Referee report for the manuscript ‘Capturing Complex Interactions in Disease Ecology with Simplicial Sets’.

The manuscript ‘Capturing Complex Interactions in Disease Ecology with Simplicial Sets’ is a timely contribution to the field of disease ecology, stressing the relevance of considering non-dyadic interactions. The subject is potentially of great interest for both ecologists using network science and network scientists focused on ecological applications.
While I find the manuscript well structured, the presentation is not always clear, particularly in explaining the key concepts of the higher-order representations. Crucially, these are precisely the concepts that might be new to the reader, in particular given the pedagogical intention of the manuscript. I tried to give some suggestions (below) keeping this in mind.
My main concern is the type of article. According to the journal guidelines, the review criteria for a 'Method' should be based on the novelty and importance of the method. Here, the authors do not propose a new method, but rather provide an accessible introduction to a plethora of existing methods that have also been already applied in the context of interest (so the application is not new either). Nevertheless, the authors also expand what has been done in this direction, proposing new ideas and future direction, but without pursuing them explicitly. For these reasons, I would see the manuscript more suited for a 'Perspective' article instead. This decision goes obviously beyond my limited vision, but I would like to draw the attention of both the authors and the editor on this point.
Overall, the approach presented in the manuscript is surely interesting and I see great value in presenting such methods to a broad audience that might be not familiar with higher-order approaches. Therefore, I would be happy to recommend the manuscript for publication --in a suitable format-- after discussing the following points:

RELEVANT LITERATURE:
- Grilli, J., Barabás, G., Michalska-Smith, M. et al. Higher-order interactions stabilize dynamics in competitive network models. Nature 548, 210–213 (2017). https://doi.org/10.1038/nature23273
- Mayfield, M., Stouffer, D. Higher-order interactions capture unexplained complexity in diverse communities. Nat Ecol Evol 1, 0062 (2017). https://doi.org/10.1038/s41559-016-0062

FIGURE 1:
-panel labels are missing.
-why does (a) have the same colours but these are different in (b)? Explanation needed.
-dyadic interaction in (a) are not easy to spot given the color/thickness.
-(c) is not very informative, since the sub-simplices are not shown. Maybe these could be 'unravelled' on the side? This is a key point that surely deserves a better visual explanation.
-The caption is mentioning social interactions, which seems a bit out of place given the ecological application of the manuscript. Why not using a concrete ecological example instead?
-More importantly: at the essence of this figure there is the fact that some representations might be better than others, depending on the specific context. Here the "ground truth" is given in the caption where the different groups are listed as {1,2,3}, etc. I would suggest adding them in the figure instead, using the caption to better stress the pros and cons of each representation. For example, what is the difference between using (a) and (b) for the considered interactions? This is not clear. More in general, the figure could be better used to support the main text, for example lines 105-109, instead of being two independent pieces.

DEFINITION OF SIMPLICIAL COMPLEX (from line 112):
There is some confusion in the explanation between simplex and complex. "To be a simplicial complex a higher-order simplex must necessarily contain all nested lower-order simplices". First of all, the use of higher-order here it's not clear, the definition should hold at any order. Second, as it is it seems like a simplicial complex can be formed only by adding all lower sub-simplices to a simplex, while in a simplicial complex the crucial requirement is that for all simplices contained all their sub-simplices are part of the complex as well. This is a key point that deserves better clarity. Also line 114 states that a two-simplex would contain one-simplices etc.. This is misleading. Is the complex that should contain the one- and zero-simplices if the two-simplex is part of it, not the simplex.

TDA:
What is the role of topological data analysis in all of this? Simplicial complexes assumptions would, quoting the authors [Line 116], "broaden the suite of mathematical tools available". I guess here the authors refer to persistent homology etc. What would be the application of these tools in the context the manuscript? The authors mention this only at the very end of the manuscript. I would recommend the authors to add 1/2 lines here as well to better support the aforementioned sentence.

SIMPLICIAL VS COMPLEX CONTAGION:
Line 160: The authors argue that for some pathogens there is a minimum dose required to have a successful infection. Why do the authors think that models based on higher-order interaction are better suited for these cases than more standard models of complex contagion? See [Guilbeault, D., Becker, J., & Centola, D. (2018). Complex contagions: A decade in review. Complex spreading phenomena in social systems, 3-25.] and references therein. For example, the classic threshold model: Watts, D. J. A simple model of global cascades on random networks. Proc. Natl Acad. Sci. USA 99, 5766–5771 (2002).
At the end of the same paragraph the authors could expand their discussion on how to account for "the density of infectious individuals in a simplex" with respect to what has been done in:
-de Arruda, Guilherme Ferraz, Giovanni Petri, and Yamir Moreno. "Social contagion models on hypergraphs." Physical Review Research 2, no. 2 (2020): 023032.
-St-Onge, Guillaume, Iacopo Iacopini, Vito Latora, Alain Barrat, Giovanni Petri, Antoine Allard, and Laurent Hébert-Dufresne. "Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs." Communications Physics 5, no. 1 (2022): 1-16.

