PCSY-D-24-00176

Efficient and accurate simulation of infectious diseases on adaptive networks

PLOS Complex Systems

Dear Dr. Gubela,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems'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.

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We look forward to receiving your revised manuscript.

Kind regards,

Anjalika Nande, Ph.D.

Academic Editor

PLOS Complex Systems

Anjalika Nande

Academic Editor

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

Journal Requirements:

1. We ask that a manuscript source file is provided at Revision. Please upload your manuscript file as a .doc, .docx, .rtf or .tex.

Additional Editor Comments (if provided):

All reviewers agree that the method presented in this manuscript is novel and is a useful addition to simulation algorithms of spreading processes on adaptive networks. They also find the manuscript to be well-written and appreciate that the code is publicly available on github. The reviewers have some comments/suggestions related to notational discrepancies, clarification of methodology and interpretability of the figures that they want addressed which would help strengthen the manuscript. In addition, I would like to mention that this work is within the scope of this journal.

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Reviewers' Comments:

Reviewer's Responses to Questions

Comments to the Author

1. Does this manuscript meet PLOS Complex Systems’s publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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

Reviewer #1: N/A

Reviewer #2: Yes

Reviewer #3: Yes

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3. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. 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

Reviewer #3: Yes

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

PLOS Complex Systems 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

Reviewer #3: 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 authors present an exact simulation algorithm for a class of interacting point processes (specifically, a class of epidemic dynamics models) on adaptive networks (i.e., networks in which the edges change dynamically in response to the state of the nodes/network). The method is based on the rejection method, which is a standard algorithm, and comparisons are made with the Gillespie algorithm, which is another standard algorithm for simulating interacting point processes on networks. Because both their proposed algorithm and Gillespie algorithms are exact (i.e., no error) by definition/construction, the focus of the comparison is (and should be) computation time (or memory usage). The paper is well written and the numerical experiments are reasonably done. The simulations seem to contain some presentation problems at least, but they are amendable (see my comments below). The contribution to the science is modest while I do not have any doubt about the correctness of the algorithm. Even though they showcased their algorithms with two models including a COVID-like epidemic model, numerical implementation is not beyond showing the algorithm is working fine, and it does not intend to discover noteworthy phenomena, in my opinion.

This is certainly a solid work. It depends on how this new journal wants to be. Because the journal is new, let me make comparisons with other journals. If this is a submission to Physical Review E/Research, Scientific Reports, New Journal of Physics, or Journal of Complex Systems, for example, I would certainly accept this manuscript with minor revision. If the journal seeks a higher impact, at the level of e.g., Journal of the Royal Society Interface, PLoS Computational Biology, PNAS Nexus, I would not accept this manuscript. This does not mean that there is anything particularly weak in this paper, despite some comments below. I locate this work as a standard contribution to the field. My specific comments are as follows:

(1) Abstract should be a single paragraph.

(2) Abstract, middle: "We proof" --> "We prove"

(3) line 43: Other key work on Gillespie algorithms in network simulations include the following:

10.1017/9781009239158

10.1371/journal.pone.0246961

10.1137/16M1055876

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042108

(4) around lines 80-82: The authors should explicitly say that they are considering undirected and unweighted networks (if so).

(5) The equation right after line 135 (and elsewhere): The right-hand side has $\lambda$ to the power of $k$. Are the authors sure? I doubt it. If $k$th lambda is what they mean, I would suggest to use the notation $\lambda^{(k)}$, for example.

(6) section 4: The demonstration of the proposed algorithm is very modest. Any new insights from these simulations?

(7) Figure 2: Both methods (theirs and Gillespie) are exact by definition. So, there is no need to compare the accuracy.

(8) Figure 4: All the three degree distributions generated have short tails. Networks with power-law-like degree distributions, which many networks have? Community structure? Some empirical networks (i.e., adaptive networks on top of static empirical networks)?

(9) Figure 5: x/y tics numbers in panel A are too frequent and do not look tidy. Also, I do not understand the scale. A tick represents 0.1 when x/y is small and 1.0 when x/y is large. It is neither logscale or non-log scale. The scale suddenly changes at x=0.5 and y=0.5. This is misleading. Furthermore, if such a discontinuously changing scale is used, one should not find a smooth phase diagram shown in the present panel A.

