Review 4: "Estimating the Reproduction Number and Transmission Heterogeneity from the Size Distribution of Clusters of Identical Pathogen Sequences"

RR:C19 Evidence Scale rating by reviewer:

• Potentially informative. The main claims made are not strongly justified by the methods and data, but may yield some insight. The results and conclusions of the study may resemble those from the hypothetical ideal study, but there is substantial room for doubt. Decision-makers should consider this evidence only with a thorough understanding of its weaknesses, alongside other evidence and theory. Decision-makers should not consider this actionable, unless the weaknesses are clearly understood and there is other theory and evidence to further support it.

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Review: The paper presents a joint inference approach to determine reproduction number and dispersion parameter from pathogen sequences. It is detailed and interesting. It may have several real-world implications, particularly the study designed to analyze the transmission advantage. However, it needs significant rewriting to improve readability. The following points will aid the revision process:

1. The line: "clusters of identical sequences are imprinted by the pathogen offspring distribution" is not clear.

2. Dispersion number should be defined in the abstract, as it is not common knowledge, unlike reproduction number.

3. The authors should make a better effort to explain the literature on dispersion parameters and the challenges associated with it.

4. While the idea of an epidemiological cluster makes theoretical sense, it's not clear how it may be determined in the real world. The authors mention a 'known epidemiological link' while explaining how the clusters are identified. I suppose more clarification is needed.

5. I recommend that the authors dedicate a separate subsection under material and methods to explaining how the dispersion parameter could be determined from available epidemiological data.

6. The section dedicated to the definition of the generating function needs a more notational explanation. For example, the reader does not know what 's' stands for, until much later in the discussion. Also, it is not clear why the distribution of individuals with the same genome should follow binomial distribution.

7. The discussions section needs to be rewritten. Instead of discussing Measles or VOCs, which are more suited for the introduction, the authors should expand the last paragraph on the applicability of the framework.