Emergence of oscillations in a simple epidemic model with demographic data

A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be sustained rather than damped, and data analysis regarding the oscillations provides insights into the same set of population data. Notably, oscillations arise naturally from the model, instead of from a periodic forcing term or other exogenous mechanism that guarantees oscillation: the model has no such mechanism. These emergent natural oscillations display appropriate periodicity for smallpox, even when the model is applied to different locations and populations. The model and datasets, in turn, offer new observations about disease dynamics and solution trajectories. These results call for renewed attention to relatively simple models, in combination with datasets from real outbreaks.

Can the authors provide mathematical expressions from data for the parameters alpha(t) and delta(t) in the forced model? Also, why is beta not considered time-dependent too?
The newest reference is from 2008. This should be fixed.

02-Sep-2019
Dear Dr Greer, The editors assigned to your paper ("Emergence of oscillations in a simple epidemic model with demographic data") have now received comments from reviewers. We would like you to revise your paper in accordance with the referee and Associate Editor suggestions which can be found below (not including confidential reports to the Editor). Please note this decision does not guarantee eventual acceptance.
Please submit a copy of your revised paper before 25-Sep-2019. Please note that the revision deadline will expire at 00.00am on this date. If we do not hear from you within this time then it will be assumed that the paper has been withdrawn. In exceptional circumstances, extensions may be possible if agreed with the Editorial Office in advance. We do not allow multiple rounds of revision so we urge you to make every effort to fully address all of the comments at this stage. If deemed necessary by the Editors, your manuscript will be sent back to one or more of the original reviewers for assessment. If the original reviewers are not available, we may invite new reviewers.
To revise your manuscript, log into http://mc.manuscriptcentral.com/rsos and enter your Author Centre, where you will find your manuscript title listed under "Manuscripts with Decisions." Under "Actions," click on "Create a Revision." Your manuscript number has been appended to denote a revision. Revise your manuscript and upload a new version through your Author Centre.
When submitting your revised manuscript, you must respond to the comments made by the referees and upload a file "Response to Referees" in "Section 6 -File Upload". Please use this to document how you have responded to the comments, and the adjustments you have made. In order to expedite the processing of the revised manuscript, please be as specific as possible in your response.
In addition to addressing all of the reviewers' and editor's comments please also ensure that your revised manuscript contains the following sections as appropriate before the reference list: • Ethics statement (if applicable) If your study uses humans or animals please include details of the ethical approval received, including the name of the committee that granted approval. For human studies please also detail whether informed consent was obtained. For field studies on animals please include details of all permissions, licences and/or approvals granted to carry out the fieldwork.
• Data accessibility It is a condition of publication that all supporting data are made available either as supplementary information or preferably in a suitable permanent repository. The data accessibility section should state where the article's supporting data can be accessed. This section should also include details, where possible of where to access other relevant research materials such as statistical tools, protocols, software etc can be accessed. If the data have been deposited in an external repository this section should list the database, accession number and link to the DOI for all data from the article that have been made publicly available. Data sets that have been deposited in an external repository and have a DOI should also be appropriately cited in the manuscript and included in the reference list.
If you wish to submit your supporting data or code to Dryad (http://datadryad.org/), or modify your current submission to dryad, please use the following link: http://datadryad.org/submit?journalID=RSOS&manu=RSOS-191187 • Competing interests Please declare any financial or non-financial competing interests, or state that you have no competing interests.
• Authors' contributions All submissions, other than those with a single author, must include an Authors' Contributions section which individually lists the specific contribution of each author. The list of Authors should meet all of the following criteria; 1) substantial contributions to conception and design, or acquisition of data, or analysis and interpretation of data; 2) drafting the article or revising it critically for important intellectual content; and 3) final approval of the version to be published.
All contributors who do not meet all of these criteria should be included in the acknowledgements.
We suggest the following format: AB carried out the molecular lab work, participated in data analysis, carried out sequence alignments, participated in the design of the study and drafted the manuscript; CD carried out the statistical analyses; EF collected field data; GH conceived of the study, designed the study, coordinated the study and helped draft the manuscript. All authors gave final approval for publication.
• Acknowledgements Please acknowledge anyone who contributed to the study but did not meet the authorship criteria.
• Funding statement Please list the source of funding for each author.
Once again, thank you for submitting your manuscript to Royal Society Open Science and I look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch. The authors should justify the assumption that births occur at a constant rate. Usually, births depend on the population size.
In Eq. (5), T is the period of "damped" oscilations. This should be clearly mentioned. Also, it shoud be proved that the autonomous model can not support self-sustained oscillations (stable limit-cycles). Hence, the forcing terms are necessary for creating oscillation.
It is not clear the meaning of the colors in the colored plots in Figures 1, 2, 3, and 6. A color bar is missing.
The authors should state that the proposed model is not adequate for describing current smallpox epidemics because a vaccination term is missing.
Can the authors provide mathematical expressions from data for the parameters alpha(t) and delta(t) in the forced model? Also, why is beta not considered time-dependent too?
The newest reference is from 2008. This should be fixed.

