PCOMPBIOL-D-26-00211

Redefining and estimating the early-phase reproduction ratio for epidemic outbreaks in spatially structured populations

PLOS Computational Biology

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Academic Editor

PLOS Computational Biology

Thomas Leitner

Section Editor

PLOS Computational Biology

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At this stage, the following Authors/Authors require contributions: Boxuan Wang, and Eugenio Valdano. Please ensure that the full contributions of each author are acknowledged in the "Add/Edit/Remove Authors" section of our submission form.

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: The review has been uploaded as an attachment

Reviewer #2: The paper is well written. The authors develop a metric that uses branching process theory and the Lambert W function to define a value that can be considered "What is the equivalent Reproduction number for a seed arising in this community (could be extended to any subtype) that takes into account connectivity with other places (types) that would have the same outbreak probability as if the community were isolated?"

I would encourage the authors to add a brief clarification distinguishing the Lambert-W solution from standard next-generation-matrix quantities such as the dominant eigenvalue (spectral radius) and eigenvector-based sensitivity measures. As written, a reader familiar with next-generation-matrix methods may wonder whether ^ _

is intended to be a community-specific analogue of the spectral radius, or perhaps something close to a sensitivity measure derived from the associated left and right eigenvectors. My understanding is that these are different objects. The Lambert- expression arises in the scalar, isolated-community case, where the outbreak probability solves =1− ^− . In the spatially coupled system, by contrast, the relevant quantity is the multitype outbreak probability

, obtained from the coupled fixed-point system, and _ ^ is then defined by mapping that seed-specific probability back through the scalar survival relation. In other words, _ ^ is a nonlinear, introduction(seed)-specific effective reproduction number, not a local dominant eigenvalue or a sensitivity derivative of the spectral radius of the next-generation matrix. This distinction seems especially important in the toy two-community example, where coupling can change _ and hence _ ^ even when the system-level spectral radius is unchanged. At the same time, there is a useful formal connection to standard next-generation-matrix theory in the homogeneous and below-threshold limits, where the manuscript shows that _ ^ collapses to _ and the dominant-eigenvector structure reappears. Making this distinction explicit would help readers connect the present framework to the standard NGM literature without conflating outbreak probability, invasion threshold, and eigenvector-based sensitivity analysis.

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Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

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Reviewer #1: None

Reviewer #2: No: Regarding the code accessibility, I congratulate the authors in making their code and data public and in a neat noteboke form. However, the code in its current format is unable to be run, with missing data files and no useful readme file information. The repo currently only exposes the notebook, one workbook, one CSV, and a one-line README, so it is not a complete runnable project by itself. A fresh Run All fails for two separate reasons:

(1) the main imports are in cell 3, but cell 0 tries to use pd.read_csv(...), so a clean notebook run hits the import-order problem first

(2) once that is fixed, the next failure is missing input data, starting with ./Italy_colocation.csv.

I suggest the authors upload the missing data files, ensure the code is runnable from a clean computer, with a top to bottom workflow and develop a useful readme file. Also Can I suggest that at least for publication that you ensure that the code is available on GitHub as a frozen tagged release representing the final analysis snapshot used for this manuscript.

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Reviewer #1: No

Reviewer #2: Yes: Emma McBryde

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Attachments

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Submitted filename: Review.docx

Dear Dr Valdano,

We are pleased to inform you that your manuscript 'Redefining and estimating the early-phase reproduction ratio for epidemic outbreaks in spatially structured populations' has been provisionally accepted for publication in PLOS Computational Biology.

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Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Computational Biology.

Best regards,

James M McCaw, PhD

Academic Editor

PLOS Computational Biology

Thomas Leitner

Section Editor

PLOS Computational Biology

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Reviewer's Responses to Questions

Comments to the Authors:

Please note here if the review is uploaded as an attachment.

Reviewer #1: The authors did a great job answering all my questions.

Reviewer #2: The authors have addressed my concerns in great detail and I am satisfied that these are now resolved.

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Have the authors made all data and (if applicable) computational code underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data and code 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 and code 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 or code —e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: None

Reviewer #2: No: They have made all code and data within their means (restricted by data sharing agreements) including toy data for demonstrating the code methods.

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Reviewer #1: No

Reviewer #2: Yes: Emma McBryde