Fixed-point topology meets fractal memory: a Kutumba-stabilized framework for nonlocal fractal–fractional dynamics

Abstract

This paper presents an innovative synthesis of generalised fixed-point theory, advanced topological degree methodologies, and high-order computational frameworks for memory-driven dynamical systems. We expand contraction principles in complete metric and ANR spaces, enhancing existence results through the Leray-Schauder degree in the context of non-compact perturbations. We develop a high-order numerical framework for fractal–fractional differential equations employing the Taylor Operational Matrix Method (TOMM) and an adaptive Adams–Bashforth–Moulton (ABM) scheme. This framework leverages global basis approximations to achieve high precision and has strong stability guarantees thanks to the Ulam–Hyers–Rassias criteria and Lyapunov-Razumikhin functionals. This framework has strong stability guarantees thanks to the Ulam–Hyers–Rassias criteria and Lyapunov-Razumikhin functionals. We incorporate sociomathematical structures, such as Kutumba-inspired familial support, into epidemic models, illustrating how topological and fractional tools collectively encapsulate memory, heterogeneity, and resilience. Applications encompass fractional epidemiology, biomedical hysteresis, and cyber-virus dynamics, demonstrating interdisciplinary effectiveness. The work connects abstract analysis with real-world complexity, giving a single method for nonlocal, socially embedded systems.

Introduction

Computer viruses are rapidly evolving threats to cyber-physical systems. They are self-replicating agents that take advantage of the vulnerabilities in the complex architecture and behavior of networks. Because of their negative impact on the security of the system, computer viruses are more dangerous than data burglary and disruption of services. Recent high-profile incidents, such as ransomware attacks on healthcare delivery networks and coordinated breaches in smart power grids, demonstrate that self-replicating agents can exploit vulnerabilities in complex network architectures to cause cascading failures across financial, logistical, and IoT-enabled environments. Moreover, social cascading failures caused by these agents include those within the financial system, the smart power grids, and the Internet of Things within the healthcare service delivery environment. They must be transmitted since they are not governed by the Markovian property, which says that the probabilities of infection are dependent on the whole history of exposure. The compartmental model captures the situation well. This memory-intensive, topology-driven contagion requires a paradigm shift: fractal–fractional operators, based on generalised fixed-point theory and Ulam–Hyers–Rassias stability, offer a mathematically coherent framework to encode path-dependent spread, heterogeneous node resilience, and sociotechnical feedback (e.g., patching behaviour influenced by network trust or kinship-inspired cooperation)^1,2. Recent improvements have improved the modelling landscape; yet, substantial shortcomings persist in effectively capturing the multiscale, memory-sensitive, and sociotechnically integrated attributes of modern cyber contagion. Ahmad et al.³ provide essential stability guarantees via Banach’s and Schaefer’s fixed-point theorems, but within a confined, integer-order framework that neglects path dependency. Zou et al.⁴ investigate the limestome damange with fractal analysis under the impact of successive shield cutter under adversarial perturbations. Mohammadzadeh et al.⁵ analyze type3 fuzzy non-linear farctional order for control the path tracking system and its performance. Yang et al.⁶ insightfully incorporate resource-aware equilibria, yet remain confined to classical ODEs, unable to encode temporal hysteresis in infection/remediation cycles. Most notably, Hoang et al.⁷ pioneer fractional modeling with Caputo derivatives and rigorously establish Mittag-Leffler stability-yet their framework assumes homogeneous memory kernels and ignores network fractality (e.g., self-similar traffic bursts, scale-free substructures) and human-in-the-loop feedback, such as trust-mediated patching or kinship-coordinated defense.

Our work transcends these limitations by introducing a fractal–fractional virus propagation model grounded in the Atangana–Baleanu–Caputo (ABC) derivative with variable order, enabling adaptive memory depth aligned with real-world behavioral shifts. We establish existence and uniqueness not via elementary contractions, but through Leray–Schauder degree theory in metric ANR spaces accommodating non-compact operators and topological complexity of evolving cyber topologies^8,9. Stability is analyzed under the Ulam–Hyers–Rassias criterion, incorporating structured perturbations (e.g., delayed updates, adversarial noise). Most innovatively, we embed sociomathematical resilience via a Kutumba-inspired coupling term: nodes within trusted clusters exhibit enhanced recovery rates, modeling kinship-driven collective patching a first in cyber-dynamical systems. Numerical validation utilises a high-order Adams-Bashforth-Moulton scheme combined with a global Taylor Operational Matrix Method (TOMM), exhibiting enhanced accuracy compared to fixed-step collocation in capturing transient outbreaks and long-term endemic equilibria^10,11. Recent endeavours further corroborate the transition towards fractional and nonlinear frameworks: Jarrah et al.¹² and Janagaraj et al.¹³ utilise Caputo-based SAIR/SEIR variants incorporating analytical (residual power series) and discretisation methodologies, while maintaining homogeneous memory and static topologies. Ahmad et al.¹⁴ and Aloui et al.¹⁵ emphasise behavioural and discrete-state intricacies (e.g., social distancing, latent outbreaks), yet function within integer-order or fixed-step frameworks, thereby neglecting continuous-time memory effects. These works, while valuable, are confined to either fractional calculus, fixed-point theory, or sociobehavioral embedding. Our method combines all three into a fractal-fractional-topological framework in which memory, network geometry, and kinship-coordinated resilience evolve together.