Reviewer #2: * Summary

The authors discuss efficient and accurate simulation of epidemics on adaptive networks. They introduce a novel method based on rejection sampling which is particularly useful for studying adaptive networks as they can 'skip' over many state changes that standard Gillespie SSA steppers cannot skip. This leads to an algorithm that is orders of magnitude faster. In addition, the authors show how it compares to the Gillespie method, both in accuracy and computation time, and discuss several relevant examples on epidemics on adaptive networks.

* Verdict

Overall, the manuscript is very well written. Both the premise of the work is clear, and the method is explained to such an extent that replication or application of their methods should be no problem for the interested reader. In addition, I highly commend the authors for sharing code (that actually runs when testing). This will undoubtedly increase the use of the method that they've put forward. Overall, I highly recommend acceptance of this paper after some minor revisions. I have detailed these minor comments below.

* Minor comments

** General

- In general, there are some small inconsistencies with using "eq. (x)" or "(x)". I would propose to use "Eq. (x)" in all cases. For this, you can use the `cleveref` LaTeX packages with the appropriate settings. The same holds for figures (i.e., [l. 250] "Fig 3" -> "Fig. 3").

** Specific

- [l. 204] Here, Fig. 1D is references, but Fig. 1D does not exist.

- [Section 4.] In Section 4.1, the specifications for the model are highlighted, yet there is no reference or detailed explanation on why these specific values were chosen. While I agree that this work is more tailored towards the methodology, and not the specifics of the examples, it would be useful if the authors could add some references (if these exist) on why some values are reasonable, and/or why some specific networks were studied.

I also wonder about the infection probability p that leads to an outbreak with I/N>0, and how this depends on the network and diagnosis. In Fig. 4B, it seems to me that awareness does not affect this 'critical' p, and that the 'critical' value seems almost independent on the network. Is this the case? Also, when larger values of N are considered, does this 'transition' become more pronounced/sharp? In addition, to me it seems interesting that the results in these adaptive networks are incredibly similar to those in static ones. That is to say, superspreaders (in the exponential networks) facilitate epidemics, while lack thereof supresses them. How does this compare with the lack of diagnoses (i.e., what does Fig. 4E look like when excluding diagnosis?). Can we use this to say something about the effect of isolation after being diagnosed (as Fig. 4B suggests)? And are there (extreme) containment strategies that completely avoid the emergence of an epidemic? Finally, the order in which Figs. 4C-E are briefly discussed in the text clashes with the order of in the figures.

In Section 4.2, how does Fig. 5A compare to the other three figures? Are the periodic returns of the epidemic considered "volatile"? Why are the labels of Fig. 5A not simply `cv_on` and `cv_off` (*)? And finally, the axes are in a very strange scale. What is the reason for this? If it is not needed, I suggest to avoid this. Finally, there is no mention on how these values were computed, nor are there labels or a description in the caption that explain the difference between the solid and the dashed lines. I feel that Fig. 5A has a lot of very useful information, but in its current state I feel it fails to convey this information.

(*) I would also suggest to simply use `c_on` and `c_off`

- [Section 5. Discussion] The authors discuss the implication of diagnosis and containment strategies, but to me it seems that here lie some missed opportunities to discuss this within the context of current containment strategies (i.e., how 'effective' are they?). Also, they state "agents at high risk of contracting the disease from others are not necessarily the ones most likely to transmit it to others". Yet, in Fig. 4 (and the relevant section), this is not explicitly shown. While it appears intuitive, I am wondering whether this is truly the case. The reason is because that one interpretation of Fig. 4E is that first all nodes with high degree get infected, after which it spreads throughout the rest of the population. This, at least to me, suggests that these high degree agents are more likely both to become infected and to infected others. Am I misinterpreting the figure?

The authors mention here that time-dependent contact rates can be implemented, and indeed this is explained in the Supplementary Material. However, it would be useful to either mention some references that put forward situations for which it is known that rates are time-dependent, or to study a (simple) system with time-dependent rates (i.e., in the Supplementary Material). Seeing that the work is generally well-written and mature, I would suggest the former.