Author's Response to Decision Letter for (RSOS-191187.R0)
See Appendix A.

RSOS-191187.R1 (Revision) Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? Yes

Comments to the Author(s)
The references should be revised. There are some typos.

02-Jan-2020
Dear Dr Greer, It is a pleasure to accept your manuscript entitled "Emergence of oscillations in a simple epidemic model with demographic data" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience_proofs@royalsociety.org) and the production office (openscience@royalsociety.org) to let us know if you are likely to be away from e-mail contact --if you are going to be away, please nominate a co-author (if available) to manage the proofing process, and ensure they are copied into your email to the journal. Due to rapid publication and an extremely tight schedule, if comments are not received, your paper may experience a delay in publication.
Please see the Royal Society Publishing guidance on how you may share your accepted author manuscript at https://royalsociety.org/journals/ethics-policies/media-embargo/.

To the editors and reviewers:
Thank you for your prompt and thoughtful reviews of our paper "Emergence of oscillations in a simple epidemic model with demographic data." We have addressed each reviewer comment, as described below, and enthusiastically resubmit this work.

Reviewer: 1
Comments to the Author(s) 1. In the model (1), I would suggest writing the departure term from I as a sum of terms for natural deaths, disease deaths, and recovery.
We appreciate this suggestion and spent a significant amount of time considering how or if to implement it as stated. Indeed, earlier drafts of our models and paper used this exact approach, writing "delta + (smallpox-only death term)" instead of the current "epsilon" that combines both. However, as we wrote, and again as we revised in response to this comment by Reviewer 1, we found that the very different approaches to data collection and parameter estimation for delta (t) and epsilon meant that we should describe these parameters entirely separately. Therefore, instead of changing parameter names, we took seriously the possibility that readers would wonder about our reasoning, and so we inserted into the paper an explanation about our approach to parameter naming. This appears in Section 3, shortly after the equations for Model (2).
2. In the paper I did not see a reference to [5], but I believe it would be appropriate to mention it along with [6].
We thank this reviewer for checking throughout for our references. Reference [5] appears halfway through the second paragraph in Section 4.1; this reference tells us that 3 out of 10 people who had smallpox died. Reference [6] appears in the first paragraph of Section 4.1 and provides values of R_0 for several past outbreaks of smallpox. We therefore keep both [5] and [6]  We thank Reviewer 2 for suggesting we comment on our model not including vaccination. We made this change in Section 3 as part of describing the parameters and assumptions in our model.
Can the authors provide mathematical expressions from data for the parameters alpha(t) and delta(t) in the forced model? Also, why is beta not considered time-dependent too?
Thank you for this comment: we realize we can create a clearer connection between raw data and our model. The formulas for determining alpha_t and delta_t appear in Equations (6) and (7) of our manuscript. The numerical values at continuous time alpha(t) and delta(t), as needed in our simulations, are computed using linear interpolation between the demographic data points. The raw data appear in Supplementary Files (see the list of all pertinent files below). The linear interpolation is done using the interp1d library in python, with the default setting for linear. See relevant program files and line numbers below.
To address the reviewer's comment on why beta is not time-dependent: we agree this is important to discuss and therefore provide reasoning for this in Section 3, just below the definitions of Models (1) and (2).

The newest reference is from 2008. This should be fixed.
We agree entirely that there should be more recent references, and we have remedied this.