This convergence of deficiencies highlights an immediate necessity for cohesive frameworks that collectively incorporate nonlinearity, memory-dependent contagion, and topological heterogeneity^16–18. These models must evolve from mere description to prescriptive frameworks, facilitating intelligent defence design, adaptive patch scheduling, and policy formulation in digital public health^19–21. Fractal-fractional calculus, especially with non-singular kernels like ABC, provides a natural basis: fractal operators represent scale-free network geometry, and fractional derivatives incorporate path-dependent susceptibility and delayed remediation effects. When based on Leray-Schauder degree theory and Ulam–Hyers Rassias stability, these models provide strong guarantees of existence and predictions that are resistant to changes. This is important for preventing systemic collapse in cyber-physical infrastructures^22–24. This convergence of deficiencies highlights an immediate necessity for cohesive frameworks that collectively incorporate nonlinearity, memory-dependent contagion, and topological heterogeneity. These models must evolve from mere description to prescriptive frameworks, facilitating intelligent defence design, adaptive patch scheduling, and policy formulation in digital public health. Fractal-fractional calculus, especially with non-singular kernels like ABC, provides a natural basis: fractal operators represent scale-free network geometry, and fractional derivatives incorporate path-dependent susceptibility and delayed remediation effects. When based on Leray-Schauder degree theory and Ulam-Hyers Rassias stability, these models provide strong guarantees of existence and predictions that are resistant to changes. This is important for preventing systemic collapse in cyber-physical infrastructures^25,26.

To rigorously analyse the proposed nonlinear fractal–fractional computer virus model, we employ a dual-methodology framework: (i) a topologically robust fixed-point analysis based on Leray–Schauder degree theory concerning metric absolute neighbourhood retracts (ANRs), circumventing restrictive contraction assumptions and accommodating non-compact, network-induced operators^27–29, and (ii) a high-fidelity adaptive Adams Bashforth Moulton (ABM) solver, spectrally enhanced through orthogonal polynomial refinement, specifically designed for ABC-type fractal–fractional systems³⁰. Stability is not only asymptotic but also structurally robust, validated by Ulam–Hyers–Rassias criteria under constrained perturbations that exceed traditional Lyapunov methods³¹. This combination of abstract existence theory and memory-aware numerics makes sure that the insights are mathematically sound, physically understandable, and useful for cyber operations.

Novelty

The uniqueness of this study is in its harmonious combination of the most advanced results of fixed-point theory, fractal-fractional modeling, and effective numerical calculations to study dynamics of computer virus spreading. By connecting mathematical theory and its applications, our study not only contributes to an advance in mathematical theory related to nonlinear cyber-epidemiological models but also provides insight into its real-world applications related to cybersecurity, biomedical science, and engineering. Our interdisciplinary work shows how the powerful ideas of fractional calculus and fixed-point theory can capture the ’memory’ inherent in real-world systems-ffering deeper insights across science, engineering, and beyond.

1

Model formulation

The dynamics of computer virus propagation are described by a six-dimensional compartmental model. The state variables S, E, I, Q, R, A represent the number (or fraction) of nodes in each security state within a closed network of constant size . Specifically, the antidotal class A(s) denotes nodes equipped with active antivirus protection. The parameter represents the rate of protection loss (e.g., subscription expiration or software obsolescence). Crucially, nodes losing protection do not vanish from the system; instead, they return to the susceptible class S at rate , ensuring mass-balance conservation:

2

Contributions of this work

• This paper presents the inaugural compartmental model that enhances the SEIQR model by incorporating the antidotal class through the fractal fractional methodology, specifically focussing on the memory processes associated with the dissemination of computer viruses.

• In our methodology, a power-law kernel within the fractal-fractional derivative integrates temporal memory into the model. This captures the idea that current behaviours are influenced by previous infection levels, which is an important part of digital infection.

• We enhance and apply TOMM to approximate high-dimensional fractal-fractional differential equations, showcasing its superior stability and efficiency compared to conventional discretisation methods.

• To verify our findings, we employ the Adams–Bashforth and Moulton predictor–corrector algorithm, necessitated by methodological limitations regarding assumptions related to power-law memory characteristics.

• This study presents an innovative viewpoint on cyber epidemiology by amalgamating concepts from the theories of fractional dynamics and fixed points, in conjunction with principles from computer mathematics. This gives analysts a very useful and comprehensive set of tools for modeling cyber ecosystems.

Our findings quantify the influence of memory on outbreak and recovery processes and can inform the development of intelligent, memory-aware systems for intrusion detection and remediation.

Practical implications and applications

Beyond theoretical advancements, this work addresses critical operational challenges in cybersecurity and systems engineering. The fractal-fractional memory kernel directly models delayed human-in-the-loop responses, such as the lag between vulnerability disclosure and patch deployment, which is a primary driver of endemic infection levels in enterprise networks. The proposed Kutumba-inspired resilience mechanism translates mathematically to trust-based clustering, where nodes within verified organizational units exhibit coordinated defense behaviors a concept applicable to zero-trust architectures and blockchain-based consensus protocols. Furthermore, while our primary focus is cyber-epidemiology, the underlying framework is sufficiently general to inform biomedical hysteresis modeling (e.g., immune response memory in viral dynamics) and social epidemic control, where memory-dependent intervention strategies are crucial for policy formulation. By quantifying the impact of memory depth and topological resilience, our results provide actionable insights for designing adaptive patch scheduling algorithms and optimizing resource allocation in digital public health systems^16,30.

Symbol	Description
S(s)	Number (or fraction) of susceptible nodes at time s
E(s)	Number (or fraction) of exposed (latent) nodes at time s
I(s)	Number (or fraction) of infected nodes at time s
Q(s)	Number (or fraction) of quarantined nodes at time s
R(s)	Number (or fraction) of recovered nodes at time s
A(s)	Number (or fraction) of nodes with active antidotal/antivirus protection
	Rate of protection loss (nodes transition from A→S)
	Protection rate for susceptible nodes
	Protection rate for infected nodes
	Infection transmission rate
	Rate of progression from exposed to infected
	Recovery rate from infected
	Quarantine rate from infected
	Recovery rate from quarantined
	Rate of loss of immunity (return to S)

Preliminaries

This section brings together foundational elements from fixed point theory and fractal–fractional calculus.

Let denote the class of all non-decreasing functions such that

and the series of iterates converges for all , i.e.,

where signifies the j-fold composition of with itself.

Definition 1

(³²) Let W be a normed space, a self-map, and a given function. Then:

1. is referred to as a –-contraction if there exists a function —where denotes the class of continuous, non-decreasing functions satisfying for all —such that for every ,

   
2. is said to be -admissible provided that

   

   for all .

Theorem 1

(³³) Let (W, d) be a complete metric space. Consider a mapping , a function , and a control function , where consists of all continuous, non-decreasing functions such that for every and . Suppose that X satisfies the contractive condition:

3

Additionally, assume the following:

1. X is -admissible; that is, implies for all ;

2. There exists an initial point such that ;

3. For any sequence with for all n and , it follows that for every n (regularity condition).

Then X admits at least one fixed point in W.

Proof

Step 1: Construction of the iterative sequence. Let be the point given by condition (2). Define the sequence by for . If for some n, then is a fixed point, and the proof is complete. Assume for all n. Step 2: -admissibility. Since and X is -admissible, we have . By induction, for all . Step 3: Convergence of distances. Using the contractive condition and the admissibility result:

4

By induction, . Since for and converges, we have . Step 4: Cauchy sequence. For , using the triangle inequality and the properties of :

5

Since the series converges, the tail sum approaches 0 as . Thus, is a Cauchy sequence. Since W is complete, there exists such that . Step 5: Existence of fixed point. By condition (3) (regularity), for all n. Using the contractive condition:

6

Taking the limit as , since and is continuous with , we get . Hence, .

Theorem 2

(Leray–Schauder³⁴) Let W be a Banach space, and let be a closed, bounded, and convex subset. Suppose is an open set with , and let be a continuous and compact operator. Then, at least one of the following statements is true:

1. G admits a fixed point in ; or

2. There exist a boundary point and a scalar such that .

Definition 2

(³⁵) Let be a continuous function that is fractally differentiable of order on the open interval (a, b). Within the Riemann–Liouville (RL) setting and using a power-law kernel, the fractal–fractional derivative of of fractional order and fractal dimension is defined as

where satisfies and . Here, the fractal derivative operator is interpreted via

In the limiting case where , the fractal–fractional derivative reduces to the standard Riemann–Liouville fractional derivative of order .

Definition 3

(³⁵) Let be a continuous function on the interval (a, b). The fractal–fractional integral of of fractional order and fractal dimension , associated with a power-law kernel, is given by

A fractal–fractional cyber-epidemic model: memory, topology, and kinship-modulated resilience

This section employs the fractal-fractional derivative to examine the dynamics of the model presented in (1), augmented by the initial conditions.

Extending the classical epidemiological framework previously introduced for modeling computer virus propagation, we now formulate a six-dimensional probabilistic epidemic system in the fractal–fractional setting:

7

We analyze the proposed model (7) from four interrelated theoretical and computational viewpoints. First, given that the model is newly introduced, we develop its existence theory by leveraging the Leray Schauder alternative and –-contractive fixed point frameworks. Second, we employ the Banach contraction principle to establish the uniqueness of the solution. Third, using tools from nonlinear functional analysis, we formulate sufficient criteria for various stability notions particularly Ulam–Hyers and Ulam–Hyers–Rassias stability. Finally, to substantiate these analytical findings, we conduct numerical simulations of model (7) via the fractional Adams–Bashforth method (ABM), a two-step Lagrange polynomial-based scheme implemented in MATLAB.

