https://orcid.org/0000-0002-7979-8064
Figures
Abstract
Wireless Sensor Networks (WSNs) are particularly vulnerable to malware attacks due to their limited processing power, memory, and energy, which makes defending against such threats especially challenging. To mitigate these serious security issues caused by malware infection, various preventive measures can be implemented, such as honeypots, robust security protocols, hardware-based protections, regular updates, firewalls, and intrusion detection systems (IDS). Considering these security concerns, we adopt an advanced version of the existing susceptibleβinfectiousβprotectedβrecovered SIPR model that incorporates a fractional-fractal derivative (FFD) defined in the Atangana-Baleanu-Caputo (ABC) sense, which offers a more realistic representation than the classical model. Furthermore, this research work introduced a new isolated nodes compartment , along with parameters
and
, defining the recovery and isolation rates of
, respectively, in the existing SIPR model. Moreover, this study focuses on the existence and uniqueness of solutions, stability analysis, control theory and numerical approximation for the proposed generalized susceptibleβinfectious isolated-protectedβrecovered
model. Additionally, nonlinear and fixed-point theory are used to obtain the results of existence and stability analysis. On the same line, Newton polynomial-based numerical scheme was established for the proposed modified model. The dynamics of desired results are visualized using MATLAB.
Citation: Shah MI, Hassan EI, Ali A, Muhyi A, Ahmed WE, Aldwoah K (2025) Controlling worm propagation in wireless sensor networks: Through fractal-fractional mathematical perspectives. PLoS One 20(11): e0335556. https://doi.org/10.1371/journal.pone.0335556
Editor: Mahmoud H. DarAssi, Princess Sumaya University for Technology, JORDAN
Received: April 14, 2025; Accepted: October 6, 2025; Published: November 20, 2025
Copyright: Β© 2025 Imad Shah et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2502).
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
This section of research work is committed to the fundamentals of mathematical modeling (MM), fractional calculus (FC), propagation of worms in wireless sensor networks (WSNs), literature survey, and the new findings of this research work.
1.1 Mathematical modeling and fractional calculus
MM plays a prominent role in contemporary mathematics because it connects real-world scenarios with their mathematical representation, enabling systematic investigation and resolution of complex problems [1]. This approach allows researchers to address current challenges while predicting future possibilities [2]. MM has shown wide applications across various disciplines, ranging from engineering to biological systems. MM plays a fundamental role in studying the dynamics of worm spread by employing methodologies in epidemiological models. These models were originally developed for the study of the spread of diseases and have shown effectiveness in understanding the transmission of malware in network systems [3]. Recent studies have demonstrated the flexibility of mathematical methods to model the dynamic propagation of malware alongside the dissemination of antiviral software by considering interactions between malicious attacks and defense mechanisms in cyber environments [4]. Consequently, to address critical vulnerabilities of malicious cybersecurity attacks to networks, the analysis of viral behavior in WSNs focuses on stability and protection of networks [5]. The modeling of dynamical systems and their numerical simulations have become a foundation for advancements in engineering and scientific inquiry [6]. In particular, fractional-order operators have been extensively studied as powerful tools to extend traditional models [7]. These operators in understanding processes that exhibit hereditary properties and memory effects, which are not addressed by classical models of integer order [8].
Fractional calculus (FC) is a field of mathematics that originates and bridges theoretical and empirical investigations with real-world applications. FC has contributed a lot by addressing fundamental mathematical questions by involving appropriate numerical schemes, analytical techniques, and suitable representations across various domains [9]. Fractional derivatives (FDs) also play a crucial role in the development of modeling for frequency-dependent damping behavior in viscoelastic materials, biological phenomena, chemical processes, and engineering problems [10]. Recently, researchers have recognized some differential and integral operators prominent in capturing the complexity of inheritance, in which fractal-differential and integral operators overcome the limitations of conventional approaches [11]. The fractal-fractional derivative (FFD) has advantages over the classical derivatives, due to its important applications in the modeling of multi-faceted real-world phenomena that can not be addressed by classical ones. Contrary to classical derivatives, which focus on local behavior, FFD applies to systems with long-range dependencies or hereditary traits because it accounts for non-local effects and memory properties [12]. The merging of fractal dimensions and fractional calculus integrate the representation of irregular structures and the processes of anomalous diffusion more accurately [13]. FFD operator is helpful in disciplines such as physics, biology, and engineering because it overcomes the inadequacy of classical derivatives for processes with scaling behavior, power-law dynamics, or multi-scale interactions [14]. Also, the non-local effects of FFD improve the modeling of complex systems such as natural and engineered systems containing discontinuous or highly variable functions [15].
