. From a theoretical perspective, ensuring the well-posedness of fractional-order models—particularly the existence and uniqueness of solutions—is essential for validating their mathematical and practical relevance, and such foundational properties have been rigorously established for various classes of fractional and nonlocal problems [36,37].

Model Scope and Limitations. This study presents a novel fractional-order computer virus propagation model based on the Caputo–Fabrizio derivative. The primary objectives are to establish a theoretical framework for incorporating memory effects into cyber-epidemiological dynamics and to develop stable numerical methods for simulating such systems. The model is validated through comprehensive numerical simulations and comparative analyses with classical integer-order and alternative fractional formulations. However, it is important to note the following limitations: (1) The model assumes homogeneous mixing within the computer population and does not explicitly account for network topology or contact heterogeneity, which are known to significantly influence real-world infection dynamics; (2) The validation is conducted numerically without calibration against empirical virus spread datasets, which limits direct applicability to specific real-world scenarios; (3) Additional factors such as patching delays, human behavior patterns, and adaptive defense mechanisms are not incorporated in the current formulation. These simplifications enable tractable mathematical analysis and clear isolation of memory effects, but future extensions should address these aspects for enhanced practical relevance.

Advancements Beyond Existing CF-Based Cyber-Epidemiological Models. While recent works have employed Caputo–Fabrizio derivatives in virus and worm propagation modeling [38], this study introduces several key theoretical and methodological advances. First, the proposed compartmental structure {S, L, B, R} explicitly separates latently infected (L) from actively breaking-out (B) computers, providing cyber-specific granularity that more accurately represents digital infection chains—unlike the generic {S, E, I, R} frameworks commonly adopted in prior CF-based cyber models. Second, we develop a novel two-step Adams–Bashforth scheme specifically adapted for CF operators, achieving proven second-order convergence with computational efficiency comparable to integer-order schemes, thereby advancing beyond the first-order discretizations typically employed in CF implementations. Third, the complete stochastic formulation, together with the derived SCF–AB scheme, enables probabilistic risk assessment and ensemble simulations, whereas most existing CF-based cyber models remain deterministic. Fourth, the integrated stability framework—including Hyers–Ulam stability analysis and comprehensive sensitivity assessment—provides robustness guarantees and quantitatively evaluates parameter influences relevant to cybersecurity policy (e.g., the transmission rate β exhibits a sensitivity of +0.82, while the memory parameter α shows a sensitivity of −0.29). These combined advances in cyber-specific modeling, numerical methodology, and practical cybersecurity applicability clearly distinguish the proposed framework from existing CF-based epidemiological approaches.

Practical Implications and Future Directions. Despite these limitations, the theoretical insights from this fractional modeling approach can inform the design of more resilient cybersecurity systems. The memory effects captured by the Caputo–Fabrizio derivative provide a mathematical basis for understanding delayed responses in cyber-epidemics, which aligns with observed behaviors in real networks. Future research could integrate this framework with network-aware epidemic models [39,40], human behavior factors [41], and adaptive defense strategies [42] to develop more accurate predictive tools. Additionally, the stochastic extension of the model supports uncertainty quantification in dynamic threat environments, which can be leveraged by machine learning models for anomaly detection [43].

Recent advances in computer virus modeling have increasingly focused on capturing memory effects, network heterogeneity, and complex dynamical behaviors that cannot be adequately represented by classical integer-order formulations [44]. Fractional-order models, in particular, have proven effective in describing nonlocal interactions and hereditary characteristics inherent in cyber systems. Notably, the Atangana–Baleanu fractional derivative with a nonsingular Mittag–Leffler kernel has been successfully employed to investigate computer virus spread in networked environments, demonstrating how fractional memory significantly influences virus persistence and control efficiency [45]. In parallel, modified compartmental virus models incorporating network attributes and key system parameters have provided deeper insights into stability regions, outbreak thresholds, and the role of connectivity in shaping virus dynamics [46]. From a theoretical standpoint, the stability analysis of fractional-order nonlinear systems remains a central concern, and Lyapunov-based approaches combined with generalized Mittag–Leffler stability theory offer rigorous criteria for assessing asymptotic behavior and robustness of equilibria in fractional dynamical systems [47]. These developments collectively motivate the use of fractional derivatives and advanced stability tools in the present study to achieve a more realistic and mathematically sound representation of computer virus propagation.

This study presents a novel computer virus propagation model utilising the Caputo–Fabrizio derivative. The model categorises the network into four compartments: susceptible, latently infected, actively spreading, and antivirus-capable systems. The use of the CF derivative facilitates the explicit incorporation of memory and hereditary effects, thereby accurately representing the dynamics of virus spread and antivirus response. We demonstrate the existence and uniqueness of solutions through fixed-point theory, contingent upon Lipschitz continuity and boundedness conditions, and we develop a two-step Adams-Bashforth numerical approach tailored for CF operators. The scheme’s convergence and stability are thoroughly examined, and comprehensive simulations are conducted for different fractional orders to evaluate the influence of memory on virus dynamics. The expanded framework integrates stochastic influences via a stochastic CF formulation and encompasses a thorough sensitivity analysis to pinpoint critical parameters affecting viral dynamics. The findings indicate that fractional-order dynamics substantially influence outbreak intensity and timing, providing insights into effective antivirus techniques and cyber defence policy.

The key contributions of this study are: (i)

The formulation of a novel CF-based computer virus propagation model with four network compartments;

The remaining parts of the paper are structured as follows: Section 2 presents the mathematical formulation of the suggested model and its characteristics. It establishes results pertaining to existence and uniqueness. Also delineates the numerical methodology and convergence assessment. It ultimately provides numerical simulations, a comparative study with classical and alternative fractional models, and a sensitivity analysis. Section 3 expands the approach to encompass stochastic situations. Section 4 offers an extensive analysis of the results, while Section 5 ends with final observations and possible extensions.