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

Abstract

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.

1. Introduction

The proliferation of digital technologies across sectors has led to an unprecedented reliance on computer systems for communication, data storage, finance, and critical infrastructure management. As a result, cybersecurity threats, particularly those involving computer viruses have become a growing concern. Much like biological pathogens, computer viruses depend on host environments such as files, applications, or operating systems to replicate and spread. While early forms of malware were relatively simple and posed limited risk, the rapid advancement of digital connectivity and software complexity has led to the emergence of highly adaptive and damaging threats, including trojans, worms, ransomware, and advanced persistent threats (APTs). These malicious programs can result in severe financial losses, data breaches, operational disruptions, and reputational damage for individuals, institutions, and governments.

Mathematical modeling plays a central role in contemporary research because it provides a bridge between real-world phenomena and their mathematical representation, allowing systematic analysis of complex systems [1]. Its flexibility has enabled researchers to address current scientific challenges and to make informed predictions about future developments [2]. Applications of mathematical modeling span a wide range of disciplines, including engineering, biological sciences, and computer networks. In cybersecurity, for example, mathematical models originally designed for epidemiological studies have been effectively adapted to examine the spread of worms and malware in digital environments [3,4]. Recent studies further highlight that such models can capture dynamic interactions between malicious attacks and defensive mechanisms, such as antivirus deployment, within network systems [5]. Consequently, understanding viral behavior in wireless sensor networks has become important for evaluating system stability and improving cybersecurity strategies [6]. Advances in dynamical system theory and numerical simulation techniques continue to support developments across engineering and scientific disciplines [7]. Within this context, fractional-order calculus has received significant attention for its ability to generalize classical models and incorporate memory or hereditary effects [8]. These fractional operators allow a more accurate description of physical, biological, and engineering processes that cannot be fully captured using integer-order derivatives [9]. Fractional derivatives have therefore become valuable tools in modeling phenomena such as viscoelastic behavior, biological processes, chemical reactions, and frequency-dependent damping in engineering systems [10,11]. Understanding the mechanisms behind malware propagation and developing effective containment strategies are critical in mitigating such risks. However, conventional mathematical models typically based on classical differential equations often fall short in capturing the memory effects and long term dependencies characteristic of malware behavior in real-world systems. To address these limitations, fractional calculus has emerged as a robust modeling tool, offering greater flexibility through the use of nonlocal operators that account for historical states of the system. The study by Yang et al. analyzed the impact of patch forwarding on the prevalence of computer viruses. Their findings highlight that the ability of patches to rapidly disseminate across computer networks and become effective immediately upon installation significantly enhances their potential to mitigate virus outbreaks [12]. A theoretical evaluation of patch forwarding was conducted, leading to the development of a malware epidemic model that accounts for its influence.

In 2017, Achuba examined the transmission of computer viruses by modifying the traditional SIRA model. His research provided insights into virus behavior within a network with antivirus software. The proposed model extended the conventional susceptible-infected-recovered-antidotal (SIRA) framework to a liable-latent-infected-recovered-antidotal (SLIRA) model [13].

Singh et al. (2018) explored computer virus propagation using fractional derivatives. Given the critical role of computer viruses in cybersecurity, understanding their spread and development is essential. The authors formulated and analyzed a fractional epidemiological model to investigate these aspects [14]. Similarly, Ali et al. (2018) developed a nonlinear model to study virus prevalence, employing the EPA scheme with a Padé approximation to convert the nonlinear system into an optimization problem [15].

In 2019, Kyurkchiev et al. proposed differential models to study virus propagation [16]. Around the same time, Alweimine et al. explored the influence of local routing protocol algorithms on virus transmission, assessing the robustness of various routing techniques [17].

Arif et al. (2020) conducted numerical simulations to compare stochastic and deterministic models of virus spread [18]. Upadhyay et al. (2020) focused on targeted cyberattacks, developing a model with two distinct classes: an attacking group and a targeted network. Their analysis examined equilibrium points and stability concerning the basic reproduction number [19]. Hoang et al. (2021) introduced a fractional-order model for virus transmission using the Caputo derivative and Lyapunov function, ensuring the positivity and boundedness of the solutions [20]. Shahini et al. (2022) applied a collocation method with Legendre–Gauss–Radau points to address an optimal control problem within a nonlinear delay differential equation framework [21].

More recently, Avcı et al. (2023) investigated the role of memory effects in virus population dynamics, utilizing a fractional derivative operator to analyze the influence of fractal size and fractional order [22]. Their research incorporated numerical simulations to explore the effects of different fractional operators on viral spread. To investigate the dynamics of malware propagation, it is essential to analyze how infections spread across computer networks, considering modes of transmission, host interactions, and system vulnerabilities. Such an approach allows for assessing the effectiveness of countermeasures and predicting long-term outcomes. For further reference, see relevant studies on malware modeling and cybersecurity dynamics [23,24,25].

This study examines the role of quarantine and recovery in controlling virus outbreaks. Isolating infected files and deploying antivirus measures contribute to mitigating the spread of malicious software. However, identifying compromised files remains challenging due to latent infections, potentially limiting the effectiveness of quarantine strategies. The structure of this paper is outlined as follows. Section 2 introduces the fundamental definitions for the subsequent analysis. Section 3 presents the development of the fractional-order virus transmission model, explaining its architecture and the relationships between different components. In Section 4, the equilibrium states of the system are explored, along with the derivation of the basic reproduction number and sensitivity analysis, which provides insights into the conditions that determine whether a virus dies out or persists. Section 5 focuses on establishing the existence of solutions, thereby confirming the mathematical soundness of the proposed model. Section 6 outlines the numerical method adopted to simulate the system’s dynamics. Section 7 displays and interprets the simulation results, emphasizing the role of model parameters and evaluating the impact of various intervention strategies. Lastly, Section 8 concludes the study by summarizing key findings, drawing final observations, and proposing directions for future research.

