Fizzle Testing: An Equation Utilizing Random Surveillance to Help Reduce COVID-19 Risks

Round 1

Reviewer 1 Report

SARS-CoV-2 analysis, modelling and forecasting are hot topics due to the current sanitary crisis. So the topic presented is very interesting and the methods are sound. Nevertheless, there is a lot of room for improvement and, as a mathematician, I can't see the novelty of the paper.

My main concern is in the Results Section. The approach has a flawn, namely, the Fizzle equation should be derived from the SEIR model instead of the SIR model, and then use eigenvalue theory to the linearized system.

Since the Basic Reproduction Number (R_0) is a key-quantity,
in the Introduction Section, I strongly suggest to add the following references:
Lin. 36 ... environmental factors [1]. See for instance, [*] and [**] for an analytical/numerical approach to $R_0$ in structured population dynamics. Average estimates ...

[*] C. Barril, \`{A}. Calsina, and J. Ripoll, A practical approach to $R_0$ in continuous-time ecological models. \textit{Math. Meth. Appl. Sci.} 41 (18), 8432--8445, 2017.

[**] D. Breda, F. Florian, J. Ripoll, R. Vermiglio: Efficient numerical computation of the basic reproduction number for structured populations, \textit{J. Comput. Appl. Math.} 384, 113165 (2021). https://doi.org/10.1016/j.cam.2020.113165

 

The two Mathematical models (Monte Carlo simulations and the Stochastic SEIR model) are not explicitly stated in the paper. In my opinion, these models should be explained to some extend in order to see similarities and differences between them. For a network epidemic model using ODE systems and Monte Carlo simulations see for instance [***].

[***] D. Juher, J. Ripoll, and J. Salda\~{n}a. Analysis and Monte Carlo simulations of a model for the spread of infectious diseases in heterogeneous metapopulations. \textit{Phys. Rev. E} 80, 041920 (2009).

In Table 1, is $\alpha_a$ a rate or a percentage?

Author Response

Please see attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors use a Fizzle Equation to predict  Severe Acute Respiratory Virus-2 dynamics, and in particular for a 4000-student university cohort.  Using this equation the authors determine the frequency and percentage of random surveillance testing required to prevent an outbreak. Thus, the institution can develop scientifically public health policies to bring the effective reproduction number of the virus below one.

 

The authors performed model permutations to evaluate the potential spread of the virus based on the level of random surveillance testing, increased viral infectivity and implementing additional safety measures.

 

The model outcomes include: required level of surveillance testing, the number of infected individuals, the number of quarantined individuals and the capability to perform large numbers of tests with limited resources. The authors illustrate expected infection load and how testing policy can prevent outbreaks in an institution.

 

I like the idea proposed by the authors and the aims of the article are very valuable for the current pandemic and future ones. However, there are many aspects that need to be improved in the article. Overall, the explanation of the models and implementation needs to be much better. Here is the list of the main aspects:

 

Starting the section Material and methods, the authors mentioned two models. After reading the whole paper, I don't see the two models that the authors mentioned. I just see a SEIR-type model.  Is this an ODE model? Further explanations are needed.  Is it a network model?  The Monte Carlo simulations are to run the SEIR stochastic model? If this is the case a full explanation of the probability distributions and how the parameter values are randomly chosen. Notice, that this aspect requires a lot of explanation and it would be better to explain it in the main manuscript and other part in the appendix.

 

Table 1.  It shows some estimated parameters based on a fit?  With the amount of parameters of the model and the amount of data, I don’t think the parameters are identifiable. Tackling this aspect would take a great amount of effort for the authors and could be another article. I suggest to just assume values since the idea is to show how surveillance and other factors impact the dynamics of Covid in a small institution. 

Table 1. Estimate for days between testing ? fit? This is very unclear. How you obtain delta from data ? This aspect is much related to the previous one, but this is more dramatic.

 

Results section.  The authors used an SIR-type model to derive the main result of the article. However, the authors implemented a SEIR-type model based on the information in the appendix. Why is this ? Moreover, since the main Eq.(7) is derived from the SIR model, I think it would fail. For instance, the authors can do simulations (show them in the manuscript) and check the validity of Eq (7).  The Ro of the classical SEIR model is different from the classical SIR model.

 

 

Fig 1. The legend means that all these result are only with R0=1.6 ?  The color blue is confusing. Is the dark blue for a different Ro? In the caption it says that assures R0<1. Please explain more details in the manuscript.

 

I suggest adding more figures in the main manuscript and appendix

 

Line 155. "Surveillance testing on asymptomatic is crucial" Fig 1. has in one axes random baseline testing. How we can check the impact of "Surveillance testing on asymptomatic ". Please explain in the manuscript more details.

 

Line 163. "implementation of contact tracing at our university supports our initial assumptions. In this instance, it is an adjunct, but not the driving factor, for outbreak control". Please explain more details in the manuscript. It seems to contradict what the the authors mentioned before and the results. Is contact tracing a factor to control the outbreaks?