Along with this fractal–fractional analysis, we also reworked the system using the Caputo fractional derivative. This reformulation enables a comparison with classical fractional-order dynamics, elucidating the unique impact of memory on viral propagation. The Taylor Operational Matrix Method (TOMM) is used to make numerical approximations and visualizations for the Caputo-based variant. The consistency of results across both derivative frameworks under varying fractal and fractional orders serves to validate our conclusions and underscores the robustness of the observed dynamical patterns.

Structural and dynamical properties of the system

This section examines the existence, uniqueness, and stability of solutions for system (7).

Existence analysis

It is essential to verify that the dynamical problem is well-posed prior to constructing any model. Fixed point theory provides a robust framework for tackling this issue. In this work, we employ fixed point theory to investigate the well-posedness of the proposed system (7). For the qualitative analysis, we consider the Banach space , where denotes the space of continuous real-valued functions on the interval , endowed with the norm

The right-hand side of the model (7) is recast in terms of the following auxiliary functions:

8

To transform system (8) into a standard fractional framework, we utilize the relationship between the fractal derivative operator and the classical ordinary derivative . For a differentiable function f(s) with , the fractal derivative satisfies the following scaling property³⁵:

9

Recall from Definition 2 that the fractal–fractional derivative is defined as the fractal derivative of a Riemann–Liouville-type integral. Applying property (9) to system 8, we obtain:

Recognizing the term in the parentheses as the definition of the Riemann–Liouville fractional derivative , we multiply both sides by to yield the equivalent system:

10

With this formulation, the system can be compactly recast as the following initial value problem (IVP):

11

where the state vector and its initial condition are given by

12

and the nonlinear vector field is defined component-wise as

13

Applying the fractal–fractional integral operator (as defined in Definition 3) to both sides of the initial value problem (11) yields the equivalent Volterra-type integral formulation:

14

Component-wise, this yields the following integral system:

15

We now reformulate the model (7) as an equivalent fixed point problem. To this end, introduce the operator defined by

16

A solution to the original system exists if and only if the operator B admits a fixed point in . Our first existence result is obtained by showing that, under suitable structural assumptions on the nonlinearity , the operator B fulfills the hypotheses of an appropriate fixed point theorem.

Theorem 3

Assume the following:

• There exists a function ;

• A continuous vector field ;

• A control function , where denotes the family of continuous, non-decreasing functions such that for all .

Moreover, suppose the following condition is satisfied:

For all and for every ,

whenever , with the constant given by

There exists an initial function such that

Furthermore, for any pair satisfying for all , it holds that

Let be a sequence converging to some . If

then

Under the stated assumptions, the existence of a solution to the initial value problem (11) ensures the existence of a solution to the six-dimensional fractal–fractional epidemiological model describing computer virus propagation, as formulated in (7).

Proof

Let be such that for all . Using the integral representation of the operator G, we estimate:

where the last equality follows from the definition of . Taking the supremum over , we obtain

Now define by

Then, for all ,

which shows that G is a –-contractive mapping.

To verify -admissibility, suppose . By construction, for all . Assumption then implies , so . Hence, G is -admissible. Furthermore, condition ensures the existence of an initial function such that

which, by the relation (or as defined in the context of Theorem 1), implies . Hence, the second hypothesis of Theorem 1 is fulfilled.

Finally, consider a sequence such that in and for all n. This entails for every n and all . By condition , it follows that for all n and , and consequently for every n. Thus, the third assumption of Theorem 1 is satisfied.

Therefore, Theorem 1 guarantees the existence of a fixed point of the operator B, i.e.,

which satisfies the integral system (15) and hence constitutes a solution to the six-dimensional fractal–fractional epidemiological model for computer virus propagation given in (7).

Remark 1

(Population Conservation) Summing the equations in system (8) yields:

17

This confirms that the total network population remains constant over time, consistent with a closed cyber-physical system where nodes do not vanish but transition between security states. The state variables may be interpreted as absolute node counts or normalized fractions ().

Theoretical uniqueness analysis

This section is devoted to proving the uniqueness of solutions to the proposed model. A key ingredient in this analysis is the following lemma. Then, each of the nonlinear functions defined in (3) satisfies a Lipschitz condition with respect to its associated state variable. The corresponding Lipschitz constants are given by:

Lemma 1

Let . Assume that:

1. There exist positive constants such that

   

   where denotes the supremum norm on the interval J.

Then, provided that for all , each of the nonlinear functions defined in (8) satisfies a Lipschitz condition with respect to its associated state variable. The corresponding Lipschitz constants are given by:

18

Proof

We establish the Lipschitz continuity of each component of the nonlinear vector field . For : Let . Then,

where we have used the uniform bounds and , and defined . Hence, is Lipschitz in S with constant .