2. Preliminaries

This section presents the essential background on the Atangana–Baleanu–Caputo (ABC) fractional derivative, which forms the foundation for its application in the later sections.

Definition 1

([26]). Let $\rho \in [0,1]$ and consider a function $f\in H^1(a,b)$ with $b>a$. The ABC fractional derivative of f is defined by

$${}_0{}^{\mathit{ABC}}D_t^{\rho }f\left(t\right)={\displaystyle \frac{M\left(\rho \right)}{1-\rho }}{\int }_a^t{f}^{\prime }\left(y\right)E_{\rho }\left(-{\displaystyle \frac{\rho {(t-y)}^{\rho }}{1-\rho }}\right)dy,$$

where $M\left(\rho \right)$ is a normalization function that satisfies $M\left(0\right)=M\left(1\right)=1$.

Definition 2

([27]). On the interval $\rho \in [0,1]$, consider the function $f\in H^1(a,b)$, $b>a$, which is not necessarily differentiable. The Atangana–Baleanu fractional derivative in the Riemann–Liouville sense is defined as:

$${}_0{}^{\mathit{ABR}}D_t^{\rho }f\left(t\right)={\displaystyle \frac{M\left(\rho \right)}{1-\rho }}{\displaystyle \frac{d}{dt}}{\int }_a^{t}f\left(y\right)E_{\rho }\left[-{\displaystyle \frac{\rho {(t-y)}^{\rho }}{1-\rho }}\right]dy.$$

Definition 3

([27]). The Atangana–Baleanu (AB) fractional integral of order ρ is defined by

$${}_0{}^{\mathit{AB}}I_t^{\rho }f\left(t\right)={\displaystyle \frac{1-\rho }{\Im \left(\rho \right)}}f\left(t\right)+{\displaystyle \frac{\rho }{\Im \left(\rho \right)\tau \left(\rho \right)}}{\int }_a^t{(t-y)}^{\rho -1}f\left(y\right)dy.$$

In the special cases where $\rho =0$ or $\rho =1$, this operator coincides with the original function and the standard integral, respectively.

3. Model Formulation

We define the state variables of the model as follows. Let $S\left(t\right)$ denote the number of susceptible computers at time t, referring to those that are neither infected nor protected. Let $A\left(t\right)$ be the number of antidotal computers at time t, including both recently updated and previously patched devices. The variable $E\left(t\right)$ represents the number of exposed computers at time t, referring to those that have come into contact with the malware and are at risk of infection. Let $I\left(t\right)$ denote the number of infected computers at time t that require disinfection. Let $Q\left(t\right)$ be the number of quarantined computers at time t, isolated to prevent further malware transmission. The variable $R\left(t\right)$ captures the number of recovered computers at time t that have regained temporary immunity. The total number of computers in the network at any given time t is denoted by $N\left(t\right)$.

The total population of the system is divided into six compartments: Susceptible (S), Antidotal (A), Exposed (E), Infected (I), Quarantined (Q), and Recovered (R). At any time t, these compartments satisfy the relation:

$$S\left(t\right)+A\left(t\right)+E\left(t\right)+I\left(t\right)+Q\left(t\right)+R\left(t\right)=N\left(t\right).$$

In the SAEIQRS framework, a portion of the susceptible individuals, $S\left(t\right)$, gain protection and move into the antidotal state, $A\left(t\right)$. Meanwhile, others shift into the exposed state, $E\left(t\right)$, where the infection is present but not yet active. Individuals in the exposed state may transition to the infected class, $I\left(t\right)$ as the infection advances. The progression from infection depends on their level of antidotal resistance with sufficient immunity to recover and move to the recovered state, $R\left(t\right)$. In contrast, others may relapse and return to the exposed state, $E\left(t\right)$, emphasizing the potential for reinfection.

Infected humanoids may either remain in the infected class $I\left(t\right)$ while contagious or be restored to the recovered class $R\left(t\right)$ after receiving updates or antivirus protection. Alternatively, some infected humanoids are moved into the quarantine class $Q\left(t\right)$, where they remain until they recover and transition to $R\left(t\right)$. However, in the cyber world, immunity is temporary, meaning recovered humanoids eventually revert to the susceptible class $S\left(t\right)$, making them vulnerable to reinfection.

To describe the evolution of the system over time, we consider the following set of differential equations,

$$\begin{array}{cc}\hfill {\displaystyle \frac{dS}{dt}}& =B-\varpi S-\xi SI-\alpha S+\chi R,\hfill \\ \hfill {\displaystyle \frac{dA}{dt}}& =\alpha S-\varpi A-{\varphi }_{2}A-{\varphi }_{1}AI,\hfill \\ \hfill {\displaystyle \frac{dE}{dt}}& =\xi SI-\varpi E-\tau E+{\varphi }_{1}AI,\hfill \\ \hfill {\displaystyle \frac{dI}{dt}}& =\tau E-(\varpi +\zeta )I-{ϵ}_{1}I-{ϵ}_{2}I,\hfill \\ \hfill {\displaystyle \frac{dQ}{dt}}& ={ϵ}_{2}I-(\varpi +\zeta )Q-\delta Q,\hfill \\ \hfill {\displaystyle \frac{dR}{dt}}& ={ϵ}_{1}I+\delta Q-\varpi R+{\varphi }_{2}A-\chi R.\hfill \end{array}$$

(1)

These equations describe the flow between compartments, capturing the dynamics of the system as individuals transition from one state to another.

To construct the ABC–fractional version of the system, the standard first-order time derivative on the left-hand side of the governing equations is replaced by the ABC–fractional operator, as introduced earlier. This adjustment results in a generalized formulation, yielding an ABC–fractional model that describes the dynamics of the computer network more accurately.