1.2 Survey literature on wireless sensor networks
The recent development in information technology has turned out all the fields of science efficiently and connectivity [16]. However, this advancement poses threats like cyber attacks, specifically to wireless sensor networks [17]. National security and safety are increasingly threatened by malicious attacks on networks [18]. Thus, ensuring the security, reliability, and efficiency of WSNs is of the highest priority. WSNs, being a key element of contemporary technological development, play a key role in achieving the desired goals, defining protocols, and gathering information [19]. Networks composed of low-cost, intelligent sensor nodes have proven highly effective in numerous applications ranging from tracking traffic flow, fuel, water conditions, seismic movements, and health parameters [20]. Similarly, these networks also play a critical role in agriculture, disaster mitigation, and environmental contaminant detection [21]. The integration of the internet with WSNs improves real-time information retrieval, facilitates quick decision-making, and improves technological efficiency [22]. However, among the several challenges that persist, one of the most significant threats is malware propagation, which compromises the confidentiality and security of data [23]. Therefore, ensuring the security of WSNs is crucial, particularly in ensuring the reliability of data, such as infrastructure management and healthcare applications. The intersection of MM, FC, and WSNs security represents a dynamic and evolving field in various research fields [24]. This convergence highlights the necessity of modern technologies for addressing the contemporary challenges related to cybersecurity and technological resilience [25].
Global models have observed noteworthy examples, including a review [26] which examines compartmental models based on the Kermack-McKendrick paradigm adapted for malware propagation in WSNs. This study determines that existing models have insufficiently considered energy and memory management, authentication schemes, and sensor mobility. Contemporary research on internet worm propagation relies primarily on epidemic theory [27] with state machine approaches. These classical models include: the Susceptible-Infected-Susceptible
model, the SIR model, the Two-factor model, and the Improved Worm Mitigation (IWM) model [28]. These differential equations provide a framework to effectively describe internet worm propagation characteristics to some extent. The
model extends the
model and is widely applied for analyzing internet worm dynamics. However, these models are not specially designed for WSNs and thus fail to account for energy consumption during propagation. In the SIS model [29], hosts exist in either susceptible or infectious states. A susceptible host may transit to infectious with a certain probability per unit time, while an infectious host may revert to susceptible with a certain probability per unit time (where the probability of Susceptible and infectious are not equal). At time t, with initial host
, the infectious host is
and the susceptible host S(t). Critically, this model neglects host death from severe infections and ignores potential immunity after disinfection. The
model [29] addresses these limitations by adding a removed state. When an infectious host is cleaned, it transitions to removed and gains immunity; susceptible hosts become infectious with a certain probability, while infectious hosts transition to removed with a certain probability. At time t, recovered hosts
complete the state set alongside
and
. While this accounts for immunization/isolation/death, it erroneously assumes susceptible host remain perpetually active. This observed a serious problem for WSNs as expire from energy depletion. This is due to the removed nodes remain vulnerable to new worm strains in large-scale deployments. During the investigation in [30] extends SIR for WSNs where fundamental limitations persist. Various researchers studied the propagation of worms through WSNs from different aspects and published a variety of articles. For a more detail study we referred [31β33] to the readers.
1.3 Generalized SII1PR model
To address the aforementioned challenges, this research work proposed a generalized SII1PR model, which overcomes the above constraints by: introducing an isolated compartment to separate infected nodes, preventing the other nodes from infection. On line by the inter-venting fractal-fractional operator instead of integer-order derivatives, which capturing memory and hereditary effects. This framework maintains mathematical tractability while accurately modeled WSNs specific dynamics like intermittent connectivity and energy-constrained propagation. The newly established proposed model of five states, where each state represents a group of nodes: which are vulnerable to malware attacks and not yet infected. These states are susceptible nodes, infected nodes
, the nodes that are already infected and capable of spreading malware, Isolated nodes
, which are isolated from infected and protected nodes
, which restrain the spread of malware in the network and
recovered nodes which spot and remove malware infections by using a detection mechanism and at any time
the network comprises
nodes. The characteristics and properties posses by all the nodes in proposed model are the same as that of model [34]. A new compartment
, representing isolated nodes, is introduced for nodes that are removed from the network to reduce the propagation worm. The isolated compartment
reflects practical WSNs security measures where infected nodes are forcefully disconnected (but retained in the network for potential recovery).
The present study formulates the generalized model by deriving the governing equations, qualitative analysis, control strategies, and employing numerical techniques to solve the fractal-fractional system. To evaluate the efficiency of the proposed modified model in compared to the classical model by [34], simulations studies will be conducted by considering key parameters such as infection rate, isolation effectiveness, and overall network recovery. The aims of this study are to demonstrate the enhanced resilience of WSNs against worm attacks through the proposed generalized model, offering valuable insights into robust epidemic modeling for network security. The introduction of
, in the classical
model, directly addresses the practical challenges related to isolating infected nodes to prevent further propagation. Finally, this study proposed a strategy not only to improve the performance of the model but also to provide meaningful insights into how to control the propagation of worms in WSNs.
1.4 Organization of the paper
This article is structured as follows: Sect 1 introduces fractional calculus, the framework for mathematical modeling, and the dynamics of WSN propagation. Sect 2 presents the fundamental materials, including definitions, lemmas, and theorems of FC, necessary for subsequent sections. Sect 3 details the existence and stability results for the generalized fractional WSN mathematical model. Sect 4 introduces the proposed WSN model, covering equilibrium points, the basic reproductive number, and its sensitivity analysis. Sect 5 constructs a numerical scheme based on Newton polynomial for the considered WSN model. Sect 6 provides graphical visualizations and discusses the results. Finally, the study presents the conclusion and future work.
2 Preliminaries
This section establishes the fundamental definitions and concepts of Fractional Calculus (FC), presenting well-known generalizations like the Gamma and Beta functions and their interconnections. We review key definitions, lemmas, and theorems relevant to our reformulated model (12) and discuss essential definitions for stability analysis. This framework supports achieving the main objectives of this investigation.
Definition 1. [35] The Gamma function is defined as:
Certain properties related to Gamma function are mentioned in Table 1.
Definition 2. [36] The Beta function is defined as:
Certain properties related to Beta function are mentioned in Table 2.
Definition 3. [37] Consider that is a continuous function, then the fractal derivative of order
of
is given as:
where its fractional generalized form is given by
Remark 1. Let , where both of its fractional-order and fractal fractional order derivatives exist, we have
Definition 4. [38] Let be continuous on
, with
and
, then the fractal fractional ABC derivative
of
is defined as:
Definition 5. [38] Let the function be continuous on
, such that
and
, then the fractal-fractional integral (FFI) in the sense of
is follows as:
where ,
, and
. The function
is defined as:
Lemma 1. For any function , the solution of
is given by
Definition 6. A function with a Lipschitz constant
is Lipschitz continuous (LC) on
, if
Theorem 1. Let be a convex closed subset of a Banach space
, then
is completely continuous and has at least one fixed point.
Assumption 1. Consider the functions and
. The following norm properties hold:
3 Investigation of generalized WSNs model
In this section, we investigate the proposed generalized model (12) regarding existence and stability. A comprehensive literature review highlights the significance of worm propagation in WSNs and reveals that this research area requires further attention to explore its various aspects. Our study integrates a FFD in the ABC sense into the existing framework, achieving enhanced realism compared to classical models. Furthermore, we extend the model [34] by introducing a new compartment
for isolated nodes, accompanied by parameters
(recovery rate) and
(isolation rate) for
. The advanced
model is formulated as follows:
with ,
,
,
,
with
.
Fig 1 represents the flow chart to describe the dynamics of our proposed model 12) is follow as:
The descriptions of the parameters used in the generalized model is given in Table 3.
3.1 Results of existence
We define the following system of operator equations for the proposed model (12) as follows:
Using the concepts of FF-order derivatives from Definition 5 with the combination of system of operator Eq 13 in proposed model (12), we get
Theorem 2. The solutions of the proposed generalized model (12) are non-negative and bounded.
Proof: From the considered model (12), we have
Thus, for R5 is a positive invariant set and therefore every hyperplane is non-negative and bounded. β‘
Theorem 3. The operators defined in system of Eq 13, namely ,
,
,
and
satisfy the Lipschitz condition (LC) and exhibit contraction properties.
Proof: Consider two solutions and
of first compartment of considered model (12) for kernel
, we have
Subtracting and
and applying the maximum norm, we obtain
Defining , it follows that the operator
satisfies the Lipschitz condition (LC).