Finally, the authors note that if the waiting times are not exponentially distribued, difficulties in using HAS arise, as HAS "requires knowledge about the waiting time distribution". Are there (realistic) cases where this is the case? And what should one do if one encounters these cases and has no access to a closed form distribution? Are there additional difficulties in applying HAS to non-Markovian processes?

** Figures

- [Fig. 1] In general; is it possible to use LaTeX fonts for all of the labels in the figure? Right now I feel that some are in the standard TeX font, but some are not, and it feels a bit odd. In A; what are the blue and red nodes? I think blue is a Susceptible and red an Infected, but some comment on this in the caption would be useful. In C; I greatly appreciate this figure, but I am having trouble grasping what it actually means. I understand the shaded curves, but what are the red crosses exactly? Is it perhaps the random uniform number u~Unif(0,1)? There is no mention of it in the caption.

- [Fig. 2] In general; I think tis figure is great, but I cannot shake the feeling that it's relatively jarring that the HAS markers overlap in this way with the SSA markers. I would personally label the SSA trajectories as red/black, and use three distinct line styles, and then for HAS use only markers in black/red. Or, perhaps simpler, simply remove the markers for the SSA algorithm. I think this will make the figure slightly more appealing. Again, note that this is highly subjective. In the caption; "Each contact is expected to last for one time step and one time step." What does this sentence mean?

- [Fig. 3] In general; I feel that this figure is a low dpi `.png`, which makes it difficult to see details (axis labels, etc.). I would highly suggest to either drastically increase the dpi, or (better), use a vector image format when saving such figures. In A; For me, A is an uninformative (sub)figure and can be omitted. The audience are people that study networks and complex systems: they'll know these distributions. In doing so, some labels for the specific networks should be added somewhere. In B; I suggest that "Final outbreak size in % of N" to be replaced by some symbol, e.g. "fraction infected I/N" or something similar.

- [Fig. S1] I am not sure what this figure adds to the work. To me it seems irrelevant, and it is also not discussed in any capacity in the main text.

** Suggested textual improvements and typo's

[!note: note that these are just suggestions. feel free to adapt as you may wish]

- [abstract] "While analytic ... realistic systems" -> "As analytic solutions can usually not be obtained, one has to resort to exact stochastic simulation algorithms, yet these have remained infeasible for the large sizes of realistic systems."

- [l. 324] "... (proof in Supplementary Materials)." -> Remove the text between parentheses.

Reviewer #3: The authors present a new algorithm to simulate efficiently spreading processes on adaptive and temporal networks. Their approach is original to my knowledge, and I believe it would be of great interests to a broad audience interested in simulating spreading dynamics on networks, especially in the context of infectious diseases.

The algorithm proposed provides significant improvements to the state of the art in computational time, while remaining exact. The authors provide a GitHub repository with their implementation of the algorithm, allowing their method to be easily implemented by other researchers in the complex systems.

I recommend this article for publication, provided the authors address the following points.

(1) The method presented simulates an SIR like spread on a temporal contact network, where each contact may lead to an infection with a fixed probability. In reality, contacts are often very heterogeneous in duration, and infection probability increases with longer contacts. Can the authors comment on this ? It would be interesting to know if this method can be modified to include contacts with heterogeneous duration.

(2) notation:

The presentation of the models, the SSA and HAS algorithms in the Methods section are not notationally consistent.

- edge creation/deletion rates, and transition rates are introduced with a node/agent index i on page 4 suggesting that these rates are individual based. On page 5, in the definition of SSA, the equation for lambda 0 now replaces the previously included indices with 0. Moreover the 0 index disappears at the end of the equation in the sum … If these rates are node dependent: this equation should reflect that. If the rates are homogeneous over the network (as used in the simulations), no need to introduce any index. For the HAS, the rates become node dependent again. Please clarify this and homogenise the notation throughout.

- The description of SSA does not include the diagnosis compartment… which appears again in the HAS. Please again here be consistent, so that the reader is able to compare the two methods.

- Line 200: tau rejection is defined, while the next equation is for tau. Please correct.

(3) Section 2.4 is not easy to follow for a few reasons, on top of the notation inconsistencies.

- When you say “Otherwise the infection is rejected and we set Delta_ij —> t + tau” (line 182). This corresponds to the case: infection does not occur, but edge is realised (so Delta_ij is updated). Is Delta_ij also updated if the infection occurs ? This aspect is not clear.