Initially, new computers introduced into the network are assumed to be virus-free and are classified as susceptible. These susceptible systems can adopt antivirus measures, either outdated or updated, at a rate $\alpha$. Susceptible and antidotal computers may come into contact with infected systems, becoming exposed at rates $\xi$ and ${\varphi }_1$, respectively. Antidotal computers that have been updated recover and move to the recovered state at a rate ${\varphi }_2$. All computers, regardless of their infection status, experience a natural failure rate denoted by $\varpi$. After an incubation period, exposed computers progress to the infected state at a rate $\tau$. Infected systems can either recover at rate ${ϵ}_1$ or be transferred to quarantine at rate ${ϵ}_2$. Quarantined computers regain their functionality and move to the recovered state at rate $\delta$. However, the immunity of recovered systems is temporary, and these units can return to the susceptible class at rate $\chi$.

The parameter B denotes the rate of introduction of new devices to the network. The natural failure rate $\varpi$ applies to all systems regardless of infection status. The rate $\zeta$ captures the rate of device breakdown caused by the infection itself. The parameter $\xi$ describes the rate at which the virus spreads from infected to susceptible units, leading to their transition from S to E. The parameter $\alpha$ governs the movement of susceptible units into the antidotal state, where $\alpha =0$ means no antivirus protection is applied. The rate ${\varphi }_1$ characterizes the transition of antidotal units that have not been updated to the exposed state after contact with infected units. The parameter ${\varphi }_2$ governs recovery of antidotal units that have been updated, allowing their movement from A to R. The rate $\tau$ describes the progression from exposed to infected status. The parameters ${ϵ}_1$ and ${ϵ}_2$ govern the recovery of infected units ($I\to R$) and their placement into quarantine ($I\to Q$), respectively. The rate $\delta$ applies to the recovery of quarantined units ($Q\to R$), while $\chi$ governs the loss of immunity, allowing recovered units to return to the susceptible class ($R\to S$).

$$\begin{array}{cc}\hfill {}_0{}^{\mathit{ABC}}D_t^{\rho }S\left(t\right)& =B-\varpi S-\xi SI-\alpha S+\chi R,\hfill \\ \hfill {}_0{}^{\mathit{ABC}}D_t^{\rho }A\left(t\right)& =\alpha S-\varpi A-{\varphi }_{2}A-{\varphi }_{1}AI,\hfill \\ \hfill {}_0{}^{\mathit{ABC}}D_t^{\rho }E\left(t\right)& =\xi SI-\varpi E-\tau E+{\varphi }_{1}AI,\hfill \\ \hfill {}_0{}^{\mathit{ABC}}D_t^{\rho }I\left(t\right)& =\tau E-(\varpi +\zeta )I-{ϵ}_{1}I-{ϵ}_{2}I,\hfill \\ \hfill {}_0{}^{\mathit{ABC}}D_t^{\rho }Q\left(t\right)& ={ϵ}_{2}I-(\varpi +\zeta )Q-\delta Q,\hfill \\ \hfill {}_0{}^{\mathit{ABC}}D_t^{\rho }R\left(t\right)& ={ϵ}_{1}I+\delta Q-\varpi R+{\varphi }_{2}A-\chi R.\hfill \end{array}$$

(2)

where the initial conditions are,

$$\begin{array}{c}\hfill S\left(0\right)=S_0,A\left(0\right)=A_0,E\left(0\right)=E_0,I\left(0\right)=I_0,Q\left(0\right)=Q_0,R\left(0\right)=R_0.\end{array}$$

4. Equilibrium Analysis and ${\Re }_0$

In this section, we analyze the equilibrium states of the SAEIQRS model governed by system (2), with particular attention to both the virus-free and endemic scenarios. The basic reproduction number, ${\Re }_0$, is derived to evaluate the conditions under which the infection may spread or die out. An equilibrium point represents a steady-state solution where all compartments remain constant over time. We set the right-hand sides of the differential equations in system (2) to zero to identify these points and solve the resulting algebraic equations.

$${\displaystyle \frac{dS}{dt}}={\displaystyle \frac{dA}{dt}}={\displaystyle \frac{dE}{dt}}={\displaystyle \frac{dI}{dt}}={\displaystyle \frac{dQ}{dt}}={\displaystyle \frac{dR}{dt}}=0$$

in Equation (2). This gives the system of equations,

$$\begin{array}{cc}\hfill B-\varpi S-\xi SI-\alpha S+\chi R& =0\hfill \\ \hfill \alpha S-\varpi A-{\varphi }_{1}AI& =0\hfill \\ \hfill \xi SI-\varpi E-\tau E-{\varphi }_{1}AI& =0\hfill \\ \hfill \tau E-\varpi I-{ϵ}_{1}I-{ϵ}_{2}I& =0\hfill \\ \hfill {ϵ}_{2}I-\varpi Q-\delta Q& =0\hfill \\ \hfill {ϵ}_{1}I+\delta Q+{\varphi }_{2}A-\varpi R-\chi R& =0\hfill \end{array}$$

4.1. Virus-Free Equilibrium

At the disease-free state, all infected compartments vanish, yielding the virus-free equilibrium point,

$$P_0=(S_0,A_0,E_0=0,I_0=0,Q_0=0,R_0).$$

The expressions for the susceptible, asymptomatic, and recovered compartments at this state are,

$$\begin{array}{cc}\hfill S_0=& {\displaystyle \frac{B(\varpi +\chi )(\varpi +{\varphi }_2)}{(\varpi +\chi )(\varpi +\alpha )-\alpha \chi (\varpi +{\varphi }_2)}},\hfill \\ \hfill A_0=& {\displaystyle \frac{B\alpha (\varpi +\chi )}{(\varpi +\chi )(\varpi +\alpha )(\varpi +{\varphi }_2)-\alpha \chi (\varpi +{\varphi }_2)}},\hfill \\ \hfill R_0=& {\displaystyle \frac{B\alpha {\varphi }_2}{(\varpi +\chi )(\varpi +\alpha )(\varpi +{\varphi }_2)-\alpha \chi (\varpi +{\varphi }_2)}}.\hfill \end{array}$$

In this state, the disease has been eradicated from the system, and only the susceptible, asymptomatic, and recovered compartments maintain their respective steady levels.