Similarly, we can show that the remaining operator also satisfies the LC. β‘
Theorem 4. The solution of proposed SII1PR model (12) is unique, if the following inequality holds true
Proof: Let us consider the first compartment of our considered model (12)
Applying the operator to both sides of Eq (17), we obtain
Similarly, we can obtain the remaining compartments of model (12).
Consider that ,
,
,
, and
are Picard operators, we have
Here, we present boundedness results for the proposed Picard operators as follows:
Since is LC, it implies that
is bounded.
Thus, there exist ,
,
,
, and
constants such that
Thus,
For simplicity, putting the maximum value of to obtain beta function.
Now, assume that , such that
:
Similarly, for , where
we have
Therefore, the operators ,
,
,
, and
are Picard bounded operators.
The operators ,
,
,
, and
satisfy the contraction principle (CP), if
Let .
Now, putting the values of and
we get
Let is bounded by
, i.e.,
. From the definition of
for any constant, we arrive at
Let
Then
If , then
satisfies the CP.
Following the same procedure, we have
For where
the operator satisfies the CP.
Similarly, for , where
the operator satisfies the CP.
For
and
where , the operators
and
satisfy the CP.
Now, to show that the solution of (12) is unique, assuming any two solutions and
for the first compartment of model (12), we have
Thus
From system of Eq 13, we have
Putting and
into (22), we obtain
Since ΞΌ is a constant and is bounded by
, we have
and
. Substituting these values into (23), we obtain
Simplifying (24), we obtain:
Thus, the term cannot be equal to zero, therefore, we have
which gives
Hence, and
are equal, which implies that the first compartment of model (12) has a unique solution.
Thus, we conclude that the operators ,
,
,
, and
are well-defined and satisfy the CPs. Therefore, the solution for our proposed WSNs
model (12) is unique in the sense of the F-ABC derivative. β‘
3.2 Stability analysis
In this section, we provide the results related to the stability analysis of our proposed model (12). To achieve this goal, we first establish the following results:
Definition 7. The solution of proposed (12) is HUS in the FFI sense, if for there exists
such that the following hold:
Let us assume that be an approximate solution for (12) as:
Thus, the system of equations model (12), is Hyers-Ulam Stable, if
Theorem 5. The solution of model (12) is Hyers-Ulam-Stable (HUS), if the system of inequalities (28) holds true.
Proof: Let us take the first compartment of (28), we have
Now, let and
, thus
which implies that the first compartment (12) is HUS. Similarly, we can obtain HUS results for the remaining compartments of the considered model. Therefore, we can say that the proposed system of Eq 12 is HUS. β‘
4 Control theory
In this section, we translate mathematical rigor into cyber-resilience, offering engineers a blueprint to outpace digital contagions. By anchoring worm dynamics to epidemiological principles like R0, the framework acts as an early warning system: if R0<1, networks can choke outbreaks before they metastasize. Sensitivity analysis reveals where defenses matter most, slowing data leaks Ξ² and amplifying proactive patching Ο become frontline tactics, akin to deploying a digital immune response. The fractional-order structure mirrors the messy reality of networks, where delays and legacy vulnerabilities linger, ensuring strategies adapt rather than crumble under uncertainty.
4.1 Equilibrium points
Here, mainly, we deal with the following equilibrium points.
4.1.1 Worm Free Equilibrium points (WFEP).
To find the WFEP of the system (12) algebraically, we set all the derivatives equal to zero by solving the equations of the system, and we get
with
.
Thus, the worm-free equilibrium is
4.1.2 Endemic Equilibrium (EE).
For finding the EE, we consider the following system of equations:
From the second equation, we have
From the third equation, we have
From the fourth equation, we have
From the fifth equation, we have
Further, from the first compartment of model (12), we get
4.2 Basic reproduction number 
The most critical threshold parameter concerning viral transmissibility is the basic reproduction number, typically stated as .
Definition 8. [39] is the average number of new infections that are transmitted by an infected individual throughout the entire period of infectiousness. If
is greater than 1, the number of infected individuals will multiply rapidly and lead to an epidemic.