 

Fig 2. Is very difficult to understand.

 

Appendix A. Is very difficult to see how all these parameters affect the simulations. I suggest to add a pseudocode to explain it. Maybe a digram can help.

 

The figures in Appendix B are very confusing since they have different colors and different number of fits.  Fig(a) Is the red the logistic fit ? How Ro and Reff were computed ? I can't see any explanation in the manuscript about the logistic fit and the Rs.

 

Fig(c) has only one red curve. Is this the SEIR model ? Since it has a SEIR type model (not exactly a SEIR) how you computed the RO ? In the manuscript it has only a SIR ODE model.

 

Explain how the fits in appendix B are used. How the logistic fit is used? What model was fitted?

 

I suggest to rewrite lines 46-47-48.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Now the current version of the manuscript deserves to be published by the Journal.

Please, check reference [2] since it should be:

2.   Barril, C., Calsina, À., Ripoll, J. A practical approach to R0 in continuous-time ecological models. Math. Meth. Appl. Sci., 2018, 41(18), pp. 8432–8445

 

Author Response

Thank you for catching our mistake with the reference, it has been updated.

Reviewer 2 Report

For the second round of the review, I will try to be more specific. The idea of the fizzle equation sounds good and useful. However, it is important to validate it. For the next review round, the authors need to focus in showing the validity of the fizzle equation. These are other concerns:

 

• The authors mentioned that the 1st approach is a MC based simulation. This needs to be explained in detail. MC based simulation is very broad and can be performed for many different models, in fact is not a model as it seems that authors suggest. The second approach is a stochastic model, and probably the authors used MC simulations to obtain results from the stochastic SEIR model. For readers without a deep knowledge the description presented in this article would be very confusing. I suggest to add diagrams for both models to describe all the states that a person can be.

 

• Line 62. How you implement that in the SEIR model ? one week or every day ?

 

• Line 6. More details are needed to validate the article.  I suggest to add the explanations at least in the Appendix. There are two references but articles should show the basic aspects in this article.

 

• Lines 68-83. Is a social network? How the connections are selected from the beginning?

 

• Line 87. line 86 where are the validations results. Explain it.

 

• Line 97. Final values. There is not such a Table "The final values of parameters used for establishing SARS98
• CoV-2 mitigation and policy measures are given in Table 2 and Appendix A"

 

• Table 2. Please add references to the values in Table 2.

 

• eqs 5 and 6 should be sigma DE and sigma DI ?. Think carefully.

 

• Eq 6. DE how can you detect the infection if they are in an incubation stage?

 

• Reff should vary over time, explain eq 10.

 

• Explain in detail eq 11. There are parameters there that were not in 10.

 

• In Fig 1. Please specify the exact values used in the fizzle eq. 11. Are the curves the fizzle equation ?

 

• I suggest to simulate and show the results for the particular case (red arrows) . In addition show simulations for both models for the case where just the fizzle eq condition is not met and other one where is met. Close to the threshold of the condition. Notice that there are stochastic effects in both models.

 

• line 190. Authors mention that contact tracing is not the driving factor. However, Fig 1 shows that it is important. Please can you add a explanation maybe considering that contact tracing parameter is zero ? can we control the epidemic ?

 

 

• Caption in Fig 2 mention hospitalized. I don't see that in the SEIR model 1-6

 

 

• The Fig 2 shows cumulative cases, but it seems is lower than the number of infected. Please explain.

 

 

• Line 238. “The reinfection rate was kept close to zero” , but the SEIR model does not have reinfection. What about the 1st approach ?

 

 

• Fig B1. gamma different ? Duration of infectiousness? Authors mentioned in the introd. that is well-known. In table 2 is fixed.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 2 Report

The authors improved the presentation and clarity of the paper. This new version looks very different than the original one.  The authors added the type of network that they used, and this changed a lot the presentation of the paper. "

Readers would have a better probability to understand how the research was done.

One final comment is that the response letter says: "Reff should vary over time, explain eq 10. Concur, statement added, page 4, lines 126-129. Surveillance testing rate is set conservatively at the outset with a fully-susceptible population."

However, I cannot see lines 126-129 in page 4. Not sure if the manuscript is complete ?

I like the idea proposed by the authors and the aims of the article are very valuable for the current pandemic and future ones. 

I recommend that the authors take care of the details of the manuscript for the future.

Author Response

One final comment is that the response letter says: "Reff should vary over time, explain eq 10. Concur, statement added, page 4, lines 126-129. Surveillance testing rate is set conservatively at the outset with a fully-susceptible population." However, I cannot see lines 126-129 in page 4. Not sure if the manuscript is complete ?

We apologize for the confusion, the statement is immediately following equation 10 on page 4 (lines 125-128) and again following equation 11 on page 4 (lines 132-138).  