4.2. Basic Reproduction Number

The basic reproduction number, denoted by ${\Re }_0$, quantifies the expected number of secondary infections produced by a single infectious individual in an entirely susceptible population. We apply the next-generation matrix approach to calculate ${\Re }_0$. Given that the model involves three infectious-related compartments-exposed (E), infected (I), and quarantined (Q)-our analysis focuses on the relevant subsystem extracted from system (2)

$$\begin{array}{cc}\hfill {\displaystyle \frac{dE}{dt}}& =\xi SI-\varpi E-\tau E-{\varphi }_{1}AI,\hfill \\ \hfill {\displaystyle \frac{dI}{dt}}& =\tau E-\varpi I-{ϵ}_{1}I-{ϵ}_{2}I,\hfill \\ \hfill {\displaystyle \frac{dQ}{dt}}& ={ϵ}_{2}I-\varpi Q-\delta Q.\hfill \end{array}$$

Defining the vector $X={(E,I,Q)}^{\top }$, the system can be expressed as,

$${\displaystyle \frac{dX}{dt}}=f\left(X\right)-v\left(X\right),$$

where,

$$f\left(X\right)=\left(\begin{array}{ccc}\xi S+{\varphi }_{1}A& 0& 0\\ \tau & 0& 0\\ 0& {ϵ}_2& 0\end{array}\right)\mathrm{and}v\left(X\right)=\left(\begin{array}{ccc}\varpi +\tau & 0& 0\\ -\tau & \varpi +{ϵ}_1+{ϵ}_2& 0\\ 0& -{ϵ}_2& \varpi +\delta \end{array}\right).$$

The basic reproduction number ${\Re }_0$ is then calculated as,

$${\Re }_0={\displaystyle \frac{\tau (\xi S+{\varphi }_{1}A)}{(\varpi +\tau )(\varpi +{ϵ}_1+{ϵ}_2)}}.$$

4.3. Sensitivity Analysis of the Basic Reproduction Number

To determine which epidemiological parameters exert the strongest influence on the transmission potential of the disease, a sensitivity analysis of the basic reproduction number ${\mathcal{R}}_0$ is carried out. Using the expression

$${\mathcal{R}}_0={\displaystyle \frac{\tau (\xi S+{\varphi }_{1}A)}{(\varpi +\tau )(\varpi +{ϵ}_1+{ϵ}_2)}},$$

we compute the normalized forward sensitivity index for each parameter. The index is defined as

$${{\rm Y}}_p^{{\mathcal{R}}_0}={\displaystyle \frac{\partial {\mathcal{R}}_0}{\partial p}}·{\displaystyle \frac{p}{{\mathcal{R}}_0}},$$

where p denotes any parameter appearing in the expression of ${\mathcal{R}}_0$. This formulation quantifies the relative change in ${\mathcal{R}}_0$ resulting from a proportional change in the parameter p, thereby indicating whether an increase in a particular parameter enhances or suppresses the transmission potential. In Figure 1, parameters with positive sensitivity indices contribute to increasing the reproduction number, while those with negative indices act to reduce it. This analysis highlights the key biological processes that play a dominant role in shaping the dynamics of infection and assists in identifying the most effective targets for intervention strategies.

4.4. Endemic Equilibrium Point

When the disease persists in the population, we have $I^{\ast }\ne 0$. The endemic equilibrium (EEP) of the system is denoted as,

$$P_e=(S^{\ast },A^{\ast },E^{\ast },I^{\ast },Q^{\ast },R^{\ast })$$

$$\begin{array}{cc}\hfill S^{\ast }& ={\displaystyle \frac{B+\chi R^{\ast }}{\varpi +\xi I^{\ast }+\alpha }}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill A^{\ast }& ={\displaystyle \frac{\alpha S^{\ast }}{\varpi +{\varphi }_2+{\varphi }_1{I}^{\ast }}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill E^{\ast }& ={\displaystyle \frac{I^{\ast }(\xi S^{\ast }+{\varphi }_1{A}^{\ast })}{\varpi +\tau }}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill Q^{\ast }& ={\displaystyle \frac{{ϵ}_1{I}^{\ast }}{\varpi +\zeta +\delta }}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill R^{\ast }& ={\displaystyle \frac{{ϵ}_2{I}^{\ast }+\delta Q^{\ast }+{\varphi }_2{A}^{\ast }}{\varpi +\chi }}\hfill \end{array}$$

(7)

Similarly, we get

$$I^{\ast }={\displaystyle \frac{\tau (\xi S^{\ast }+{\varphi }_1{A}^{\ast })}{(\varpi +\tau )(\varpi +\zeta +{ϵ}_1+{ϵ}_2)-\tau (\xi S^{\ast }+{\varphi }_1{A}^{\ast })}}$$

Hence, the Endemic Equilibrium satisfies,

$$\begin{array}{cc}\hfill P_e=& (S^{\ast },A^{\ast },E^{\ast },I^{\ast },Q^{\ast },R^{\ast })\hfill \\ \hfill =& ({\displaystyle \frac{B+\chi R^{\ast }}{\varpi +\xi I^{\ast }+\alpha }},{\displaystyle \frac{\alpha S^{\ast }}{\varpi +{\varphi }_2+{\varphi }_1{I}^{\ast }}},{\displaystyle \frac{I^{\ast }(\xi S^{\ast }+{\varphi }_1{A}^{\ast })}{\varpi +\tau }},\hfill \\ & {\displaystyle \frac{\tau (\xi S^{\ast }+{\varphi }_1{A}^{\ast })}{(\varpi +\tau )(\varpi +\zeta +{ϵ}_1+{ϵ}_2)-\tau (\xi S^{\ast }+{\varphi }_1{A}^{\ast })}},{\displaystyle \frac{{ϵ}_1{I}^{\ast }}{\varpi +\zeta +\delta }},{\displaystyle \frac{{ϵ}_2{I}^{\ast }+\delta Q^{\ast }+{\varphi }_2{A}^{\ast }}{\varpi +\chi }}).\hfill \end{array}$$

5. Existence of Solution

This section establishes the existence of solutions for the computer virus transmission model governed by the ABC-fractional derivative using fixed-point theory.

Definition 4.

Let ϕ be a function defined on time and $\rho \in (0,1)$. The Atangana–Baleanu (AB) fractional integral of order ρ is defined by,

$${}_0{}^{\mathit{AB}}I_t^{\rho }\varphi \left(t\right)=t_1\left(\rho \right)\varphi \left(t\right)+t_2\left(\rho \right){\int }_0^t{(t-\vartheta )}^{\rho -1}\varphi \left(\vartheta \right)d\vartheta ,$$

(8)

where the coefficients $t_1\left(\rho \right)$ and $t_2\left(\rho \right)$ are given as,

$$t_1\left(\rho \right)={\displaystyle \frac{1-\rho }{B\left(\rho \right)}},t_2\left(\rho \right)={\displaystyle \frac{\rho }{B\left(\rho \right)\tau \left(\rho \right)}}.$$

Consider the Banach space $F\left(j\right)$ of all continuous functions on the interval $\tau =[0,b]$. Define the product space,

$$Q=F\left(j\right)\times F\left(j\right)\times F\left(j\right)\times F\left(j\right)\times F\left(j\right)\times F\left(j\right),$$

by using the norm,

$$\parallel (S,A,E,I,Q,R)\parallel =\parallel S\parallel +\parallel A\parallel +\parallel E\parallel +\parallel I\parallel +\parallel Q\parallel +\parallel R\parallel ,$$

where,

$$\parallel S\parallel =\underset{t\in \tau }{sup}\left|S\left(t\right)\right|,\parallel A\parallel =\underset{t\in \tau }{sup}\left|A\left(t\right)\right|,\parallel E\parallel =\underset{t\in \tau }{sup}\left|E\left(t\right)\right|,$$

$$\parallel I\parallel =\underset{t\in \tau }{sup}\left|I\left(t\right)\right|,\parallel Q\parallel =\underset{t\in \tau }{sup}\left|Q\left(t\right)\right|,\parallel R\parallel =\underset{t\in \tau }{sup}\left|R\left(t\right)\right|.$$

Applying the fractional integral operator to system (2) results in the following integral equations,

$$\begin{array}{cc}\hfill S\left(t\right)-S\left(0\right)& ={}_0{}^{\mathit{AB}}I_t^{\rho }\left[B-\varpi S-\xi SI-\alpha S+\chi R\right],\hfill \\ \hfill A\left(t\right)-A\left(0\right)& ={}_0{}^{\mathit{AB}}I_t^{\rho }\left[\alpha S-\varpi A-{\varphi }_{2}A-{\varphi }_{1}AI\right],\hfill \\ \hfill E\left(t\right)-E\left(0\right)& ={}_0{}^{\mathit{AB}}I_t^{\rho }\left[\xi SI-\varpi E-\tau E+{\varphi }_{1}AI\right],\hfill \\ \hfill I\left(t\right)-I\left(0\right)& ={}_0{}^{\mathit{AB}}I_t^{\rho }\left[\tau E-(\varpi +\zeta )I-{ϵ}_{1}I-{ϵ}_{2}I\right],\hfill \\ \hfill Q\left(t\right)-Q\left(0\right)& ={}_0{}^{\mathit{AB}}I_t^{\rho }\left[{ϵ}_{2}I-(\varpi +\zeta )Q-\delta Q\right],\hfill \\ \hfill R\left(t\right)-R\left(0\right)& ={}_0{}^{\mathit{AB}}I_t^{\rho }\left[{ϵ}_{1}I+\delta Q-\varpi R+{\varphi }_{2}A-\chi R\right].\hfill \end{array}$$

(9)

Using the definition (8), the above can be rewritten as,

$$\begin{array}{cc}\hfill S\left(t\right)-S\left(0\right)& =t_1\left(\rho \right)\zeta (t,S\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }\zeta (t,S\left(t\right)),\hfill \\ \hfill A\left(t\right)-A\left(0\right)& =t_1\left(\rho \right)K_2(t,A\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_2(t,A\left(t\right)),\hfill \\ \hfill E\left(t\right)-E\left(0\right)& =t_1\left(\rho \right)K_3(t,E\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_3(t,E\left(t\right)),\hfill \\ \hfill I\left(t\right)-I\left(0\right)& =t_1\left(\rho \right)K_4(t,I\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_4(t,I\left(t\right)),\hfill \\ \hfill Q\left(t\right)-Q\left(0\right)& =t_1\left(\rho \right)K_5(t,Q\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_5(t,Q\left(t\right)),\hfill \\ \hfill R\left(t\right)-R\left(0\right)& =t_1\left(\rho \right)K_6(t,R\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_6(t,R\left(t\right)),\hfill \end{array}$$

(10)

where the functions $K_i$, $i=1,\dots ,6$, are defined as,

$$\begin{array}{cc}\hfill K_1(t,S)& =B-\varpi S-\xi SI-\alpha S+\chi R,\hfill \\ \hfill K_2(t,A)& =\alpha S-\varpi A-{\varphi }_{2}A-{\varphi }_{1}AI,\hfill \\ \hfill K_3(t,E)& =\xi SI-\varpi E-\tau E+{\varphi }_{1}AI,\hfill \\ \hfill K_4(t,I)& =\tau E-(\varpi +\zeta )I-{ϵ}_{1}I-{ϵ}_{2}I,\hfill \\ \hfill K_5(t,Q)& ={ϵ}_{2}I-(\varpi +\zeta )Q-\delta Q,\hfill \\ \hfill K_6(t,R)& ={ϵ}_{1}I+\delta Q-\varpi R+{\varphi }_{2}A-\chi R.\hfill \end{array}$$

Each $K_i$ satisfies the Lipschitz condition provided that $S,A,E,I,Q,R$ remain bounded.

