-->PONE-D-26-09719-->-->A Hybrid AI-Mathematical Approach for Epidemic Threshold Prediction in Metapopulation Networks: Integrating Physics-Informed Neural Networks with Spectral Graph Theory-->-->PLOS One

Dear Dr. Kouokam,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

==============================

The manuscript still requires substantial revision before it can be considered for publication. In particular, you must ensure that the methodological foundations of the study are clearly defined and rigorously justified. The definition of the epidemic threshold used as ground truth needs to be explained in detail and supported with appropriate justification, including how it is computed and how uncertainty is handled. In addition, the formulation of the learning problem must be clarified to avoid ambiguity and potential target leakage, especially regarding the use of input features that are directly related to the predicted quantity.

You should also revisit the characterization of the model as “physics-informed,” ensuring that this claim is either rigorously supported by theory or appropriately moderated. The experimental validation must be strengthened by including additional baseline methods, ablation analyses, and, where possible, statistical assessment of performance. Furthermore, the scope of the study should be reconsidered, as validation on real-world data or a more cautious interpretation of applicability is necessary.

Finally, to improve transparency and reproducibility, the code and datasets should be made available during the review process. Addressing these points is essential for the manuscript to meet the standards required for publication.

==============================

Please submit your revised manuscript by May 15 2026 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:-->

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We look forward to receiving your revised manuscript.

Kind regards,

Americo Cunha Jr

Academic Editor

PLOS One

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Additional Editor Comments :

The manuscript still requires substantial revision before it can be considered for publication. In particular, you must ensure that the methodological foundations of the study are clearly defined and rigorously justified. The definition of the epidemic threshold used as ground truth needs to be explained in detail and supported with appropriate justification, including how it is computed and how uncertainty is handled. In addition, the formulation of the learning problem must be clarified to avoid ambiguity and potential target leakage, especially regarding the use of input features that are directly related to the predicted quantity.

You should also revisit the characterization of the model as “physics-informed,” ensuring that this claim is either rigorously supported by theory or appropriately moderated. The experimental validation must be strengthened by including additional baseline methods, ablation analyses, and, where possible, statistical assessment of performance. Furthermore, the scope of the study should be reconsidered, as validation on real-world data or a more cautious interpretation of applicability is necessary.

Finally, to improve transparency and reproducibility, the code and datasets should be made available during the review process. Addressing these points is essential for the manuscript to meet the standards required for publication.

[Note: HTML markup is below. Please do not edit.]

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: No

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

Reviewer #1: Yes

Reviewer #2: No

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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: No

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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: Yes

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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: The manuscript presents a technically sound and well-structured hybrid framework integrating physics-informed neural networks with spectral graph theory for epidemic threshold prediction. The methodology is appropriate and experiments are conducted on a reasonably diverse set of synthetic networks, showing clear improvements over established baselines. The inclusion of interpretability analysis further strengthens the practical relevance of the work.

However, the novelty is somewhat incremental, and the evaluation relies entirely on synthetic data with simulation-based ground truth, limiting evidence of real-world applicability. The manuscript would benefit from validation on empirical contact network data and discussion of generalization to real-world scenarios.

Overall, the work meets the methodological and reproducibility standards of PLOS ONE, and I recommend acceptance and a minor revisions addressing the above points will benefits the work more

Reviewer #2: This manuscript addresses an important problem in computational epidemiology, namely the prediction of epidemic thresholds in complex networks, and proposes a hybrid approach combining spectral graph features with a neural model described as physics-informed. The topic is appropriate for a multidisciplinary journal such as PLOS ONE, and the paper is generally well structured, clearly written, and supported by a coherent experimental pipeline. However, in its current form, the manuscript does not yet meet the level of methodological rigor and technical soundness required for publication, and substantial revisions are necessary before it can be considered suitable.

The most critical issue concerns the definition of the target variable used throughout the study. The “true” epidemic threshold is obtained via stochastic SIS simulations using a heuristic criterion based on a transition in outbreak probability between 10% and 90%. This procedure is not sufficiently justified, and key details are missing, including how the threshold is numerically identified, the time horizon of simulations, the role of initial conditions, and the handling of absorbing states. Moreover, no uncertainty quantification is provided, despite the acknowledged presence of Monte Carlo noise. Since all reported performance metrics depend on this estimated ground truth, the lack of a rigorous and validated definition undermines the credibility of the results and constitutes a major obstacle to publication.