The Basic Reproduction Number quantifies the inherent βspreadability" of a phenomenon in a system, answering: How widely will one source propagate its influence if left unchecked? In epidemiology, the principles of
apply universally to measure the average downstream impact of a single unit (for example, a node) within a vulnerable network. If
is greater than 1, the effect cascades exponentially, demanding mitigation to prevent system overload. If
is less than 1, the influence disappears naturally. This metric helps design resilient systems by identifying critical thresholds for intervention, optimizing resource distribution, and predicting failure points. Although simplified, the logic of
bridges theoretical models and real-world stability, offering a framework to balance efficiency and robustness in dynamic networks.
In the next-generation matrix (NGM) method, matrix specifically describes the rate of new infections entering the system. In
, we only take
(primary infected nodes) due to biological and mathematical reasons considered within the framework of this model.
is not epidemiologically related to new infections; rather, it is derived from
. The dynamics of the infected compartments
and
are:
Constructing and
, the new infections (
) and transitions (
) matrices are:
New infections matrix :
Transitions matrix :
Spectral radius and , the eigenvalues of
are
Therefore, the basic reproduction number is the dominant eigenvalue of and hence:
4.2.1 Sensitivity analysis of 
.
The basic reproduction number is given by
To quantify the relative impact of parameters on , we compute the normalized sensitivity indices (elasticity) using:
where is a parameter of interest. The following are the sensitivity indices for each parameter.
Transmission Rate ()
Biologically this means that a 1% increase in increases
by 1%.
Protection Rate ()
A 1% increase in decreases
by
.
Parameters in the denominator (
).
For any parameter :
For example: ββ , βββ
.
The key observations for the sensitivity indices of R0 from Table 4 are the following:
- The most sensitive parameter is
, with a direct 1:1 effect on R0.
- The protection (
) is effective for reducing
, especially when
is large.
- Parameters in the denominator (
) reduce
, but their impact depends on their relative magnitudes.
4.2.2 Stability analysis on the bases of R0.
Here, we present the result of stability and visualization of wormβs transmission dynamics for the proposed model based on R0, using the concepts of LaSalleβs principle.
Theorem 6. The worm-free equilibrium (WFE) is locally asymptotically stable when R0<1 and unstable when R0>1.
Proof: We prove this via Jacobian analysis and the Lyapunov method. The Jacobian at WFE is:
The eigenvalues are:
All eigenvalues have negative real parts iff:
For R0<1, define:
Its derivative along trajectories satisfies:
Equality holds only when . By LaSalleβs invariance principle, WFE is globally attractive. When R0>1,
makes
unstable. The transcritical bifurcation at R0β=β1 is verified by computing:
β‘
The bifurcation is forward (a > 0), implying a stable endemic equilibrium for R0>1.
To visualize how the spread of the worm depends on its contagiousness, the following Figs 2 and 3, show the transmission dynamics within the WSNs. They illustrate three distinct outbreak scenarios determined by the basic reproduction number, R0. The value of R0 critically influences whether the infection persists, grows, or dies out.
Understanding how a worm spreads in a WSNs is crucial for developing defenses. The basic reproduction number R0, acts as a key predictor for this spread. Fig 2 illustrates the critical threshold scenario where R0 equals 1; here, each infected node transmits the worm to exactly one new node on average, leading to a persistent but non-expanding infection level. When R0 exceeds 1, as shown in Fig 3, the outbreak grows exponentially because each infection causes more than one subsequent infection, posing a significant threat to the network. Conversely, Fig 4 depicts the safer situation where R0 is less than 1; the worm struggles to spread effectively, causing infections to quickly decline and eventually fizzle out. These dynamics clearly demonstrate how the value of R0 determines the ultimate fate of a worm within the WSNs.
5 Newtonβs polynomial-based numerical scheme
This section details the numerical approximation of model (12) via Newtonβs polynomial approach, integrating interpolation techniques with Newtonβs method using Newton-form polynomials. These methodologies find extensive application across diverse fields such as physics, economics, and computer science.
For model (12), we delineate the numerical implementation scheme through the following steps:
As we know from the Atangana-Baleanu integral, the solution for the model (12) takes a specific form, which can be derived as follows:
Substituting with
, in (32), we have
Now, applying the two step Lagrange polynomial to , gives
where the difference of and
is represented by
.
Solving the integrals involving in Eq 34, we obtain
After simplification, we get
Replacing , we get
Now, applying the numerical scheme to model (12), we have
6 Numerical simulations
This section provides graphical visualizations and a concise discussion of the approximate solutions obtained for our proposed generalized fractional-order WSNs model.