For instance, for two functions $S\left(t\right)$ and $S^{\ast }\left(t\right)$, the following estimate holds,

$$\begin{array}{cc}\hfill \parallel K_1(t,S)-K_1(t,S^{\ast })\parallel & \le {\delta }_1\parallel S-S^{\ast }\parallel ,\hfill \end{array}$$

where ${\delta }_1$ is a positive constant depending on parameters and variables. Similarly, the other $K_i$ satisfy,

$$\parallel K_i(t,X)-K_i(t,X^{\ast })\parallel \le {\delta }_i\parallel X-X^{\ast }\parallel ,i=2,\dots ,6,$$

with suitable constants ${\delta }_i$. Defining iterative sequences,

$$\begin{array}{cc}\hfill S_n\left(t\right)& =t_1\left(\rho \right)K_1(t,S_{n-1}\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_1(t,S_{n-1}\left(t\right)),\hfill \\ \hfill A_n\left(t\right)& =t_1\left(\rho \right)K_2(t,A_{n-1}\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_2(t,A_{n-1}\left(t\right)),\hfill \\ \hfill E_n\left(t\right)& =t_1\left(\rho \right)K_3(t,E_{n-1}\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_3(t,E_{n-1}\left(t\right)),\hfill \\ \hfill I_n\left(t\right)& =t_1\left(\rho \right)K_4(t,I_{n-1}\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_4(t,I_{n-1}\left(t\right)),\hfill \\ \hfill Q_n\left(t\right)& =t_1\left(\rho \right)K_5(t,Q_{n-1}\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_5(t,Q_{n-1}\left(t\right)),\hfill \\ \hfill R_n\left(t\right)& =t_1\left(\rho \right)K_6(t,R_{n-1}\left(t\right))+t_2{\left(\rho \right)}_0{I}^{\rho }{K}_6(t,R_{n-1}\left(t\right)),\hfill \end{array}$$

(11)

starting from initial data $S_0\left(t\right)=S\left(0\right),\dots ,R_0\left(t\right)=R\left(0\right)$, the differences satisfy,

$$\begin{array}{cc}\hfill \psi S_n\left(t\right)& =S_n\left(t\right)-S_{n-1}\left(t\right),\hfill \\ \hfill \psi A_n\left(t\right)& =A_n\left(t\right)-A_{n-1}\left(t\right),\hfill \\ \hfill \psi E_n\left(t\right)& =E_n\left(t\right)-E_{n-1}\left(t\right),\hfill \\ \hfill \psi I_n\left(t\right)& =I_n\left(t\right)-I_{n-1}\left(t\right),\hfill \\ \hfill \psi Q_n\left(t\right)& =Q_n\left(t\right)-Q_{n-1}\left(t\right),\hfill \\ \hfill \psi R_n\left(t\right)& =R_n\left(t\right)-R_{n-1}\left(t\right).\hfill \end{array}$$

(12)

By applying the Lipschitz conditions and fractional integral properties, one obtains the inequalities,

$$\begin{array}{cc}\hfill \parallel \psi S_n\left(t\right)\parallel & \le t_1\left(\rho \right){\delta }_1\parallel \psi S_{n-1}\left(t\right)\parallel +t_2\left(\rho \right){\delta }_{10}{I}^{\rho }\parallel \psi S_{n-1}\left(t\right)\parallel ,\hfill \\ \hfill \parallel \psi A_n\left(t\right)\parallel & \le t_1\left(\rho \right){\delta }_2\parallel \psi A_{n-1}\left(t\right)\parallel +t_2\left(\rho \right){\delta }_{20}{I}^{\rho }\parallel \psi A_{n-1}\left(t\right)\parallel ,\hfill \\ \hfill \dots & \\ \hfill \parallel \psi R_n\left(t\right)\parallel & \le t_1\left(\rho \right){\delta }_6\parallel \psi R_{n-1}\left(t\right)\parallel +t_2\left(\rho \right){\delta }_{60}{I}^{\rho }\parallel \psi R_{n-1}\left(t\right)\parallel .\hfill \end{array}$$

(13)

Theorem 1.

If the following condition is met for each $i=1,2,\dots ,6$,

$$t_1\left(\rho \right){\delta }_i+{\displaystyle \frac{t_2\left(\rho \right)}{\rho }}{t}^{\rho }{\delta }_i\le 1,$$

(14)

then for $t\in [0,1]$, there exists a unique solution to the fractional epidemic system (2).

Proof. 

Since the functions $S\left(t\right),A\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right),R\left(t\right)$ are bounded and the functions $K_i$ satisfy the Lipschitz condition, applying the recursive inequality (13) yields geometric convergence of the successive approximations. Specifically,

$$\parallel \psi S_n\left(t\right)\parallel \to 0,\parallel \psi A_n\left(t\right)\parallel \to 0,\dots ,\parallel \psi R_n\left(t\right)\parallel \to 0\mathrm{as}n\to \infty ,$$

which implies the sequences $\left\{S_n\right\},\left\{A_n\right\},\dots ,\left\{R_n\right\}$ are Cauchy and hence converge uniformly in $C\left(\tau \right)$. By passing to the limit in (11), the limit functions satisfy system (2), guaranteeing existence and uniqueness of the solution. □

6. Numerical Scheme

Toufik and Atangana [26] developed a highly efficient numerical scheme for solving fractional differential equations with non-singular, non-local kernels. Their approach delivers superior accuracy and rapid convergence, effectively overcoming the limitations of traditional methods such as Euler and Adams–Bashforth when dealing with complex fractional dynamics.