A second major concern is the conceptual formulation of the learning problem. The manuscript defines the epidemic threshold as the critical ratio β/γ, yet simultaneously includes β, γ, and β/γ among the input features used to predict that same quantity. This creates ambiguity and raises the possibility of redundancy or partial target leakage. The paper does not clearly distinguish between the parameters used to generate simulations and the quantity being predicted, and this lack of clarity weakens both the interpretation of the model and the validity of the comparisons. A precise and consistent formulation of the prediction task is required.

The characterization of the method as a physics-informed neural network is also not convincingly supported. The constraints incorporated into the loss function are heuristic (monotonicity, boundedness, and a relation to the spectral radius) and are not rigorously derived from the governing equations of the Ross–Macdonald or related epidemic models. While these constraints may act as useful regularizers, the connection to underlying epidemiological theory is largely qualitative rather than formal. As a result, the current presentation overstates the theoretical grounding of the approach. Either the constraints should be derived more rigorously, or the claims regarding “physics-informed” modeling should be significantly moderated.

The experimental validation is also insufficient for a study of this type. The comparison is limited to a small set of baselines, and important alternatives—such as tree-based regression methods or other nonparametric models using the same feature set—are not considered. There is no ablation analysis to assess the contribution of individual features or constraints, and no statistical assessment of the reported improvements. In addition, the model uses τ_QMF as an input feature while being compared against QMF itself, which requires careful interpretation and explicit discussion, as it effectively embeds one of the baselines into the proposed method. Without a more comprehensive and carefully controlled benchmarking strategy, the reported performance gains are difficult to evaluate.

The scope of the study is further limited by its exclusive reliance on synthetic networks. Although the manuscript acknowledges this limitation, the absence of validation on empirical contact networks significantly reduces the practical relevance of the results, particularly given the strong claims regarding applications to real-time surveillance and public health decision-making. At a minimum, the claims about applicability should be tempered; ideally, the study should include at least one real dataset or a more detailed discussion of generalization beyond the synthetic setting.

Reproducibility is another concern. The manuscript states that code and data will be made available upon acceptance, but for a computational study of this nature, availability during the review process is highly desirable and often expected. Making the implementation and datasets accessible at submission would substantially strengthen the paper.

Finally, the manuscript would benefit from a broader and deeper engagement with the existing literature. The number of references is limited relative to the scope of the topic, and the positioning of the work within current research on epidemic processes on networks, machine learning surrogates, and physics-informed modeling remains somewhat narrow.

In summary, while the paper addresses a relevant problem and presents an interesting combination of ideas, it currently falls short of PLOS ONE’s standards for technical soundness and completeness. To become suitable for publication, the authors should provide a rigorous and well-justified definition of the epidemic threshold used as ground truth, clarify the formulation of the learning problem to avoid ambiguity or leakage, either substantiate or moderate the claims regarding physics-informed modeling, substantially strengthen the experimental evaluation with additional baselines and ablations, improve reproducibility by releasing code and data, and, if possible, include validation on empirical networks or temper claims accordingly. With these improvements, the manuscript could evolve into a solid and publishable contribution.

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Reviewer #1: No

Reviewer #2: No

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-->-->PONE-D-26-09719R1-->-->A Hybrid AI-Mathematical Approach for Epidemic Threshold Prediction in Metapopulation Networks: .Integrating Physics-Guided Neural Networks with Spectral Graph Theory-->-->PLOS One

Dear Dr. Kouokam,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

==============================

Take into account the minor changes asked by referee 1.

==============================

Please submit your revised manuscript by Jul 05 2026 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:-->

• A letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.
• A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.
• An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

-->

As the corresponding author, your ORCID iD is verified in the submission system and will appear in the published article. PLOS supports the use of ORCID, and we encourage all coauthors to register for an ORCID iD and use it as well. Please encourage your coauthors to verify their ORCID iD within the submission system before final acceptance, as unverified ORCID iDs will not appear in the published article. Only     the individual author can complete the verification step; PLOS staff cannot     verify ORCID iDs on behalf of authors.

We look forward to receiving your revised manuscript.

Kind regards,

Americo Cunha Jr

Academic Editor

PLOS One

Journal Requirements:

1. If the reviewer comments include a recommendation to cite specific previously published works, please review and evaluate these publications to determine whether they are relevant and should be cited. There is no requirement to cite these works unless the editor has indicated otherwise.

2. Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

Additional Editor Comments:

Please take into account the minor comments.

[Note: HTML markup is below. Please do not edit.]

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 #1: All comments have been addressed

Reviewer #2: (No Response)

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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 #1: Yes

Reviewer #2: Yes

**********

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

Reviewer #1: Yes

Reviewer #2: Yes

**********

-->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 #1: Yes

Reviewer #2: Yes

**********

-->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 #1: Yes

Reviewer #2: Yes

**********

-->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 #1: The manuscript presents a technically sound and reproducible comparative study and addressed previous commentss. The experimental evaluation updated and is thorough and transparent, particularly the uncertainty analysis, ablation study, and public code/data availability and author also clarify it's litigation while doing this study.