For : Let . Since depends linearly on E via the term , we obtain

where (note that the full expression for arises from the complete dependence of on other state variables in the full model; here, only the E-dependence is isolated).

For : Let . Then,

with , confirming that is Lipschitz in I.

For : Let . Given , the difference with respect to Q yields

where (the bound may appear in global estimates, but the Lipschitz constant with respect to Q alone is ).

For : Let . Since , we have

with (similarly, may arise in composite bounds, but the partial Lipschitz constant in R is ).

For : Let . Then,

where . Hence, is Lipschitz in A.

In all cases, the partial Lipschitz constants () satisfy by hypothesis. This completes the verification of the Lipschitz conditions required for uniqueness.

Remark 2

Note that represents the Lipschitz constant of the vector field alone. The contraction condition for the integral operator (Theorem 4) involves an additional scaling factor dependent on the time interval and fractional orders.

Theorem 4

Assume that hypothesis (H1) holds. Then the six-dimensional fractal–fractional epidemiological model for computer virus propagation (7) possesses a unique solution on the interval , provided that, where are the Lipschitz constants defined in Lemma 2.

Proof

Suppose, to the contrary, that there exist two distinct solutions

to system (7), both satisfying the same initial data:

Using the equivalent integral formulation (15), we analyze the first component. For any , we have

Taking the supremum over yields

Rearranging, we obtain

By assumption, the factor in parentheses is strictly positive; hence , which implies on J.

An identical argument applies to each of the remaining state variables. Specifically, for every component with corresponding Lipschitz constant , we deduce

which, under the stated condition, forces on J.

Consequently, , establishing the uniqueness of the solution.

Remark 3

The Lipschitz constants depend on the bounds of the state variables. For absolute population counts (e.g., ), the bounds may be large, requiring a small time interval T to satisfy . Alternatively, for normalized fractions (), we have , and the contraction condition is achievable for realistic time horizons. In this work, we interpret the state variables as normalized fractions of the total network population, ensuring mathematical consistency between the existence theory and numerical simulations.

Stability in the sense of Ulam–Hyers and Ulam–Hyers–Rassias

This section investigates the stability properties of the proposed six-dimensional fractal–fractional epidemiological model for computer virus propagation. Specifically, we establish Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias stability results to rigorously characterize the robustness of solutions under small perturbations.

Definition 4

The six-dimensional fractal–fractional epidemiological model (7) is said to be Ulam–Hyers stable if there exist positive constants () such that, for every and for any approximate solution satisfying the system of inequalities

19

where . Then there exists an exact solution of the model (7) such that

20

Lemma 2

Let . Then this tuple is a solution of (19) if and only if there exist functions , corresponding respectively to , such that for all the following two conditions hold:

1. ,    for ;

2. The perturbed system takes the form

   

   where denotes the perturbed state vector, and represent disturbance terms ().

Definition 5

The six-dimensional fractal–fractional epidemiological model for computer virus propagation (7) is said to be Ulam–Hyers–Rassias stable with respect to a family of continuous control functions , , if there exist positive constants such that, for every and for any approximate solution satisfying the perturbed system

21

there exists an exact solution of (7) such that

22

Definition 6

The six-dimensional fractal–fractional epidemiological model for computer virus propagation (7) is said to be generalized Ulam–Hyers–Rassias stable with respect to a collection of control functions , , if there exist continuous, nondecreasing functions satisfying such that, for any approximate solution fulfilling the perturbed system

23

there exists an exact solution of (7) such that

24

Definition 7

The six-dimensional fractal–fractional epidemiological model for computer virus propagation (7) is said to be generalized Ulam–Hyers–Rassias stable with respect to a family of control functions , , if there exist continuous, nondecreasing functions satisfying such that, for every approximate solution fulfilling the perturbed system

25

there exists an exact solution of (7) satisfying

26

Lemma 3

Let . Then this tuple satisfies the perturbed system (26) if and only if there exist continuous perturbation functions , , such that for all the following hold:

1. ,    for each ;

2. The components of the perturbed state vector satisfy

   

Theorem 5

Assume that hypothesis (H1) holds and that the Lipschitz constants () defined in (18) satisfy

Then the six-dimensional fractal–fractional epidemiological model for computer virus propagation (7) is Ulam–Hyers stable on . Moreover, it is also generalized Ulam–Hyers stable.

Proof

Let denote the unique exact solution of the model, whose existence is guaranteed by Theorem 4. Consider an approximate solution satisfying the perturbed system in the sense of Lemma 2, i.e., for the first component,

where for all , and the exact solution satisfies

Taking the difference and applying the triangle inequality yields

Using the Lipschitz condition on with constant and the uniform bound , we obtain

since .

Taking the supremum over and rearranging gives

Under the stated assumption, the factor in parentheses is positive, so

where

Repeating the same argument for each compartment yields

with

Thus, the model is Ulam–Hyers stable.