6.1 Graphical analysis and discussion
We have used MATLAB to analyze and generate the graphs given below. The parameter values were obtained in order to get a graphical representation of the dynamics of our proposed model.The model parameters consist of and
, each with specific values and corresponding units. Let us assume the following initial conditions:
Thus, the data presented here lay the foundation for developing the model we are examining. The simulation results of the SII1PR model are shown in Figs 5β20, illustrating the dynamical representation of the time evolution of each compartment. These figures demonstrate to exhibit the dynamics of the system under different parameter values. For example, as the parameters and
increase, there is a notable reduction in the susceptible population S(t) shown in Fig 7. The behavioral pattern of S(t), P(t), and R(t) in Figs 5, 16β19 also substantiates the theoretical findings, affirming the boundedness, convergence, and long-term influence of protection mechanisms. We see that the infected nodes I(t) in Fig 10 first rise and then fall as a result of the impact of isolation and recovery parameters. In addition, the infected I(t) presented and isolated I1(t) illustrated in Figs 10 and 13 exhibit sharp oscillations, reaching their peak before rapidly declining. This behavior suggests an initial aggressive infection that can be mitigated through strategic interventions. Likewise, the isolated class I1(t) in Fig 13 presents an increase followed by stabilization, establishing the efficacy of the containment strategy. Such graphical interpretations justify the modelβs ability to catch worm propagation patterns in WSNs. These dynamics emphasize the modelβs sensitivity to parameter changes and demonstrate the impact of
and
on infection control in wireless sensor networks. The Figs 21, 22, 26, 27, 28 and 29 reveal how cybersecurity βknobs" shape worm outbreaks in WSNs. Infected nodes Fig 21 surge faster with higher transmission rates
, but flatten with stronger protection
. Isolated nodes in Fig 22 peak earlier when isolation is swift
, while faster recovery
shortens isolation periods, but node death
permanently reduces network capacity. Protected nodes Figs 23β25 grow rapidly with preemptive hardening
, though new nodes
introduce vulnerabilities. Recovered nodes Fig 26 thrive with efficient remediation
, but node loss
forces costly replacements. Finally, combined dynamics Figs 27β29 show fractional parameters
capturing real world delays: infections ripple through
, while protection P curbs susceptibility. Together, they prove that early isolation and proactive patching are critical and that fractional calculus models real world quirks like βmemory effects" in nodes.
The following are the graphs for different values of parameters.
In general, the analysis indicates that increasing these parameters leads to the following:
- A faster decline in the susceptible population S(t) shown in Fig 7.
- Rapid fluctuations illustrated in infection I(t) presented in Fig 10 and isolated I1(t) in Fig 13.
- A steady rise in protection P(t) shown in Fig 16, and recovered nodes R(t) presented in Fig 19.
This underscores the effectiveness of strategic interventions in controlling the spread of worms in WSNs.
6.2 Comparative study
To evaluate the effectiveness of the proposed numerical scheme, we conducted comparative simulations with classical methods:
- 4th-order Runge-Kutta (RK-4) and modified Euler method (EM)
- Conventional integer-order case (
)
- MATLAB R2015a with fixed step size hβ=β0.5
- CPU time comparison for the susceptible compartment
We compared the computational efficiency of the numerical schemes by measuring their CPU times, as shown in Table 1. The results demonstrate that our proposed method is more reliable than both RK-4 and EM approaches. Moreover, the NP scheme maintains its effectiveness for fractional-order cases (), where traditional methods like RK-4 fail to apply. This adaptability makes the NP method particularly valuable for modeling real-world phenomena exhibiting memory effects and non-local behavior, such as contaminant transport in aquatic systems.
7 Conclusions
This study presented an advanced mathematical model for analyzing worm-spreading dynamics in WSNs.
- This work extends the traditional (
) model by incorporating an isolated compartment (
), introducing new parameters
(isolation rate) and
(recovery rate of
), and utilizing FFD in the ABC sense. Incorporating an isolated compartment into the model improves its alignment with real-world scenarios by acknowledging nodes that are purposefully isolated to reduce the spread of worms. The ABC fractional derivative is employed to represent memory-influenced behavior and network-wide interactions, which are critical characteristics of malware propagation.
- By studying the
compartmental framework, researchers can identify how the parameters
and
, the measurable distinct and measure functions, are used in managing worm outbreaks within WSNs.
