Periodic Behaviour of HIV Dynamics with Three Infection Routes

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

    This paper is devoted to studies of the important problem of HIV dynamics with three infection routes. Here authors extended the HIV dynamics model explored in [AlShamrani, N.H.; Halawani, R.H.; Shammakh, W.; Elaiw, A.M. Global Properties of HIV-1 Dynamics Models with CTL Immune Impairment and Latent Cell-to-Cell Spread. Mathematics 2023, 11. https://doi.org/10.3390/math11173743]. This article is interesting and pleasant to read. The main contribution of authors consists in studies of various aspects of ultimate dynamics of autonomous/ nonautonomous HIV infection models in cases of fixed environment/ variable environment respectively. Authors explored 3 routes of infection including the HIV-to-cell contact, latently infected cell-to-cell contact, and actively infected cell-to-cell contact. Among the most interesting parts of this work, I noted the study of periodic solutions.

    My minor criticism is this.

    1. Hundreds of works have been devoted to the study of the dynamics of infectious diseases. Therefore, I recommend reworking the Introduction and making it more specialized, focused only on research of the dynamic problems of HIV infection that began at the end of the 20th century. Appropriate changes should be made to the list of references.

    2. The model proposed by the authors does not take into account the fact that HIV viruses mutate quickly. Since the authors study various properties of the ultimate dynamics such as positive invariance, stability, etc., which are considered over an infinite time interval, the practical meaning of such a study should be clarified. 

    In my opinion, in this regard, it would be appropriate if the authors mention the relevance of their paper with the following articles, which examined HIV models with viral mutations:

    i) Stengel, R.F. (2008). Mutation and control of the human immunodeficiency virus. Math. Biosci., 213, 93--102.

    ii) Starkov, K. E., & Kanatnikov, A. N. (2021). Eradication conditions of infected cell populations in the 7-order HIV model with viral mutations and related results. Mathematics, 9(16), 1862.

    3. Please carefully check your entire manuscript. See some found typos. 

    Page 10, above line 169

    a) z_{n} instead of z_{m} under the root; b) with positive integer n

    Therefore, I believe that this study needs minor revision.

    

Comments on the Quality of English Language

no special comments

Author Response

Ref: mathematics-2775205.

Title: Periodic behaviour of HIV dynamics with three infection routes.

Journal: Mathematics.

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 below.

1. Hundreds of works have been devoted to the study of the dynamics of infectious diseases. Therefore, I recommend reworking the Introduction and making it more specialized, focused only on research of the dynamic problems of HIV infection that began at the end of the 20th century. Appropriate changes should be made to the list of references.

The Introduction was improved to be more specialized and focus on mathematical modeling for HIV infection and to highlight the novelty and the findings of this study.

2. The model proposed by the authors does not take into account the fact that HIV viruses mutate quickly. Since the authors study various properties of the ultimate dynamics such as positive invariance, stability, etc., which are considered over an infinite time interval, the practical meaning of such a study should be clarified.
   In my opinion, in this regard, it would be appropriate if the authors mention the relevance of their paper with the following articles, which examined HIV models with viral mutations:

i) Stengel, R.F. (2008). Mutation and control of the human immunodeficiency virus. Math. Biosci., 213, 93--102.

ii) Starkov, K. E., & Kanatnikov, A. N. (2021). Eradication conditions of infected cell populations in the 7-order HIV model with viral mutations and related results. Mathematics, 9(16), 1862.

In this paper, we focus on seasonal environment. The fact that HIV viruses mutate quickly is an important question that can be approached by slow-fast dynamics and could be a good future work.

The references were added and discussed inside the text.

3. Please carefully check your entire manuscript. See some found typos. Page 10, above line 169: a) z_{n} instead of z_{m} under the root; b) with positive integer n. Therefore, I believe that this study needs minor revision.

Many typos and missed quotation marks were corrected.

Reviewer 2 Report

Comments and Suggestions for Authors

see the attached file.

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

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript provides a thorough exploration of HIV-1 dynamics through a well-constructed nonlinear differential equation model. The division of infected cells and the consideration of various infection routes add realism to the model, enhancing its applicability. The proof of nonnegativity and boundedness of trajectories strengthens the credibility of the proposed system, and the determination of the basic reproduction number is a crucial contribution to the field.

The stability analyses conducted using Lyapunov theory and LaSalle’s invariance principle are comprehensive, addressing both fixed and variable environments. The proof of global asymptotic stability under specific conditions in a variable environment is a key finding, providing valuable insights into the potential control and management of HIV-1.

The inclusion of numerical examples is commendable, offering a practical demonstration of the theoretical concepts. The manuscript is generally well-written and effectively communicates complex mathematical concepts to a broad audience.

However, to enhance the manuscript further, I recommend a careful review of the presentation and clarity of certain sections, such as on pages 4, 5, 11, and 12. There are large matrices within the text, and I suggest presenting them in equation form without numbering. Additionally, the alignment of some equations needs to be modified; at times, they are aligned to the left, while in other instances, they are aligned to the right. Furthermore, providing more context on the practical implications of the findings and potential applications of the model would be beneficial.

In conclusion, this manuscript makes a substantial contribution to the field of HIV modelling. With some minor revisions to improve clarity and additional contextualization of practical implications, the manuscript is well-prepared for publication.

Author Response

Ref: mathematics-2775205.

Title: Periodic behaviour of HIV dynamics with three infection routes.

Journal: Mathematics.

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 below.

However, to enhance the manuscript further, I recommend a careful review of the presentation and clarity of certain sections, such as on pages 4, 5, 11, and 12. There are large matrices within the text, and I suggest presenting them in equation form without numbering. Additionally, the alignment of some equations needs to be modified; at times, they are aligned to the left, while in other instances, they are aligned to the right. Furthermore, providing more context on the practical implications of the findings and potential applications of the model would be beneficial.

The presentation and clarity of certain sections were carefully reviewed according to your suggestions. Please see the highlighted text in the 'pdf' file.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have made a response to the comments.

The paper can be considered for publication.