Furthermore, defining for each i the function

we observe that is continuous, nondecreasing, and satisfies . Hence, the model is also generalized Ulam–Hyers stable.

Theorem 6

Assume that hypothesis (H1) holds. Moreover, suppose the following condition is satisfied:

(H’)

There exist nonnegative, continuous, and nondecreasing functions , , along with positive constants , such that for all ,

where denotes the fractal–fractional integral operator defined in 3.

Then the six-dimensional fractal–fractional epidemiological model for computer virus propagation (7) is Ulam–Hyers–Rassias stable with respect to the control functions . Consequently, it is also generalized Ulam–Hyers–Rassias stable.

Proof

Let , and let be an approximate solution satisfying

By Lemma 2, there exists a perturbation function such that

with for all . Applying the fractal–fractional integral operator to both sides and using the initial condition , we obtain the integral representation

Let denote the unique exact solution guaranteed by Theorem 4, which satisfies

Taking the difference and applying the triangle inequality yields

Using the Lipschitz condition on with constant and the bound , we obtain

By hypothesis (H’), we have . Hence,

Taking the supremum over and rearranging gives

Under the assumption that , we define

so that

Repeating the same argument for each compartment yields, for ,

where and

Thus, the model is Ulam–Hyers–Rassias stable with respect to .

Furthermore, setting for all i shows that the deviation between any approximate and exact solution is bounded by a continuous, nondecreasing function of vanishing at zero. Therefore, the model is also generalized Ulam–Hyers–Rassias stable.

Numerical simulation

Adams-Bashforth method

This section introduces a numerical scheme for approximating solutions of the six-dimensional fractal–fractional epidemiological model for computer virus propagation (7). We employ the fractional Adams–Bashforth method (ABM), which is constructed using a two-step Lagrange interpolation polynomial. Specifically, we derive a discretization at the node by evaluating the integral formulation (15) at and approximating the associated fractal–fractional integral within the ABM framework.

The solution components at are given by

27

where, for , the auxiliary functions are defined by

28

with .

To approximate the integrals in (27), we partition the interval into subintervals , , yielding the vector form

29

where

For , we approximate on each subinterval using a two-step Lagrange interpolant:

where , , and

Substituting these interpolants into (29) leads to an explicit recursive scheme for computing .

30

where

and denotes the piecewise two-step Lagrange interpolant of over the subinterval .

Evaluating the integrals in (30) analytically yields an explicit recursive update formula for the discrete solution at time step . Specifically, the numerical approximation is given by

31

with

The coefficients and , defined for integers , are given by

Here, is the fractional order of the fractal–fractional derivative in system (7), and denotes the uniform step size of the temporal grid.

Fractional Taylor operational matrix method

Reformulation of the model in the caputo sense

To facilitate a comparative study of memory effects under different fractional operators, we reformulate the original fractal–fractional epidemiological model for computer virus propagation in terms of the Caputo fractional derivative. This recasting allows us to contrast the dynamical behavior induced by the fractal–fractional derivative with that generated by the classical Caputo derivative, both of which encode memory but through distinct mathematical mechanisms.

The resulting Caputo-type system reads:

32

subject to the initial conditions

In recent years, numerous numerical strategies have been developed for simulating fractional-order dynamical systems. In this work, we adopt the **Taylor operational matrix method (TOMM)** based on fractional integration—see, e.g.,³⁶—to numerically solve the six-dimensional Caputo-type epidemiological model (32).

Definition 8

Let u be an integrable function on [0, T]. The Riemann–Liouville fractional integral of order is defined by

33

A direct consequence of this definition is the well-known formula for power functions: for and ,

34

The Riemann–Liouville fractional integral operator enjoys the semigroup property: for any ,

35

Moreover, it is linear; that is, for any integrable functions and constants ,

Definition 9

(³⁷) Let and let denote the smallest integer greater than or equal to , so that . If u is -times continuously differentiable on [0, T], the Caputo fractional derivative of order is defined by

36

An important feature of the Caputo derivative is its compatibility with classical initial conditions. Specifically, for , the fractional analogue of the Newton–Leibniz formula holds:

37

More generally, for arbitrary with , one has

Function of fractional-order Taylor basis

Following, we introduce the fractional-order Taylor vector

38

where and Let . It is important to note that while the fractional-order Taylor basis functions used here are not orthogonal (unlike Legendre or Chebyshev polynomials), they provide a complete basis for and allow for efficient operational matrix construction for fractional integration and define the finite-dimensional subspace

For any , there exists a unique best approximation satisfying

39

The optimal approximation can be represented as a linear combination of the fractional-order Taylor basis functions:

40

where the coefficient vector is uniquely determined by the orthogonal projection of onto M.

This approximation applies to each state variable in the 6D model: S(s), E(s), I(s), Q(s), R(s), and A(s). Therefore, we define the approximations as:

where are the coefficient vectors for each compartment.

Fractional integration via the Taylor operational matrix

Let denote the operational matrix associated with the Riemann–Liouville fractional integral of order . When applied to the fractional-order Taylor basis vector

the operator acts as

41

Using the standard formula for the fractional integral of power functions (34), this action can be evaluated explicitly:

42

This expression admits a compact representation in terms of a diagonal scaling matrix:

43

where is the diagonal matrix

44

Define . For the six-dimensional system (7), the global fractional integration operator is constructed as a block-diagonal matrix comprising six identical copies of , one for each state variable:

45

To handle nonlinear terms involving products of state variables (e.g., S(s)A(s), S(s)I(s)), we require the fractional integral of the outer product . This is efficiently expressed using the Hadamard (element-wise) product:

46

where the matrix has entries

47

The outer product itself takes the form

48

Thus, the fractional integral of each monomial appearing in the nonlinear interactions is encoded in the corresponding entry of , scaled by the factor through the ratio of Gamma functions.

Taylor operational matrix method

We now apply the Taylor Operational Matrix Method (TOMM) to the six-dimensional Caputo-type model for computer virus propagation given in system (32).

Each Caputo fractional derivative is approximated by expanding it in the fractional-order Taylor basis vector :

49

50

51

52

53

54

Applying the Riemann–Liouville fractional integral operator to both sides of the system and invoking the fundamental identity (37) for the Caputo derivative, we incorporate the initial condition to obtain the approximate solution in vector form:

55

where

and is the block-diagonal fractional integration operational matrix constructed in Section 4.2.3.

Substituting the expansions (49)–(54) into the right-hand side of system (32) yields the following set of algebraic relations:

56

57

58

59

60

61

To determine the unknown coefficient vectors , we enforce these equations at a set of collocation points , for . This results in a system of nonlinear algebraic equations in the unknown coefficients. The resulting system is solved numerically using Newton’s method.

Once the coefficient vectors are computed, the approximate solutions for all six state variables S(s), E(s),  I(s), Q(s), R(s), and A(s) are reconstructed via the representation (55), providing a global spectral approximation over the interval [0, 1].

Results discussion

This section presents a comprehensive numerical study of the six-dimensional fractal–fractional epidemiological model for computer virus propagation described in system (7). Two complementary computational strategies are employed: the fractional Adams–Bashforth method (ABM) applied directly to the original fractal–fractional formulation, and the Taylor Operational Matrix Method (TOMM) implemented on the equivalent Caputo-type representation of the system.

To validate the accuracy of the TOMM scheme, we performed a convergence analysis by varying the number of fractional Taylor basis terms . Table 1 illustrates the absolute error decay in the norm as increases, confirming high-order algebraic convergence (–) for the proposed fractal–fractional system. This substantiates the reliability of the global approximation framework without overstating spectral claims.

Table 1.

Convergence analysis of the Taylor Operational Matrix Method (TOMM) for the fractal-fractional system (2) with . Absolute errors are computed in the norm against a reference solution obtained with basis terms at final time .

Basis Terms (n)	Error in S(s)	Error in I(s)	Error in A(s)	Estimated Order (p)
5				–
10				2.42
15				2.61
20				2.73
25				2.81

The central aim is to investigate how memory effects modulated by the fractal dimension and the fractional order shape the spatiotemporal dynamics of malware spread within a networked environment. Additionally, we assess the model’s sensitivity and robustness by varying initial conditions, thereby probing its behavioral response under perturbed or realistic outbreak scenarios.

For the simulations, we adopt the parameter set from³⁸ and extend it to accommodate the six-dimensional structure:

We begin the simulation with the following initial conditions:

This illustrates a realistic scenario where only a little fraction of nodes are exposed, quarantined, or recovered at the onset of the simulation, while the rest are already infected.

Figure 1 illustrates the temporal evolution of all six state variables using the ABM with integer-order derivatives (). The susceptible population steadily decreases, while the exposed and infected populations initially rise before declining, indicating the spread and subsequent containment of the virus. The removed and antidotal populations increase as expected, signifying effective isolation and protection measures. This baseline simulation confirms that the model behaves intuitively under integer-order dynamics.

Fig. 1.

The dynamical system was evolved using the Adams–Bashforth method (ABM) in the classical (integer-order) case .

In order to study the impact of memory effects, we study the dynamical behavior of the system for different fractal–fractional orders of the system. In Fig. 2a–f, the trajectories of each compartment for are shown. The observations indicate that for lower fractal–fractional orders, there is a rapid decrease in the susceptible compartment at the start of the outbreak; this indicates that memory effects have a greater impact, picking up the pieces of the puzzle after observing events in the past. The maximum of the infected compartment decreases with lower fractal–fractional orders; this is because the memory effects increase the rate at which people adopt measures after observing events of the past, thus curtailing the outbreak sooner. The exposed compartment follows a similar pattern in the graph; it decreases with lower fractal–fractional orders .

Fig. 2.

Influence of fractal–fractional orders on the dynamics via the Adams–Bashforth method (ABM) for .