DISEASE AVOIDANCE AND GROUP CONSENSUS: (from line 184)
Re. defensive behaviours the authors might be interested in:
-Scarpino, S., Allard, A. & Hébert-Dufresne, L. The effect of a prudent adaptive behaviour on disease transmission. Nature Phys 12, 1042–1046 (2016). https://doi.org/10.1038/nphys3832
In the same paragraph, I like the idea of the authors of modelling individual behaviour and opinions based on the the one of the groups an individual is part of. Can the authors integrate this idea with what has been partially observed? For example, some work on this line has been done in opinion and majority rule models, while the effects of group sizes have been studied in models of social convention:
-Noonan, James, and Renaud Lambiotte. "Dynamics of majority rule on hypergraphs." Physical Review E 104, no. 2 (2021): 024316.
-Neuhäuser, Leonie, Andrew Mellor, and Renaud Lambiotte. "Multibody interactions and nonlinear consensus dynamics on networked systems." Physical Review E 101, no. 3 (2020): 032310.
-Iacopini, I., Petri, G., Baronchelli, A. et al. Group interactions modulate critical mass dynamics in social convention. Commun Phys 5, 64 (2022). https://doi.org/10.1038/s42005-022-00845-y

SPATIAL AND MOVEMENT NETWORKS:
I very much enjoyed reading the ideas proposed in this section. Could the authors provide some concrete examples in which "there is a strongly non-linear dose-response curve associated with transmission"? I think this would strengthen the argument and put it in perspective.

I think that Fig. 2A would be more clear if the home ranges and the two different ways of constructing the network were shown in separate panels. Again, this is very clear to me, but I'm thinking in terms of a reader not familiar with higher-order structures and their application. Separating the network from the simplicial complexes might help.
The curves Fig. 2B are hard to distinguish. For example, how many blue curves are shown? This panel could improved a lot.
Also, how does the SIS work? This is not clear. I assume the authors are using the one in [Iacopini et al.], cited there, where a 2-simplex contributes to the spreading if 2/3 of its nodes are infectious, but this obscure to me and it should be specified.

ECOLOGICAL NETWORKS:
Line 363: "Some methods have already been developed" should be supported by Refs., as well as "their application in its infancy" --if this is the case.
The last sentence of the section (Line 363) should be supported by some refs, or it should be linked to the following section --if that's what the authors intended.

MEASURES:
When taking about degree the authors should include also one of the first ones on the topic:
-Courtney, Owen T., and Ginestra Bianconi. "Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes." Physical Review E 93, no. 6 (2016): 062311.

When talking about homology the foundational work by Christ should be included as well:
-Ghrist, Robert W. Elementary applied topology. Vol. 1. Seattle: Createspace, 2014.

FUTURE DIRECTIONS/BOXES
The claim in Line 450 should be softened. Many softwares are available, as the authors well known. Some of them are very accessible and the variety accounts for different programming languages. What is surely missing is a solid comprehensive software that well integrates with other libraries for network analysis and machine learning (networkx and scikit-learn for example). Some early efforts in this direction are on their way: https://scikit-tda.org/, https://github.com/ComplexGroupInteractions/xgi.

Line 457: the claim that only low-order simplices have been considered it is not true, see for example the master equation approach in:
- St-Onge, Guillaume, Vincent Thibeault, Antoine Allard, Louis J. Dubé, and Laurent Hébert-Dufresne. "Master equation analysis of mesoscopic localization in contagion dynamics on higher-order networks." Physical Review E 103, no. 3 (2021): 032301.
- St-Onge, Guillaume, Iacopo Iacopini, Vito Latora, Alain Barrat, Giovanni Petri, Antoine Allard, and Laurent Hébert-Dufresne. "Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs." Communications Physics 5, no. 1 (2022): 1-16.

Box1, from line 788: This entire discussion seems interesting, but not clear. It is also not clear what kind of novelties it provides with respect to the master equation approach cited above.
As it is written, it is halfway between really explaining the idea and giving just a little glimpse. Since I find it interesting, I would advise the authors to expand on that.

Box 2: the authors might want to include the following references:
- Otter, Nina, Mason A. Porter, Ulrike Tillmann, Peter Grindrod, and Heather A. Harrington. "A roadmap for the computation of persistent homology." EPJ Data Science 6 (2017): 1-38.
- Patania, Alice, Francesco Vaccarino, and Giovanni Petri. "Topological analysis of data." EPJ Data Science 6, no. 1 (2017): 1-6.

MINOR POINTS:
Line 33: ecology, social
Line 38: 'where' a context is sounds a bit weird to me.
Line 71: realisation->representation?
Line 77: higher-order
Line 124: the sentence 124-127 is long and a bit hard to follow.
Line 178: simplicity "calculation" could be rephrased.
From Line 166: Cech should be Čech complex [in TeX is \v{C}ech I think].
Line 166: Čech OR Vietoris-Rips.
Line 331: higher order -> higher-order
Fig. 3 caption: higher order -> higher-order
Fig. 3: Borders of simplices (in particular the yellow and gray ones) are very hard to spot.
Why does Box2 come before Box1?

I would like to thank the authors for thoroughly addressing the points raised.

I find the revised manuscript much more readable and in line with the intention of providing guidelines to researchers interested in including higher-order approaches into their research. I also find the examples provided very interesting and stimulating.

I am very happy to recommend publication.

Iacopo Iacopini