- This analysis helps quantify the efficacy of containment strategies and track the restoration of compromised nodes to normal operation, offering insights into both preventive and reactive measures.
- These parameters, representing transmission dynamics and control interventions, respectively have a significant impact across all compartments. Fixing
and
values produces a typical epidemic curve, whereas varying these parameters scales outbreak severity and duration. Higher values of
accelerate transmission, while increasing
reduces infection peaks, indicating its role in mitigation measures.
- The Isolation patterns inversely reflect infection rates, with
and
influencing the timeliness and severity of isolation measures. The combined patterns
show that the balanced modifications to
and
, such as maximizing transmission suppression while enhancing vaccination efforts, can effectively flatten the infection curves.
Future works
This study established a theoretical foundation for modeling worm propagation in WSNs using fractal-fractional operators. Subsequent research should focus on validating the model empirically through test bed implementations with commercial sensor nodes, examining real-world constraints like dynamic topologies and intermittent connectivity. Further extensions could incorporate multi-vector threat scenarios (e.g., simultaneous malware strains) and energy-aware mitigation strategies that optimize security-energy tradeoffs. Developing machine learning frameworks for real-time parameter calibration and exploring hardware-level isolation mechanisms would enhance practical deployability. Finally, integrating cross-layer network vulnerabilities and post-quantum security considerations would strengthen the modelβs resilience against evolving cyber-physical threats.
References
- 1.
Dym C. Principles of mathematical modeling. Elsevier; 2004.
- 2. Strielkowski W, Vlasov A, Selivanov K, Muraviev K, Shakhnov V. Prospects and challenges of the machine learning and data-driven methods for the predictive analysis of power systems: a review. Energies. 2023;16(10):4025.
- 3. Nwokoye CH, Madhusudanan V. Epidemic models of malicious-code propagation and control in wireless sensor networks: an indepth review. Wireless Pers Commun. 2022;125(2):1827β56.
- 4. Ferdous J, Islam R, Mahboubi A, Islam MdZ. A review of state-of-the-art malware attack trends and defense mechanisms. IEEE Access. 2023;11:121118β41.
- 5. Yu J-Y, Lee E, Oh S-R, Seo Y-D, Kim Y-G. A survey on security requirements for WSNs: focusing on the characteristics related to security. IEEE Access. 2020;8:45304β24.
- 6.
van den Bosch PP, van der Klauw AC. Modeling, identification and simulation of dynamical systems. CRC Press; 2020.
- 7. Patnaik S, Hollkamp JP, Semperlotti F. Applications of variable-order fractional operators: a review. Proc Math Phys Eng Sci. 2020;476(2234):20190498. pmid:32201475
- 8. Acay B, Inc M, Mustapha UT, Yusuf A. Fractional dynamics and analysis for a lana fever infectious ailment with Caputo operator. Chaos, Solitons & Fractals. 2021;153:111605.
- 9. Baleanu D, Karaca Y, Vazquez L, MacΓas-DΓaz JE. Advanced fractional calculus, differential equations and neural networks: analysis, modeling and numerical computations. Physica Scripta. 2023;98(11):110201.
- 10.
Chen W, Sun H, Li X. Fractional derivative modeling in mechanics and engineering. Berlin/Heidelberg, Germany: Springer Nature; 2022.
- 11. Chen W, Liang Y. New methodologies in fractional and fractal derivatives modeling. Chaos, Solitons & Fractals. 2017;102:72β7.
- 12.
Leizerman S. Fractional-derivative field theory of cognitive memory: a unified mathematical framework for biological and artificial intelligence. In: SSRN Preprint. 2025.
- 13. Sun H, Li Z, Zhang Y, Chen W. Fractional and fractal derivative models for transient anomalous diffusion: model comparison. Chaos, Solitons & Fractals. 2017;102:346β53.
- 14.
Chen W, Sun H, Li X. Fractional derivative modeling in mechanics and engineering. Berlin/Heidelberg, Germany: Springer; 2022.
- 15. Kavallaris NI, Suzuki T. Non-local partial differential equations for engineering and biology. Mathematical Modeling and Analysis. 2018;31.
- 16. Taherdoost H. An overview of trends in information systems: emerging technologies that transform the information technology industry. Cloud Computing and Data Science. 2022;:1β16.
- 17. Bhushan B, Sahoo G. Recent advances in attacks, technical challenges, vulnerabilities and their countermeasures in wireless sensor networks. Wireless Pers Commun. 2017;98(2):2037β77.
- 18. White J. Cyber threats and cyber security: national security issues, policy and strategies. Global Security Studies. 2016;7(4).
- 19. Jamshed MA, Ali K, Abbasi QH, Imran MA, Ur-Rehman M. Challenges, applications, and future of wireless sensors in Internet of Things: a review. IEEE Sensors J. 2022;22(6):5482β94.
- 20. Sofi A, Jane Regita J, Rane B, Lau HH. Structural health monitoring using wireless smart sensor network β an overview. Mechanical Systems and Signal Processing. 2022;163:108113.
- 21.
Mubeen M, Shabbir K, Hanif A, Ali M, Hussain S, Ahmad S. Role of environmental science for disaster risk reduction in agriculture. Disaster risk reduction in agriculture. Singapore: Springer; 2023. p. 131β45.
- 22. Kumar P, Motia S, Reddy SRN. Integrating wireless sensing and decision support technologies for real-time farmland monitoring and support for effective decision making. Int j inf tecnol. 2018;15(2):1081β99.
- 23. Aslan Γ, AktuΔ SS, Ozkan-Okay M, Yilmaz AA, Akin E. A comprehensive review of cyber security vulnerabilities, threats, attacks, and solutions. Electronics. 2023;12(6):1333.
- 24. Achar SJ, Baishya C, Kaabar MKA. Dynamics of the worm transmission in wireless sensor network in the framework of fractional derivatives. Math Methods in App Sciences. 2021;45(8):4278β94.
- 25. Saeed S, Altamimi SA, Alkayyal NA, Alshehri E, Alabbad DA. Digital transformation and cybersecurity challenges for businesses resilience: issues and recommendations. Sensors (Basel). 2023;23(15):6666. pmid:37571451
- 26. Nwokoye CH, Madhusudanan V. Epidemic models of malicious-code propagation and control in wireless sensor networks: an indepth review. Wireless Pers Commun. 2022;125(2):1827β56.
- 27.
Frauenthal JC. Mathematical modeling in epidemiology. Springer Science & Business Media; 2012.
- 28. Dantu R, Cangussu JW, Patwardhan S. Fast worm containment using feedback control. IEEE Trans Dependable and Secure Comput. 2007;4(2):119β36.
- 29.
Kim J, Radhakrishnan S, Dhall SK. Measurement and analysis of worm propagation on Internet network topology. In: Proceedings of the 13th International Conference on Computer Communications and Networks (ICCCN). 2004.
- 30. Wang X, Li Y. An improved SIR model for analyzing the dynamics of worm propagation in wireless sensor networks. Chinese Journal of Electronics. 2009;18(1):8β12.
- 31. Liu P, Din A. Comprehensive analysis of a stochastic wireless sensor network motivated by Black-Karasinski process. Sci Rep. 2024;14(1):8799. pmid:38627447
- 32. Wu Y, Pu C, Zhang G, Li L, Xia Y, Xia C. Epidemic spreading in wireless sensor networks with node sleep scheduling. Physica A: Statistical Mechanics and its Applications. 2023;629:129204.
- 33. Awasthi S, Srivastava PK, Kumar N, Ojha RP, Pandey PS, Singh R, et al. An epidemic model for the investigation of multi-malware attack in wireless sensor network. IET Communications. 2023;17(11):1274β87.
- 34. Tahir H, Din A, Shah K, Abdalla B, Abdeljawad T. Advances in stochastic epidemic modeling: tackling worm transmission in wireless sensor networks. Mathematical and Computer Modelling of Dynamical Systems. 2024;30(1):658β82.
- 35. Cao X, Datta A, Al Basir F, Roy PK. Fractional-order model of the disease Psoriasis: a control based mathematical approach. J Syst Sci Complex. 2016;29(6):1565β84.
- 36.
France J, Thornley JH, Thornley JHM. Mathematical models in agriculture. Butterworths; 1984.
- 37.
Atangana A. Fractional operators with constant and variable order with application to geo-hydrology. Academic Press. 2017.
- 38. Atangana A. Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons & Fractals. 2017;102:396β406.
- 39. Li H, Zhang H, Ding K, Wang X-H, Sun G-Y, Liu Z-X, et al. The evolving epidemiology of monkeypox virus. Cytokine Growth Factor Rev. 2022;68:1β12. pmid:36244878