I recommend that the authors take care of the details of the manuscript for the future.

Thank you for the suggestion. We have gone back through and corrected some grammatical and spelling errors.

The Authors would like to thank the Reviewer for a thorough review process. The manuscript is greatly improved thanks to your efforts.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

See the attached file.

Comments for author File: Comments.pdf

Author Response

The authors thank the reviewer for taking the time to consider their work for publication. The following revisions have been made, and we hope you find them adequate to address your concerns.

Reviewer Correction 1:

By the closed form SEIR equation, the required testing rate is depicted in Figure 1. More details and explanations should be added. Different color presents
different R0, but for the same color, why in some regions the color is deeper
than others? Such as, in the green region, the color around φ = 0.2 is deeper
than that around φ = 0.3. In addition, for the same R0, curve changes with
different αa, what does the curve in the deepest color imply? Figure 1 should
be more clear, it is hard to distinguish αa.

Author Response 1:

The reviewer is correct in that the existing fan chart is difficult to discern. An easier to decipher chart has been added (see page 6), with more thorough explanation (Lines 158-163) and a more robust caption for Figure 1, lines 165-173. Additionally, an alternative figure without the colors may be used if the shading is distracting, rather than adding to the readers ability to easily discern between testing levels.

Reviewer Correction 2: 

It’s better to add details on how to obtain Eq. (8) by Eq. (7). By Table 3, both
Φ in Eq. (8) and φ in Eq. (7) are the contact tracing effectiveness. It’s better
to use the same notation for the same parameter to avoid misunderstanding.

Author Response 2:

The reviewer is correct and the manuscript has been updated in Table 3 (Page 5, Line 164), and equation 7 (page 5, Line 150-154).

Reviewer Correction 3:

In this paper, authors state that the model tailored to specific populations
provides an appropriate and adequate level of testing. How to scale the amount
of the surveillance testing by the mathematical models proposed in this work?

Author Response 3:

The authors agree with the reviewer to include a more robust implementation strategy. This proposed methodology may be seen on page 7, lines 175-225

Reviewer Correction 4:

There are also some minor errors, such as ϕ after Eq. (4) should be ψ, αa in
Eq. (7) should be αs, and so on.

Author Response 4:

The authors have gone through and done a more robust editing process to catch the minor errors. To the specific errors, for Eq 4 and E7, please see Page 5, Table 3, Line 164.  For the remainder, please see the tracked changes throughout the document.

 

Again, the authors thank the reviewer for improving their manuscript.

Reviewer 2 Report

This study by Cullenbine et al. lacks a strong methodology as well as a strong message. The model proposed is way too simple with no deaths accounted for infected individuals and the role of contact tracing in protecting susceptibles. Moreover, R0 should be replaced by Reff and subsequently, a new fizzle condition needs to be derived. 

From the Introduction, I was also expecting some data fitting to outbreaks on USS Theodore Roosevelt, French aircraft carrier Charles de Gaulle and the Diamond Princess cruise ship; however, did not see anything on that front. The results section is also very limited and adds very little to the already existing COVID-19 epidemiology literature.

 

 

Author Response

The authors thank the reviewer for taking the time to review their manuscript. The following changes have been made to meet the critiques of the reviewer and improve the overall message of the manuscript.

Reviewer comment 1:

This study by Cullenbine et al. lacks a strong methodology as well as a strong message. The model proposed is way too simple with no deaths accounted for infected individuals and the role of contact tracing in protecting susceptibles. Moreover, R0 should be replaced by Reff and subsequently, a new fizzle condition needs to be derived. 

Author comment 1:

The authors thank the reviewer for highlighting the need to expand on these topics. Specifically, these concerns have been addressed in the following places: 

For the no deaths, the information is incorporated into line 68, lines 75-93, lines 112-113, lines 146-152, and lines 184-188, where the utility of the Fizzle Equation is elaborated on.

The role of contact tracing is incorporated into lines 188-203.

Additionally, to meet the intent of an R0 should be replace by Rt, verbiage was added at lines 146-152.

Reviewer comment 2:

From the Introduction, I was also expecting some data fitting to outbreaks on USS Theodore Roosevelt, French aircraft carrier Charles de Gaulle and the Diamond Princess cruise ship; however, did not see anything on that front. The results section is also very limited and adds very little to the already existing COVID-19 epidemiology literature.

Author comment 2:

The authors elaborated on the validation and verification process in lines 75-93. Additionally, the results and discussion were ramped up to highlight the utility of the Fizzle Equation, to include how it may be utilized by an institution or a community. This narrative may be read from lines 158-163 and 175-225.

 

The authors thank the reviewer for taking the time review their manuscript.

Round 2

Reviewer 2 Report

Based on the quick view of the response provided by the authors, I can only make a comment that none of the scientific concerns were addressed by authors in their revised manuscript. They opted for a more verbal inclusion of definitions and explanations instead, which improves readability but does not improve the scientific hold.