Nonlinear Dynamics of the Introduction of a New SARS-CoV-2 Variant with Different Infectiousness

Round 1

Reviewer 1 Report

Thank you for the opportunity to review your work. This paper introduces a theoretical model using differential equations to study the effect of SARS-COV-2 variants. 

1. As a general statement, there are a number of grammatical errors throughout this manuscript. I recommend having an editorial contractor assist with fixing this. Though understandable, it is distracting from your methods and conclusions.
2. In the introduction, you mention that multiple models have been proposed to study the spread of the SARS-COV-2 virus. How does your model compare to these?
3. Why was a compartmental model based on a deterministic system of nonlinear differential equations chosen?
4. Is it reasonable to assume that a member of the population can't become infected with both variants, albeit at different times?
5. Figure 1 is a helpful summary of the first two paragraphs of the methods section. However, it is confusing not knowing what any of the terms mean without referencing back to the text. Is it possible to add some sort of legend to this figure?
6. From Lines 102-104, I am understanding that this model allows for 'new' people to enter the system (essentially allowing births to occur). Is it accurate to say that it only allows deaths from SARS-COV-2 or is the population allowed to die from other causes, and therefore able to 'leave' the system?
7. Why was the next generation matrix chosen over other potential methods for the computation of the basic reproduction number?
8. In the simulations, it is mentioned that time-varying components could be added to represent the introduction (or elimination) or shut-downs or other public health measures that affect the spread of the virus. Additionally, the inclusion of vaccinated individuals. With vaccination programs in particular, would this require a totally new model or is it simply another component that could be added to the existing model?
9. In the discussion, how does this technique contrast to those used by others for the same (or similar) purposes?

Author Response

Thank you for your nice and helpful comments that improved the manuscript. We really appreciate it.

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

This is an interesting and relevant paper containing mathematical analysis of a COVID dynamic model with variants. This is a rapidly developing popular area nowadays, where a number of papers with different approaches (mathematical, engineering, epidemological etc.) appear.

The Authors fail to recognize, that from the mathematical and system theoretical point of view the analyzed ODE model belongs to the class of positive polynomial systems for which part of the results described in the manuscript are already present. This literature should be read, taken into account and referred in the Introduction of the paper. Some of the relevant references are as follows:
[1] T. Péni, B. Csutak, G. Szederkényi, and G. Röst. Nonlinear model
predictive control with logic constraints for covid-19 management.
Nonlinear Dynamics, 102(4):1965–1986, 2020.
[2] G. Szederkényi, A. Magyar, K.M. Hangos. Analysis and control of polynomial dynamic models with biological applications. Elsevier, 2018.
[3] W.M. Haddad, V.S. Cellabonia, Q. Hiu. Nonnegative and compartmental dynamical systems. Princeton University Press, 2010.

In addition, the detailed discussion on the modelling assumptions and their effect on the possible conclusions on how to apply the findings to support decisions on healths polities are badly missing. In particular, the following questions should be answered:
a. How does the effect of vaccination may appear in the model?
b. How does the finite duration of the resistent period of the vaccinated and infected people may be included in the model?

Some minor suggestions and remarks are as follows:
1. Please, give a reference to the analyzed compartment model (1).
2. Theorem 1 (positivity and boundedness) follows from the kinetic property of the model (see [2] and [3] for details). Therefore the proof can safely omitted and replaced by referencing the literature.
3. Page 5 line 101: What does "vital dynamics" mean? Please, explain.
4. Page 5 line 112: Which are the "certain conditions"? This needs explanation.
5. The two first lines on page 9 need a lot of corrections, please, do it.
6. Page 14 lines 187-191: These sentences need a more detailed discussion perhaps in the Conclusion.
7. The caption of Figure 2 should be corrected to have "disease free equilibrium"

Author Response

Thank you for the good comments and your review.

Please see the attachment

Author Response File: Author Response.docx

Reviewer 3 Report

Please refer to the attached file.

Comments for author File: Comments.pdf

Author Response

Thank you for the revision and comments.

Please see the attachment

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

All of my important suggestions and remarks have been properly taken into account, thank you.

Author Response

Thank you for your comments and time to review our article.

Reviewer 3 Report

The authors do not seem to respond to all the comments presented by the reviewer.

Author Response

Reviewer 3 Comments and Suggestions for Authors

The authors do not seem to respond to all the comments presented by the reviewer.

R: Sorry we didn’t see the last part of the comments raised by the reviewer 3, since there was a large blank space in the first page. We apologized for this and now we addressing the second part of the comments. Thank you for the extensive review and details.

--------------------------

2.1 Positivity and … :

P.3, Line 2 from bottom: “Therefore, over the time study we need to guarantee that the solutions exist, are positive and bounded.”

Does this sentence contain mistakes? It's hard to understand.

R: We modified the phrase to be clearer.

 

I can't find a description about Λ. Not even in Table 1.

R: Thank you for noticing that !  We added a description for the parameter Lambda.

 

3. Mathematical stability analysis

3.1: Equilibrium points and R0

I understand the derivation of the equilibrium point (Equation 7), but I am not sure because the rest is omitted too much.

Line109: What is the next generation matrix? The "next generation matrix" is described as an important method, but there is no definition and the derivation is complicated. It is completely unknown where the matrix of Eq.(7) or less was derived. I would like a little more detailed explanation, not just the citations [68,77]. The derivation of up to Eq.(9) should be explained so that many readers including the unfamiliar can understand. 

R: We added some paragraphs and steps explaining more in detail the method based on the next generation matrix. It is based on a linearization of the nonlinear ODE model, but gives an easier way to prove local stability in comparison with the classical method of using the Jacobian and finding the eigenvalues in terms of the parameters. We agree with the reviewer that many readers might not be used to this methodology.

 

3.2: Local stability of disease-free equilibrium point

Line126: “theorem theorem” Delete either. 

R: Thank you for noticing that! We removed the duplicated one.

 

Line 130: This proof is too rough. The reviewer wonders if it is necessary to give it as a theorem. 3

R: Well we are not sure, what the reviewer is suggesting. The proof is presented in a classical way using Lypaunov theorem, and the proof is not easy to do.

 

3.4: Global stability of new SARS-CoV-2 variant endemic point

Line139: There is a mistake in this sentence. 

R: Thank you for noticing that. We fixed.

 

4. Numerical simulation results

The two paragraphs in this section describe the simulation settings and are too long. Since it is difficult to understand how it relates to the simulation results, it is necessary to revise the sentence structure.

R: Thank you. We revised the sentence and reduced the length of it. It was difficult since we think each phrase has some message related to the article. We removed some sentences with messages that are discussed in the conclusions in order to reduce the paragraphs.

 

5. conclusions

Lines213-217: It's a repeat, it's an unnecessary paragraph.

R: We removed that paragraph even though some reviewers from different backgrounds sometimes ask for some message along these removed lines. For some reviewers this message explains more the aims and usefulness of the article.

 

Line228: The “LaSalle invariant principle” suddenly appears here. Where did you use it in GAS analysis?

R: Yes, the LaSalle invariant principle is use in the GAS analysis. We added two phrases to explicit mention where we used. Thank you!

Round 3

Reviewer 3 Report

The authors have done a good job revising the manuscript to address concerns.