Reviewer #2: The revised manuscript shows clear and substantive improvement compared to the previous version. The author has taken the earlier comments seriously and has strengthened the work in several important respects. In particular, the definition of the ground-truth epidemic threshold is now much better documented, with a transparent simulation protocol and the addition of Monte Carlo–based uncertainty quantification, which significantly enhances the methodological credibility of the study. The formulation of the learning problem has also been clarified, notably through the removal of the β/γ ratio from the feature set and a clearer distinction between input parameters and the prediction target, resolving the earlier concern about potential leakage and ambiguity.

The repositioning of the method as “physics-guided” rather than “physics-informed” is appropriate and reflects a more accurate interpretation of the role of the constraints. More broadly, the author has improved the scientific balance of the manuscript by moderating the claims and explicitly acknowledging the limitations of the proposed model. The experimental section has been substantially strengthened, with the inclusion of additional baselines, repeated cross-validation, statistical assessment, and a thorough ablation study. The decision to make code and data publicly available is also a very positive development that greatly improves reproducibility.

Despite these improvements, a few aspects would still benefit from further refinement. The definition of the epidemic threshold, while now clearly specified, remains based on a heuristic transition criterion, and the manuscript would be strengthened by a brief discussion of its robustness or sensitivity. The use of τ_QMF as an input feature, while now acknowledged, continues to raise some interpretational questions in relation to the benchmarking against QMF itself, and this point could be clarified further. In addition, the constraints introduced in the model, although useful as regularizers, remain only loosely connected to formal theoretical derivations, and their scope of validity could be discussed more explicitly. Finally, the study is still largely restricted to synthetic networks, and although this limitation is recognized, the claims regarding applicability should remain cautious in the absence of validation on empirical data.

Overall, the manuscript is now much more solid, transparent, and scientifically balanced. The main concerns raised in the initial review have been largely addressed, and the remaining issues are relatively minor and do not undermine the technical soundness of the work.

**********

-->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 #1: No

Reviewer #2: No

**********

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files.]

To ensure your figures meet our technical requirements, please review our figure guidelines: https://journals.plos.org/plosone/s/figures

You may also use PLOS’s free figure tool, NAAS, to help you prepare publication quality figures: https://journals.plos.org/plosone/s/figures#loc-tools-for-figure-preparation.

NAAS will assess whether your figures meet our technical requirements by comparing each figure against our figure specifications.

-->-->

A Hybrid AI-Mathematical Approach for Epidemic Threshold Prediction in Metapopulation Networks: .Integrating Physics-Guided Neural Networks with Spectral Graph Theory

PONE-D-26-09719R2

Dear Dr. Kouokam,

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.

An invoice will be generated when your article is formally accepted. Please note, if your institution has a publishing partnership with PLOS and your article meets the relevant criteria, all or part of your publication costs will be covered. Please make sure your user information is up-to-date by logging into Editorial Manager at Editorial Manager® and clicking the ‘Update My Information' link at the top of the page. For questions related to billing, please contact billing support.

If your institution or institutions have a press office, please notify them about your upcoming paper to help maximize its impact. If they’ll be preparing press materials, please inform our press team as soon as possible -- no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

Kind regards,

Americo Cunha Jr

Academic Editor

PLOS One

Additional Editor Comments (optional):

The revised manuscript has undergone substantial improvement and now satisfactorily addresses the major concerns raised during the review process. The methodological foundations of the study have been significantly strengthened, particularly through the clearer definition of the epidemic threshold ground truth, the addition of Monte Carlo–based uncertainty quantification, and the clarification of the learning problem formulation to avoid ambiguity and potential leakage. The repositioning of the approach as a physics-guided rather than physics-informed framework results in a more accurate and scientifically balanced presentation of the work.

The experimental evaluation has also been considerably expanded and strengthened through the inclusion of additional baselines, repeated cross-validation, statistical assessment, ablation studies, interpretability analysis, and public availability of code and data, substantially improving transparency and reproducibility. The remaining limitations of the study—particularly regarding the heuristic nature of the threshold estimator, the use of synthetic networks, and the scope of applicability of the constraints—are now explicitly acknowledged and discussed appropriately.

Overall, the manuscript is now methodologically sound, substantially more transparent, and scientifically balanced. The remaining issues are minor, do not compromise the validity of the work, and are acceptable within the scope and editorial standards of PLOS ONE.

Reviewers' comments: