PONE-D-25-20163

Controlling Worm Propagation in Wireless Sensor Networks: Through Fractal-Fractional Mathematical Perspectives

PLOS ONE

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Mahmoud H. DarAssi, Ph.D

Academic Editor

PLOS ONE

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Additional Editor Comments:

Controlling Worm Propagation in Wireless Sensor Networks: Through Fractal-Fractional Mathematical Perspectives

This paper proposes a generalized fractional-order model (SII₁PR) for analyzing the propagation and containment of worm-based malware in Wireless Sensor Networks (WSNs). The authors build on an earlier SIPR model by introducing an additional "Isolated" compartment (I₁) and incorporating fractal-fractional derivatives in the Atangana-Baleanu-Caputo (ABC) sense to capture memory and hereditary effects. They provide theoretical analysis (existence, uniqueness, stability) and a Newton-based numerical scheme to solve the system.

I have the following comments:

• Lack of real-world validation; no simulations based on actual WSN malware data.

• Inadequate explanation of modeling assumptions, particularly for the isolated compartment I₁.

• Overemphasis on formal proofs of standard results, minimal insight or interpretation.

• Shallow R₀ analysis, no stability theorem or global analysis.

• Numerical method is symbol-heavy without convergence/error analysis or benchmarks.

• Repetitive phrasing and verbose writing add unnecessary length.

• Inconsistent or unclear notation (e.g., I∞ vs. I₁).

• No visual outputs, lacks simulation graphs or figures.

• Needs proofreading, grammatical issues and awkward sentences throughout.

Recommendations for Improvement

• Add simulation results under various parameter regimes.

• Clarify and justify all modeling assumptions.

• Revise and streamline the numerical method; include convergence or comparative benchmarks.

• Proofread the manuscript carefully for grammar and clarity.

• Include visualizations (e.g., S, I, I₁, P, R dynamics for various R₀ values).

• Strengthen the R₀ analysis with a formal stability result.

Final Recommendation

Major Revision. The paper has solid mathematical groundwork but lacks practical validation, clear exposition, and visual support. Needs substantial improvement before it can be considered for publication.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Partly

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: N/A

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3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: No

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Manuscript ID: PONE-D-25-20163

Title: Controlling Worm Propagation in Wireless Sensor Networks: Through Fractal-Fractional Mathematical Perspectives

This manuscript presents a mathematical modeling approach to understand and control worm propagation in Wireless Sensor Networks (WSNs) by extending the classical SIPR (Susceptible-Infectious-Protected-Recovered) model. The authors incorporate a novel compartment for isolated nodes (I₁) and apply a fractal-fractional operator (in the Atangana-Baleanu-Caputo sense) to better capture memory and hereditary characteristics of the system. The model is analyzed through fixed-point theory, existence and uniqueness proofs, stability analysis, and is numerically solved using a Newton polynomial-based scheme. The work also computes the basic reproduction number R0 and provides sensitivity analysis for key parameters. The proposed model adds valuable complexity to the classical epidemic framework in WSNs by incorporating isolation dynamics and fractal-fractional derivatives. This is a meaningful extension, particularly for systems with memory effects. The model is relevant and timely for enhancing cybersecurity in sensor networks.

While the work demonstrates potentially valuable contributions, the manuscript requires significant revisions before it can be considered for publication.

1. The manuscript is generally well-structured but would benefit from more clarity in certain sections:

� Consider breaking down complex derivations into subsections or appending them in supplementary material.

� Not all variables or acronyms (e.g., FFD, ABC, HUS) are defined upon first use; ensure full clarity for readers less familiar with fractional calculus.

o The introduction provides a broad overview of WSNs and malware threats but lacks a focused literature review. The novelty of the proposed approach compared to previous models should be articulated more clearly.

2. The current literature review, though rich in references, is more descriptive than analytical. It would be strengthened by thematically grouping prior models (e.g., classical ODE models, fractional models, models with isolated compartments) and explicitly stating what limitations each had that the current study addresses.

3. The addition of the isolated compartment I1 is intuitive, but its real-world implementation should be justified. For example, how practical is it to isolate infected nodes in a WSN context, and what mechanisms (software, hardware) could facilitate this?

4. The numerical scheme using Newton interpolation is appropriate, though a brief justification for selecting this method over alternatives (e.g., finite difference or Runge-Kutta methods) would be helpful.

5. The simulations are not detailed in this excerpt. If present in the full version, ensure they clearly demonstrate the dynamic differences between classical and proposed models.

6. The manuscript requires moderate English language editing. There are several syntactic and grammatical issues that may hinder comprehension. For example, the phrasing is at times redundant or overly formal. A professional language revision is strongly recommended.

7. Clearly define all parameters in Table 1 and ensure consistent notation throughout.

8. Add a paragraph at the end of the introduction to outline the paper’s structure.

9. Ensure all equations are properly referenced and integrated into the discussion.

10. Figures and tables, must be explicitly cited and discussed in the main text.

11. Consider including a summary of key contributions as bullet points for clarity.

Although, the manuscript demonstrates significant potential and rigor, it requires substantial revisions to enhance its clarity, contextual grounding, and presentation. I encourage the authors to address the above comments carefully.

Reviewer #2: This study has presented a mathematical model for analyzing worm-spreading dynamics in wireless sensor networks. It has extended the traditional Susceptible-Infected-Protected-Recovered model by incorporating an isolated compartment. The ABC fractional derivative has been employed to represent memory-influenced behavior and network-wide interactions. I would recommend publication after the issues below are addressed.

1. Huge editing is required for the English language throughout the manuscript due to numerous grammatical and spelling mistakes.

2. The full title and short title of the manuscript are the same; however, the short title must be shorter than the full one.

3. Two authors are marked as corresponding authors on the title page. One of them should be removed. Also, two affiliations are numbered as 3; the second one should be numbered as 4.

4. The manuscript is prepared using the Elsevier LaTeX template. However, PLOS ONE is not an Elsevier journal, and the phrase at the bottom left of the title page should be deleted.

5. No citations should be inserted in the abstract. Also, no more than three citations should be inserted in a single statement. The extra ones should be removed or moved to other appropriate sentences. Besides, the in-text citations should be numbered in the order of their appearance.

6. All citations of the third and fourth paragraphs of the introduction are inserted at the end of the paragraph. However, they should be distributed in the paragraph and inserted in relevant places. In addition, reference 25 is not cited in the text but is listed in the bibliography. This issue should be solved.

7. All abbreviations should be accompanied by their complete forms when first used in the abstract and the main text. Afterward, all complete forms should be replaced with their abbreviations.

8. Equation (1), Table 1, and their associated text should be removed from the introduction and inserted in Section 3, with any necessary changes made.

9. A paragraph should be added to the end of the introduction that describes the manuscript’s structure section by section.

10. Table captions should be inserted above them, not below them.

11. Only the mathematical symbols representing matrices should be written in bold; the rest should not be in bold.

12. The numbering of all Assumptions, Definitions, Lemmas, Remarks, and Theorems should be double-checked.

13. All ‘I_\infty’ should be replaced by ‘I_1’. The right-hand side of Equation (13) should be identical to Equation (1). In the last line of Equation (15), a ‘)’ is misplaced. In the proof of Theorem 5, when subtracting the kernels, the fourth line should be omitted, and the first minus sign in the fifth line should be deleted. In the fourth line of Equation (19), the ‘||.||’ should be applied only to the kernel (like other lines), and the ‘{}’ should be removed. In all equations, the beta function should be denoted by ‘\beta’, not ‘B’. In the equation above Equation (21), the ‘l_1’ on the right-hand side should have the same subscript as the ‘F’ on the left-hand side. In the equation below, ‘Now, putting …’ on page 11 and the third line of Equation (23), ‘S_1’ should be corrected as ’S_2’. In the fifth equation on page 12, ‘Y_3’ should be corrected as ‘Y_4’ or ‘Y_5’, and the equations for ‘K_4’ and ‘K_5’ should be separated. The first minus in the fourth line of Equation (23) should be removed. On the right-hand side of Equation (24), a ‘+’ is missing after ‘\mu’. There is an extra ‘,’ in the equation below Equation (34).

14. The second line of Equation (29) and Equation (30) do not agree with the second line of Equation (1). Which one is correct? If Equation (29) is correct, Equation (1) and all following equations should be revised.

15. There are two equations in Equation (34). Which one is correct? The correct one should be kept, and the other should be omitted. Additionally, it seems that the sigma operator should be applied to all expressions until the end of the equation. Consequently, the parentheses of this equation and the following ones should be reordered.

16. The plotted titles should be removed from the figures. Moreover, the x-axis label of all figures should be ‘Time (days)’. Furthermore, the font name and size of all figures should be identical.

17. In Figure 2a, the y-axis labels should be changed to ‘I’. All the subscripts in the legends of panel b are 1; some of them should be changed to 2. The y-axis labels of all panels in Figure 4 should be ‘P’.

18. Figure 6 is a single panel, so the ‘(a)’ in its caption should be removed and its text should be merged into the figure’s caption.

19. What do the dashed lines in Figure 6 mean?

20. ‘Future Works’ and ‘Authors Contributions’ sections should be included in the manuscript.

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Reviewer #1: No

Reviewer #2: No

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<p>Controlling worm propagation in wireless sensor networks: Through fractal-fractional mathematical perspectives

PONE-D-25-20163R1

Dear Dr. Muhyi,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Mahmoud H. DarAssi, Ph.D

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: All comments have been addressed

Reviewer #3: All comments have been addressed

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2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Yes

Reviewer #3: Yes

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3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: N/A

Reviewer #3: N/A

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4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #2: Yes

Reviewer #3: Yes

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5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #2: Yes

Reviewer #3: Yes

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6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #2: (No Response)

Reviewer #3: The authors have carefully incorporated all the suggested revisions, and the overall quality of the manuscript has significantly improved. The mathematical arguments are now clearer, the numerical results are well supported, and the presentation is more consistent. In light of these improvements, I am satisfied with the current version of the paper and recommend it for publication in this esteemed journal.

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7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #2: No

Reviewer #3: No

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