- Some of the information included in the naive bound paragraph is repeated in the tight bound section. For example, line 178 to 182. This comes back lines 186, with slight differences, unclear why. What else than the bound changes ?

- (Related to above) Why is Delta_ij set to 0 when we consider the tight bound in equations for P(a_ij(t)) (line 187, and following)? i.e. why is it that “the previous state of the edge Delta_ij is only one if the edge was initialised to exist at the beginning of the simulation” (l. 190). This does not seem so intuitive, and needs explaining.

- It is not clear what you mean by “is greater than a uniformly random number” on line 181.

- When an event “k” is drawn (line 208)… : is it just the type of transition drawn ? Or also the individual/edge. I assume the latter given equation 2 and 3. Please be specific (this goes back to point (1)).

- The algorithm is partially introduced in the text, together with some elements of justification. It becomes clearer with the bullet list at the end. For example step 5 has already partially been introduced, but not fully. I believe more clarity could be reached by introducing the algorithm earlier on in the section to guide the reader, possibly referring back to each step as additional details are provided.

(4) Section 2.2 partially introduces behavioural adaptations of the agents in response to the outbreak. This paragraph is unclear. beta is undefined, and the mechanistic effect of lambda reduce is not presented. This information appears in section 4.1. Instead, alpha is introduced, but never referred to again (from the text it appears it is set to 0?).

Please reorganise these elements so that the behaviour adaptations are all introduced in one place only, and are coherently presented.

(5) Why the factor of 10 in the definition of lambda reduce on page 9 ?

(6) typos/minor points:

- Lines 19-20: “Importantly, the evolution of the network also depends on the state of the infection process”. Please soften this sentence, eg. Using “may depend on the infection process” or “sometime also depends”.

- Line 141: contact transitions —> contact dependent transitions.

- Line 252: Should the O(N^2) not be O(N) ? As currently written the sentence does not make sense.

- Line 268: “Observed” here is not right, since the reference is a modelling study showing that behaviour changes were behind the decline of cases. I would use either “postulated”, or a reference to survey data that actually denotes observations of reduced contacts: see eg https://www.thelancet.com/journals/laninf/article/PIIS1473-3099(24)00531-0/fulltext .

- We proof —> we prove (throughout)

- Fig 1:

- there is no colour code for the dots. I can deduce they are S and I nodes.

- Last line: where —> were

- Fig 2: Each contact is supposed… the sentence is incorrect.

- Fig 5: please indicate on the phase diagram in panel A the exact locations of the three curves in panels B-D .

- In the SI. Page 2, lemma A.3: step 4: “Sample the state if the edge (i, j) using” is not right.

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Reviewer #1: No

Reviewer #2: Yes: Johannes Nauta

Reviewer #3: No

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Efficient and accurate simulation of infectious diseases on adaptive networks

PCSY-D-24-00176R1

Dear Dr. Gubela,

We are pleased to inform you that your manuscript 'Efficient and accurate simulation of infectious diseases on adaptive networks' has been provisionally accepted for publication in PLOS Complex Systems.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow-up email from a member of our team. 

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.

IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.

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 complexsystems@plos.org.

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

Best regards,

Anjalika Nande, Ph.D.

Academic Editor

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

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Reviewer Comments (if any, and for reference):

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: All comments have been addressed

Reviewer #3: All comments have been addressed

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2. Does this manuscript meet PLOS Complex Systems's publication criteria? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

Reviewer #3: Yes

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

Reviewer #1: N/A

Reviewer #2: Yes

Reviewer #3: Yes

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4. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. 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

Reviewer #3: Yes

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

PLOS Complex Systems 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

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 #1: (No Response)

Reviewer #2: The authors have adequately responded to all my comments, and I recommend this article for publication.

Reviewer #3: I thank the authors for the attention they gave my comments. The description of the HAS algorithm is now much clearer, and I have enjoyed reading the new version of the discussion, which now includes interesting additions following comments from the other reviewers.

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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.

Do you want your identity to be public for this peer review? If you choose “no”, your identity will remain anonymous but your review may still be made public.

For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: Yes: Johannes Nauta

Reviewer #3: No

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