Consider the general nonlinear ABC fractional differential equation,

$${}_0{}^{\mathit{ABC}}D_t^{\rho }\varphi \left(t\right)=f\left(t,\varphi \left(t\right)\right),\varphi \left(0\right)={\varphi }_0.$$

(15)

Using fractional integral theory, this can be equivalently expressed as,

$$\varphi \left(t\right)-\varphi \left(0\right)={\displaystyle \frac{1-\rho }{\Im \left(\rho \right)}}f\left(t,\varphi \left(t\right)\right)+{\displaystyle \frac{\rho }{\Im \left(\rho \right)\tau \left(\rho \right)}}{\int }_0^{t}f\left(y,\varphi \left(y\right)\right){(t-y)}^{\rho -1}dy.$$

(16)

At the discrete grid points $t=t_{\kappa +1}$ for $\kappa =0,1,2,\dots$, the above equation takes the form,

$$\varphi \left(t_{\kappa +1}\right)-\varphi \left(0\right)={\displaystyle \frac{1-\rho }{\Im \left(\rho \right)}}f\left(t_{\kappa },\varphi \left(t_{\kappa }\right)\right)+{\displaystyle \frac{\rho }{\Im \left(\rho \right)\tau \left(\rho \right)}}{\int }_0^{t_{\kappa +1}}f\left(y,\varphi \left(y\right)\right){(t_{\kappa +1}-y)}^{\rho -1}dy.$$

(17)

On each subinterval $[t_l,t_{l+1}]$, the function $f(y,\varphi (y\left)\right)$ is approximated by the linear interpolant,

$$f(y,\varphi \left(y\right))\approx {{\rm Y}}_l\left(y\right)={\displaystyle \frac{y-t_{l-1}}{h}}f(t_l,{\varphi }_l)-{\displaystyle \frac{y-t_l}{h}}f(t_{l-1},{\varphi }_{l-1}),$$

where $h=t_l-t_{l-1}$. Substituting this into (17) and performing the integration yields,

$$\begin{array}{cc}\hfill {\varphi }_{\kappa +1}& ={\displaystyle \frac{1-\rho }{\Im \left(\rho \right)}}f(t_{\kappa },{\varphi }_{\kappa })+{\displaystyle \frac{\rho }{\Im \left(\rho \right)\tau \left(\rho \right)}}\sum _{l=0}^{\kappa }[{\displaystyle \frac{f(t_l,{\varphi }_l)}{h}}{\int }_{t_l}^{t_{l+1}}(y-t_{l-1}){(t_{\kappa +1}-y)}^{\rho -1}dy\hfill \\ \hfill & -{\displaystyle \frac{f(t_{l-1},{\varphi }_{l-1})}{h}}{\int }_{t_l}^{t_{l+1}}(y-t_l){(t_{\kappa +1}-y)}^{\rho -1}dy]+{\varphi }_0.\hfill \end{array}$$

(18)

Define the integrals,

$$\begin{array}{cc}\hfill {{\rm Y}}_{\rho ,l,1}& ={\int }_{t_l}^{t_{l+1}}(y-t_{l-1}){(t_{\kappa +1}-y)}^{\rho -1}dy,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {{\rm Y}}_{\rho ,l,2}& ={\int }_{t_l}^{t_{l+1}}(y-t_l){(t_{\kappa +1}-y)}^{\rho -1}dy.\hfill \end{array}$$

(20)

Closed-form solutions for these integrals are given by,

$$\begin{array}{cc}\hfill {{\rm Y}}_{\rho ,l,1}& =h^{\rho +1}{\displaystyle \frac{{(\kappa -l+1)}^{\rho }(\kappa -l+\rho +2)-{(\kappa -l)}^{\rho }(\kappa -l+2+2\rho )}{\rho (\rho +1)}},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {{\rm Y}}_{\rho ,l,2}& =h^{\rho +1}{\displaystyle \frac{{(\kappa -l+1)}^{\rho +1}-{(\kappa -l)}^{\rho }(\kappa -l+1+\rho )}{\rho (\rho +1)}}.\hfill \end{array}$$

(22)

By substituting these results into (18), we obtain the fully discrete scheme,

$$\begin{array}{cc}\hfill {\varphi }_{\kappa +1}& ={\displaystyle \frac{1-\rho }{\Im \left(\rho \right)}}f(t_{\kappa },{\varphi }_{\kappa })+{\displaystyle \frac{\rho }{\Im \left(\rho \right)}}\sum _{l=0}^{\kappa }[{\displaystyle \frac{h^{\rho }}{\tau (\rho +2)}}f(t_l,{\varphi }_l)\hfill \\ & \left[{(\kappa -l+1)}^{\rho }(\kappa -l+\rho +2)-{(\kappa -l)}^{\rho }(\kappa -l+2+2\rho )\right]\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{h^{\rho }}{\tau (\rho +2)}}f(t_{l-1},{\varphi }_{l-1})\left[{(\kappa -l+1)}^{\rho +1}-{(\kappa -l)}^{\rho }(\kappa -l+1+\rho )\right]]+{\varphi }_0.\hfill \end{array}$$

(24)

Applying the scheme (23) to each compartment $X\in \{S,A,E,I,Q,R\}$ in system (2), governed by,

$${}_0{}^{\mathit{ABC}}D_t^{\rho }X\left(t\right)=f_X(t,X\left(t\right)),$$

leads to the update formula,

$$X_{\kappa +1}={\displaystyle \frac{1-\rho }{\Im \left(\rho \right)}}{f}_X(t_{\kappa },X_{\kappa })+{\displaystyle \frac{\rho }{\Im \left(\rho \right)}}\sum _{l=0}^{\kappa }{\displaystyle \frac{h^{\rho }}{\tau (\rho +2)}}\left[f_X(t_l,X_l)Z{\kappa ,l}^{\left(1\right)}-f_X(t_{l-1},X_{l-1})Z{\kappa ,l}^{\left(2\right)}\right]+X_0,$$

where,

$$\begin{array}{cc}\hfill Z{\kappa ,l}^{\left(1\right)}& ={(\kappa -l+1)}^{\rho }(\kappa -l+\rho +2)-{(\kappa -l)}^{\rho }(\kappa -l+2+2\rho ),\hfill \\ \hfill Z{\kappa ,l}^{\left(2\right)}& ={(\kappa -l+1)}^{\rho +1}-{(\kappa -l)}^{\rho }(\kappa -l+1+\rho ),\hfill \end{array}$$

