Periodic Behaviour of an Epidemic in a Seasonal Environment with Vaccination

Round 1

Reviewer 1 Report

The authors' study of a five-dimensional system modelling a fatal disease in a seasonal environment presents significant contributions in establishing the existence, uniqueness, and dynamics of a periodic orbit. The use of the basic reproduction number (R_0) to determine the global stability and persistence of the disease is noteworthy. The inclusion of numerical investigations supporting the theoretical findings adds credibility to the study. Overall, I recommend this paper for publication with major revisions.

I have some comments and suggestions to improve the manuscript.

Abstract

- I would suggest adding a short sentence about the results of the autonomous system.

Introduction

- Lines 44-58: add references.

- Lines 68–76: summarize this paragraph.

- Line 108–111: extend this paragraph with a strong description of the new model and the results you would expect to obtain.

Mathematical model and properties

- Line 128: "which can be applied to the case of dengue fever" and also in Line 377:  "This model can describe the seasonal flu behaviour". I don't see any application of the model, the authors just use some general periodic functions to validate the theoretical results by numerical simulations. I don't think these functions represent the real seasonal variation of the parameters.

The case of constant parameters

- Line 170: Is the function F_0 the Lyapunov function?

- Throughout the manuscript, some quotation marks are missing, such as "." and ",". For example in the equation before Line 185.

Seasonal environment and periodic solution

- Line 203: Please add a reference.

- Please define $\partial \Gamma_0$ in the equation after Line 282.

Examples and numerical results

- Equation (22): How did you choose the parameters? Do you have any references? Please cite some references here. 

References

Can be extended as mentioned above.

 

 

Author Response

Thank you for the opportunity to revise our manuscript. We appreciate the careful review and constructive suggestions and comments. We have revised the manuscript accordingly and provide specific answers in the attached pdf file.

Author Response File: Author Response.pdf

Reviewer 2 Report

The introduction is neglected. It opens obvious and known aspects of the disease that the model does not consider afterward. It refers to a seasonal behavior of influenza, but with statements that are only valid in the northern hemisphere. It makes use of biological terms in a non-precise way. It incorporates a notation for state variables unusual for the mathematical epidemiology community. I notice several grammatical errors, particularly in punctuation. In general, it can become a good article of medium interest. Still, it needs to improve its presentation in the aspects of writing and ways to present mathematics (to reach more readers). The manuscript must improve substantially to be published. Is reference [10] essential?

Author Response

Thank you for the opportunity to revise our manuscript. We appreciate the careful review and constructive suggestions and comments. We have revised the manuscript accordingly and provide specific answers in the attached pdf file.

Author Response File: Author Response.pdf

Reviewer 3 Report

In this article, the authors have studied SVEIR epidemic model taking into account the seasonal environment. This model can describe the seasonal flu behaviour. In a first step they have studied the case of autonomous system where all parameters are supposed to be constants. In a second step, the authors have considered the non-autonomous system. Some basic results and basic reproduction number have been obtained. Some numerical examples are performed that illustrate the theoretical findings, including the autonomous system, the partially non-autonomous system and the full non-autonomous system. It is seen that if the system is autonomous, the trajectories converge to one of the equilibrium of the system (2) according to theorems 1 and 2. However, if at least one of the model parameters is periodic, the trajectories converge o a limit cycle according to theorems 4 and 5. 

It may be publishable.

Author Response

Thank you for the opportunity to revise our manuscript. We appreciate the careful review and constructive comments.  

Reviewer 4 Report

Comments for author File: Comments.pdf

Author Response

Thank you for the opportunity to revise our manuscript. We appreciate the careful review and constructive suggestions and comments. We have revised the manuscript accordingly and provide specific answers in the attached pdf file.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have made the necessary changes in the revised version.

Author Response

Thank you for the opportunity to revise our manuscript. We appreciate the careful review and constructive comments.  

Reviewer 2 Report

Thank you for considering my previous suggestions.

Reserve lowercase letters for parameters and variable uppercase, i.e., change S_in.

Please do not start sentences with symbols; there are several cases in assumptions, lemmas and Theo.1.

 

 

Author Response

Thank you for the opportunity to revise our manuscript. We appreciate the careful review and constructive suggestions and comments. We have revised the manuscript accordingly.

1- We reserved lowercase letters for parameters and variable uppercase, in particulared we changed S_{in} to s_{in}.

2- We modified the sentences according to your suggestions.