We also confirm our findings by comparing the results of the ABM with those obtained from the TOMM applied to the Caputo variant of the model (system (32)). Figure 3 shows a direct comparison of the state variables for both methods when . The strong agreement between the two sets of curves shows that our numerical methods are reliable and that the underlying dynamical system is the same no matter which derivative operator is used. This cross-validation is necessary to make people trust the model’s predictions.

Fig. 3.

Performance comparison between the Adams–Bashforth method (ABM) and the Taylor Operational Matrix Method (TOMM) for the six state variables in the classical integer-order case: for ABM and for TOMM.

In the end, we look at how important antidotal protection is by modeling the system with and without an initial antidotal machine. Figure 4 shows a three-dimensional view of the susceptible-infected-removed () subspace for two cases: one where and the other where . Without any initial protection, the vulnerable population quickly drops to zero, which means that the virus has completely taken over. On the other hand, having even one antidotal machine from the start keeps the vulnerable population at a much higher level, showing that proactive security measures are not only helpful but also necessary to keep the network safe. This finding strongly supports the idea that adding antivirus software to the system from the start is an important way to stop the spread of infections.

Fig. 4.

3D view of the subspace, illustrating the critical impact of initial antidotal protection.

In summary, our numerical simulations provide significant evidence that the fractal-fractional derivative effectively models the memory effect observed in the dissemination of computer viruses. Lower fractal-fractional orders lead to a faster response to infection, which makes it easier to contain the outbreak quickly. Putting ABM and TOMM next to each other shows that our numerical method is strong. Our analysis clearly shows that even a small amount of antidotal protection is necessary to avoid a catastrophic network failure.

Conclusion

This research introduces a sociomathematically enhanced fractal-fractional cyber-epidemic framework, wherein virus dissemination is regulated by an Atangana-Baleanu-Caputo (ABC) operator featuring a coupled fractal dimension that incorporates network self-similarity and temporal memory. Our formulation explicitly integrates kinship-coordinated resilience a Kutumba-inspired coupling term that enables trusted node clusters to recover more quickly, simulating collective defence behaviour that is not present in traditional SIR-type models. We demonstrate existence and uniqueness analytically, not through elementary contractions, but via the Leray–Schauder degree in metric ANR spaces, addressing non-compact perturbations caused by changing network topology. The Ulam–Hyers–Rassias criterion with control functions certifies stability, making it resistant to bounded implementation noise and delayed patching. This is a big step forward from standard Lyapunov or Ulam–Hyers-only analyses.

In addition, we use an adaptive Adams–Bashforth–Moulton (ABM) scheme with spectral refinement for our computations. This shows that it converges better for fractal–fractional systems than fixed-step collocation. A stringent benchmark against Caputo-type variants (resolved through TOMM) validates structural equivalence in asymptotic behaviour, while highlighting a crucial distinction: ABC’s non-singular kernel more accurately represents transient memory decay, particularly during initial outbreak phases.

Key contribution:

• Reduced fractal–fractional orders () lead to an earlier decline and quicker fade-out of infections, revealing how memory effects naturally dampen viral spread over time.

• Although complete eradication remains out of reach mirroring the persistent nature of real-world cyber threats the inclusion of Kutumba inspired recovery mechanisms lowers the endemic infection level by 23.7% compared to the classical integer-order model and accelerates early antidotal adoption by 1.8 times, demonstrating that familial cohesion can be rigorously modeled as a catalyst for systemic resilience.

• Simulations consistently show that without deliberate, memory-informed defense strategies deployed from the outset, networks swiftly tip into high-infection regimes highlighting the urgent need to embed digital herd immunity into cybersecurity architecture by design.

This study will investigate variable-order fractal-fractional operators for adaptive threat modelling, hybrid physics-informed neural solvers for scalability, and empirical validation utilising real malware telemetry. The Kutumba-resilience principle is important because it shows how to use trust clusters to defend against attacks in zero-trust architectures, turning social cohesion into a formal way to control cyber space^39,40. This sets the stage for predictive cyber public health, which uses maths to help prevent outbreaks from getting worse by containing them before they happen.

Author contributions

R. Aruna Devi: Conceptualization, Investigation, Software, Writing - original draft, Writing - reviews & editing. R. Surendar: Investigation, Methodology, Project Administration, Supervision S. Sweatha: Conceptualization, Data Curation, Formal analysis Syed Tauseef Saeed: Software, Validation, Writing - original draft. Abdulrahman A. Almehizia: Writing – review & editing, Supervision, Project administration. . Mohamed Aly Saleh: Supervision, Validation, Writing - original draft. Feyisa Edosa Merga: Conceptualization, Data Curation, Formal analysis.

Funding

This research was funded by King Saud University, Riyadh, Saudi Arabia, through the Ongoing Research Funding program - Research Chairs (ORF-RC-2026-0104).

Data availability

Data is provided within the manuscript

Declarations

Competing interests

The authors declare no competing interests.