and the functions for each compartment are,

$$\begin{array}{cc}\hfill f_S& =B-\varpi S-\xi SI-\alpha S+\chi R,\hfill \\ \hfill f_A& =\alpha S-\varpi A-{\varphi }_{2}A-{\varphi }_{1}AI,\hfill \\ \hfill f_E& =\xi SI-\varpi E-\tau E+{\varphi }_{1}AI,\hfill \\ \hfill f_I& =\tau E-(\varpi +\zeta )I-{ϵ}_{1}I-{ϵ}_{2}I,\hfill \\ \hfill f_Q& ={ϵ}_{2}I-(\varpi +\zeta )Q-\delta Q,\hfill \\ \hfill f_R& ={ϵ}_{1}I+\delta Q-\varpi R+{\varphi }_{2}A-\chi R.\hfill \end{array}$$

7. Numerical Analysis

Extensive numerical simulations were conducted to explore the transmission dynamics of the computer virus using the ABC fractional-order model. The system of differential equations was solved using a Runge–Kutta method to gain preliminary insights into the system’s behavior before considering the complexities introduced by fractional calculus. The initial conditions were set to represent a largely susceptible network as follows: $S\left(0\right)=20$, $A\left(0\right)=10$, $E\left(0\right)=0$, $I\left(0\right)=1$, $Q\left(0\right)=0$, $R\left(0\right)=0$. Model-stimulated parameters, informed by realistic network and infection dynamics, were assigned as: $B=20$, $\varpi =0.01$, $\xi =0.02$, $\chi =0.03$, ${\varphi }_1=0.1$, ${\varphi }_2=0.02$, $\tau =0.07$, ${ϵ}_1=0.003$, ${ϵ}_2=0.2$, and $\delta =0.001$. The simulations were run over an adequate time span to capture the complete progression of the virus spread, including infection peaks and subsequent recovery phases. The results, displayed across several figures, provide detailed insights into the evolution of the system across different states of vulnerability and protection within the network. Figure 2 represents the population of susceptible computers and devices at risk of infection. Initially, their numbers declined due to exposure to infected systems and the implementation of antivirus defenses. Transmission coefficients and cybersecurity measures in place influence the rate of decline. Figure 3 highlights the antidotal computers, referring to systems that have received antivirus updates. The graph demonstrates how security patches mitigate infection risks and enhance network resilience.

Figure 4 illustrates the exposed computer devices that have encountered infected systems but are not yet compromised. Latency factors and the network interaction frequency govern the transition from exposure to infection. Figure 5 portrays the infected computers, emphasizing the virus’s progression without timely security updates. The peak infection levels indicate the outbreak’s severity and the effectiveness of containment strategies. Figure 6 depicts quarantined computers, which are isolated to prevent further viral transmission. A growing number in this category reflects the success of containment efforts in reducing the spread. Figure 7 showcases the recovered computers, which have undergone disinfection or received security updates. Due to the temporary nature of immunity, some systems may revert to the susceptible state, thereby sustaining the risk of reinfection. The 3D numerical simulations in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show how time and the fractional-order parameter influence the behavior of all compartments in the computer virus model. Figure 8 describes the surface plot for the susceptible class, showing a gradual decrease as time progresses, due to the systems losing susceptibility because of exposure and infection. Figure 9 describes the surface plot for the antidotal (patched) class, showing that initially, the surface rises before leveling off. This reflects the buildup of patched systems, which stabilizes once the rates of patching and transition are equal. Figure 10 describes the exposed class, where the surface plot takes the form of a hump-shaped surface. This shows that systems linger in an incubation-like stage before going into an infected status. The width of the hump changes with the fractional parameter, reflecting memory effects on the length of time systems stay exposed. Figure 11 shows the infected class as a function of time and fractional-order parameter as a large peak. The fractional order influences the peak’s width and height, where lower orders produce wider, smoother peaks (outbreaks), and higher orders produce narrower spikes. Figure 12 shows the quarantined class in the form of a delayed ridge that increases after the peak of infected classes has risen. It is representative of a natural delay between the point at which an infection is detected and isolation occurs. Finally, Figure 13 describes the recovery class as a surface that steadily rises and eventually approaches a plateau. It characterizes the gradual stabilization of the system as the infected or quarantined machines get back to a safe status. These 3D plots taken together thus illustrate how the timing and fractional-order memory affect the process of malware transmission across all compartments. The findings from this computational study validate the proposed model, demonstrating how key factors such as antivirus interventions, quarantine measures, and recovery mechanisms influence virus transmission dynamics in computer networks. By analyzing these trends, cybersecurity strategies can be refined to enhance protection against digital threats and improve overall system security.

8. Conclusions

In this study, we formulate and analyze a mathematical model describing computer viruses’ spread across connected digital systems. This work presents a comprehensive analysis of computer virus propagation using an Atangana–Baleanu–Caputo fractional-order framework, which effectively captures the memory-dependent nature of digital infections. By integrating six distinct components into the SAEIQRS model, the study reflects the nuanced dynamics of virus spread and antivirus responses within a cyber-physical environment. The derivation of the basic reproduction number, ${\Re }_0$, allows for assessing infection thresholds. At the same time, analysis of sensitivity and the equilibrium states provides insight into the potential long-term behavior of the system. Rigorous mathematical analysis is employed to establish the existence and uniqueness of the model solutions. A suitable numerical scheme is implemented to validate the theoretical findings, highlighting the significant impact of antidotal updates, quarantine strategies, and reinfection risks. Although this work is exploratory in nature, it demonstrates the potential of fractional-order models to capture memory-driven effects that classical approaches often overlook. The study highlights how such models provide a stronger mathematical foundation for analyzing and mitigating malware spread in digital networks. Future research incorporating empirical data and advanced fractional dynamics will further validate and enhance their applicability in IoT and AI-driven environments.